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In mathematics, a quasifield is an algebraic structure ( Q , + , ⋅ ) {\displaystyle (Q,+,\cdot )} where + {\displaystyle +} and ⋅ {\displaystyle \cdot } are binary operations on Q {\displaystyle Q} , much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields.
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https://en.wikipedia.org/wiki/Quasifield
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In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far. The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1).
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https://en.wikipedia.org/wiki/Quasiperfect_number
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In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function. A function f {\displaystyle f} is quasiperiodic with quasiperiod ω {\displaystyle \omega } if f ( z + ω ) = g ( z , f ( z ) ) {\displaystyle f(z+\omega )=g(z,f(z))} , where g {\displaystyle g} is a "simpler" function than f {\displaystyle f} . What it means to be "simpler" is vague. A simple case (sometimes called arithmetic quasiperiodic) is if the function obeys the equation: f ( z + ω ) = f ( z ) + C {\displaystyle f(z+\omega )=f(z)+C} Another case (sometimes called geometric quasiperiodic) is if the function obeys the equation: f ( z + ω ) = C f ( z ) {\displaystyle f(z+\omega )=Cf(z)} An example of this is the Jacobi theta function, where ϑ ( z + τ ; τ ) = e − 2 π i z − π i τ ϑ ( z ; τ ) , {\displaystyle \vartheta (z+\tau ;\tau )=e^{-2\pi iz-\pi i\tau }\vartheta (z;\tau ),} shows that for fixed τ {\displaystyle \tau } it has quasiperiod τ {\displaystyle \tau } ; it also is periodic with period one.
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https://en.wikipedia.org/wiki/Quasiperiodic_function
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Another example is provided by the Weierstrass sigma function, which is quasiperiodic in two independent quasiperiods, the periods of the corresponding Weierstrass ℘ function. Functions with an additive functional equation f ( z + ω ) = f ( z ) + a z + b {\displaystyle f(z+\omega )=f(z)+az+b\ } are also called quasiperiodic. An example of this is the Weierstrass zeta function, where ζ ( z + ω , Λ ) = ζ ( z , Λ ) + η ( ω , Λ ) {\displaystyle \zeta (z+\omega ,\Lambda )=\zeta (z,\Lambda )+\eta (\omega ,\Lambda )\ } for a z-independent η when ω is a period of the corresponding Weierstrass ℘ function. In the special case where f ( z + ω ) = f ( z ) {\displaystyle f(z+\omega )=f(z)\ } we say f is periodic with period ω in the period lattice Λ {\displaystyle \Lambda } .
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https://en.wikipedia.org/wiki/Quasiperiodic_function
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In mathematics, a quasirandom group is a group that does not contain a large product-free subset. Such groups are precisely those without a small non-trivial irreducible representation. The namesake of these groups stems from their connection to graph theory: bipartite Cayley graphs over any subset of a quasirandom group are always bipartite quasirandom graphs.
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https://en.wikipedia.org/wiki/Quasirandom_group
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In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension E of a simple group S. In other words, there is a short exact sequence 1 → Z ( E ) → E → S → 1 {\displaystyle 1\to Z(E)\to E\to S\to 1} such that E = {\displaystyle E=} , where Z ( E ) {\displaystyle Z(E)} denotes the center of E and denotes the commutator.Equivalently, a group is quasisimple if it is equal to its commutator subgroup and its inner automorphism group Inn(G) (its quotient by its center) is simple (and it follows Inn(G) must be non-abelian simple, as inner automorphism groups are never non-trivial cyclic). All non-abelian simple groups are quasisimple. The subnormal quasisimple subgroups of a group control the structure of a finite insoluble group in much the same way as the minimal normal subgroups of a finite soluble group do, and so are given a name, component. The subgroup generated by the subnormal quasisimple subgroups is called the layer, and along with the minimal normal soluble subgroups generates a subgroup called the generalized Fitting subgroup. The quasisimple groups are often studied alongside the simple groups and groups related to their automorphism groups, the almost simple groups. The representation theory of the quasisimple groups is nearly identical to the projective representation theory of the simple groups.
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https://en.wikipedia.org/wiki/Quasisimple_group
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In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets A and B have diameters t and are no more than distance t apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent.
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https://en.wikipedia.org/wiki/Quasisymmetric_map
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In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an abelian group of odd order normalizing a non-trivial 2-subgroup of G. When G is a group of Lie type of characteristic 2 type, the width is usually the rank (the dimension of a maximal torus of the algebraic group).
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https://en.wikipedia.org/wiki/Quasithin_group
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In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth 2 n {\displaystyle 2n} -dimensional manifold is a quasitoric manifold if it admits a smooth, locally standard action of an n {\displaystyle n} -dimensional torus, with orbit space an n {\displaystyle n} -dimensional simple convex polytope. Quasitoric manifolds were introduced in 1991 by M. Davis and T. Januszkiewicz, who called them "toric manifolds". However, the term "quasitoric manifold" was eventually adopted to avoid confusion with the class of compact smooth toric varieties, which are known to algebraic geometers as toric manifolds.Quasitoric manifolds are studied in a variety of contexts in algebraic topology, such as complex cobordism theory, and the other oriented cohomology theories.
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https://en.wikipedia.org/wiki/Quasitoric_manifold
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In mathematics, a quasivariety is a class of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class.
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https://en.wikipedia.org/wiki/Quasivariety
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In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros form a cubic surface in 3-dimensional projective space.
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https://en.wikipedia.org/wiki/Sylvester_pentahedron
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In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, A ⊗ F K {\displaystyle A\otimes _{F}K} is isomorphic to the 2 × 2 matrix algebra over K. The notion of a quaternion algebra can be seen as a generalization of Hamilton's quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over F = R {\displaystyle F=\mathbb {R} } , and indeed the only one over R {\displaystyle \mathbb {R} } apart from the 2 × 2 real matrix algebra, up to isomorphism. When F = C {\displaystyle F=\mathbb {C} } , then the biquaternions form the quaternion algebra over F.
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https://en.wikipedia.org/wiki/Quaternion_algebra
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In mathematics, a quaternionic discrete series representation is a discrete series representation of a semisimple Lie group G associated with a quaternionic structure on the symmetric space of G. They were introduced by Gross and Wallach (1994, 1996). Quaternionic discrete series representations exist when the maximal compact subgroup of the group G has a normal subgroup isomorphic to SU(2). Every complex simple Lie group has a real form with quaternionic discrete series representations. In particular the classical groups SU(2,n), SO(4,n), and Sp(1,n) have quaternionic discrete series representations. Quaternionic representations are analogous to holomorphic discrete series representations, which exist when the symmetric space of the group has a complex structure. The groups SU(2,n) have both holomorphic and quaternionic discrete series representations.
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https://en.wikipedia.org/wiki/Quaternionic_discrete_series_representation
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In mathematics, a quaternionic structure or Q-structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field. A quaternionic structure is a triple (G, Q, q) where G is an elementary abelian group of exponent 2 with a distinguished element −1, Q is a pointed set with distinguished element 1, and q is a symmetric surjection G×G → Q satisfying axioms 1. q ( a , ( − 1 ) a ) = 1 , 2. q ( a , b ) = q ( a , c ) ⇔ q ( a , b c ) = 1 , 3. q ( a , b ) = q ( c , d ) ⇒ ∃ x ∣ q ( a , b ) = q ( a , x ) , q ( c , d ) = q ( c , x ) . {\displaystyle {\begin{aligned}{\text{1. }}\quad &q(a,(-1)a)=1,\\{\text{2.
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https://en.wikipedia.org/wiki/Quaternionic_structure
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}}\quad &q(a,b)=q(a,c)\Leftrightarrow q(a,bc)=1,\\{\text{3. }}\quad &q(a,b)=q(c,d)\Rightarrow \exists x\mid q(a,b)=q(a,x),q(c,d)=q(c,x)\end{aligned}}.} Every field F gives rise to a Q-structure by taking G to be F∗/F∗2, Q the set of Brauer classes of quaternion algebras in the Brauer group of F with the split quaternion algebra as distinguished element and q(a,b) the quaternion algebra (a,b)F.
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https://en.wikipedia.org/wiki/Quaternionic_structure
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In mathematics, a queen's graph is an undirected graph that represents all legal moves of the queen—a chess piece—on a chessboard. In the graph, each vertex represents a square on a chessboard, and each edge is a legal move the queen can make, that is, a horizontal, vertical or diagonal move by any number of squares. If the chessboard has dimensions m × n {\displaystyle m\times n} , then the induced graph is called the m × n {\displaystyle m\times n} queen's graph. Independent sets of the graphs correspond to placements of multiple queens where no two queens are attacking each other.
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https://en.wikipedia.org/wiki/Queen's_graph
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They are studied in the eight queens puzzle, where eight non-attacking queens are placed on a standard 8 × 8 {\displaystyle 8\times 8} chessboard. Dominating sets represent arrangements of queens where every square is attacked or occupied by a queen; five queens, but no fewer, can dominate the 8 × 8 {\displaystyle 8\times 8} chessboard. Colourings of the graphs represent ways to colour each square so that a queen cannot move between any two squares of the same colour; at least n colours are needed for an n × n {\displaystyle n\times n} chessboard, but 9 colours are needed for the 8 × 8 {\displaystyle 8\times 8} board.
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https://en.wikipedia.org/wiki/Queen's_graph
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In mathematics, a quintic function is a function of the form g ( x ) = a x 5 + b x 4 + c x 3 + d x 2 + e x + f , {\displaystyle g(x)=ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f,\,} where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero. In other words, a quintic function is defined by a polynomial of degree five. Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum.
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https://en.wikipedia.org/wiki/Quintic_equation
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The derivative of a quintic function is a quartic function. Setting g(x) = 0 and assuming a ≠ 0 produces a quintic equation of the form: a x 5 + b x 4 + c x 3 + d x 2 + e x + f = 0. {\displaystyle ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0.\,} Solving quintic equations in terms of radicals (nth roots) was a major problem in algebra from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the Abel–Ruffini theorem.
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https://en.wikipedia.org/wiki/Quintic_equation
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In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space P 4 {\displaystyle \mathbb {P} ^{4}} . Non-singular quintic threefolds are Calabi–Yau manifolds. The Hodge diamond of a non-singular quintic 3-fold is Mathematician Robbert Dijkgraaf said "One number which every algebraic geometer knows is the number 2,875 because obviously, that is the number of lines on a quintic."
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https://en.wikipedia.org/wiki/Quintic_threefold
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In mathematics, a quippian is a degree 5 class 3 contravariant of a plane cubic introduced by Arthur Cayley (1857) and discussed by Igor Dolgachev (2012, p.157). In the same paper Cayley also introduced another similar invariant that he called the pippian, now called the Cayleyan.
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https://en.wikipedia.org/wiki/Quippian
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In mathematics, a quotient algebra is the result of partitioning the elements of an algebraic structure using a congruence relation. Quotient algebras are also called factor algebras. Here, the congruence relation must be an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense described below. Its equivalence classes partition the elements of the given algebraic structure. The quotient algebra has these classes as its elements, and the compatibility conditions are used to give the classes an algebraic structure.The idea of the quotient algebra abstracts into one common notion the quotient structure of quotient rings of ring theory, quotient groups of group theory, the quotient spaces of linear algebra and the quotient modules of representation theory into a common framework.
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https://en.wikipedia.org/wiki/Maltsev_conditions
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In mathematics, a quotient category is a category obtained from another category by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but in the categorical setting.
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https://en.wikipedia.org/wiki/Quotient_category
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In mathematics, a radial function is a real-valued function defined on a Euclidean space Rn whose value at each point depends only on the distance between that point and the origin. The distance is usually the Euclidean distance. For example, a radial function Φ in two dimensions has the form Φ ( x , y ) = φ ( r ) , r = x 2 + y 2 {\displaystyle \Phi (x,y)=\varphi (r),\quad r={\sqrt {x^{2}+y^{2}}}} where φ is a function of a single non-negative real variable. Radial functions are contrasted with spherical functions, and any descent function (e.g., continuous and rapidly decreasing) on Euclidean space can be decomposed into a series consisting of radial and spherical parts: the solid spherical harmonic expansion.
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https://en.wikipedia.org/wiki/Radial_function
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A function is radial if and only if it is invariant under all rotations leaving the origin fixed. That is, ƒ is radial if and only if f ∘ ρ = f {\displaystyle f\circ \rho =f\,} for all ρ ∈ SO(n), the special orthogonal group in n dimensions. This characterization of radial functions makes it possible also to define radial distributions.
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https://en.wikipedia.org/wiki/Radial_function
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These are distributions S on Rn such that S = S {\displaystyle S=S} for every test function φ and rotation ρ. Given any (locally integrable) function ƒ, its radial part is given by averaging over spheres centered at the origin. To wit, ϕ ( x ) = 1 ω n − 1 ∫ S n − 1 f ( r x ′ ) d x ′ {\displaystyle \phi (x)={\frac {1}{\omega _{n-1}}}\int _{S^{n-1}}f(rx')\,dx'} where ωn−1 is the surface area of the (n−1)-sphere Sn−1, and r = |x|, x′ = x/r. It follows essentially by Fubini's theorem that a locally integrable function has a well-defined radial part at almost every r. The Fourier transform of a radial function is also radial, and so radial functions play a vital role in Fourier analysis. Furthermore, the Fourier transform of a radial function typically has stronger decay behavior at infinity than non-radial functions: for radial functions bounded in a neighborhood of the origin, the Fourier transform decays faster than R−(n−1)/2. The Bessel functions are a special class of radial function that arise naturally in Fourier analysis as the radial eigenfunctions of the Laplacian; as such they appear naturally as the radial portion of the Fourier transform.
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https://en.wikipedia.org/wiki/Radial_function
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In mathematics, a radially unbounded function is a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } for which Or equivalently, Such functions are applied in control theory and required in optimization for determination of compact spaces. Notice that the norm used in the definition can be any norm defined on R n {\displaystyle \mathbb {R} ^{n}} , and that the behavior of the function along the axes does not necessarily reveal that it is radially unbounded or not; i.e. to be radially unbounded the condition must be verified along any path that results in: For example, the functions are not radially unbounded since along the line x 1 = x 2 {\displaystyle x_{1}=x_{2}} , the condition is not verified even though the second function is globally positive definite. == References ==
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https://en.wikipedia.org/wiki/Radially_unbounded_function
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In mathematics, a random compact set is essentially a compact set-valued random variable. Random compact sets are useful in the study of attractors for random dynamical systems.
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https://en.wikipedia.org/wiki/Random_compact_set
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In mathematics, a random minimum spanning tree may be formed by assigning random weights from some distribution to the edges of an undirected graph, and then constructing the minimum spanning tree of the graph. When the given graph is a complete graph on n vertices, and the edge weights have a continuous distribution function whose derivative at zero is D > 0, then the expected weight of its random minimum spanning trees is bounded by a constant, rather than growing as a function of n. More precisely, this constant tends in the limit (as n goes to infinity) to ζ(3)/D, where ζ is the Riemann zeta function and ζ(3) is Apéry's constant. For instance, for edge weights that are uniformly distributed on the unit interval, the derivative is D = 1, and the limit is just ζ(3).In contrast to uniformly random spanning trees of complete graphs, for which the typical diameter is proportional to the square root of the number of vertices, random minimum spanning trees of complete graphs have typical diameter proportional to the cube root.Random minimum spanning trees of grid graphs may be used for invasion percolation models of liquid flow through a porous medium, and for maze generation. == References ==
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https://en.wikipedia.org/wiki/Random_minimal_spanning_tree
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In mathematics, a random polytope is a structure commonly used in convex analysis and the analysis of linear programs in d-dimensional Euclidean space R d {\displaystyle \mathbb {R} ^{d}} . Depending on use the construction and definition, random polytopes may differ.
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https://en.wikipedia.org/wiki/Random_polytope
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In mathematics, a random walk, sometimes known as a drunkard's walk, is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line Z {\displaystyle \mathbb {Z} } which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler. Random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. The term random walk was first introduced by Karl Pearson in 1905.
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https://en.wikipedia.org/wiki/Random_walk_with_drift
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In mathematics, a rank ring is a ring with a real-valued rank function behaving like the rank of an endomorphism. John von Neumann (1998) introduced rank rings in his work on continuous geometry, and showed that the ring associated to a continuous geometry is a rank ring.
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https://en.wikipedia.org/wiki/Rank_ring
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In mathematics, a ranked partially ordered set or ranked poset may be either: a graded poset, or a poset with the property that for every element x, all maximal chains among those with x as greatest element have the same finite length, or a poset in which all maximal chains have the same finite length.The second definition differs from the first in that it requires all minimal elements to have the same rank; for posets with a least element, however, the two requirements are equivalent. The third definition is even more strict in that it excludes posets with infinite chains and also requires all maximal elements to have the same rank. Richard P. Stanley defines a graded poset of length n as one in which all maximal chains have length n. == References ==
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https://en.wikipedia.org/wiki/Ranked_poset
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In mathematics, a rate is the quotient of two quantities in different units of measurement, often represented as a fraction. If the divisor (or fraction denominator) in the rate is equal to one expressed as a single unit, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the dividend (the fraction numerator) of the rate expresses the corresponding rate of change in the other (dependent) variable. One common type of rate is "per unit of time", such as speed, heart rate, and flux. In fact, often rate is a synonym of rhythm or frequency, a count per second (i.e., hertz); e.g., radio frequencies, heart rates, or sample rates.
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https://en.wikipedia.org/wiki/Temporal_rate_of_change
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In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate; for example, a heart rate is expressed as "beats per minute". Rates that have a non-time divisor or denominator include exchange rates, literacy rates, and electric field (in volts per meter). A rate defined using two numbers of the same units will result in a dimensionless quantity, also known as ratio or simply as a rate (such as tax rates) or counts (such as literacy rate). Dimensionless rates can be expressed as a percentage (for example, the global literacy rate in 1998 was 80%), fraction, or multiple.
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https://en.wikipedia.org/wiki/Temporal_rate_of_change
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In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7). The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive.
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https://en.wikipedia.org/wiki/Ratio_analysis
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A ratio may be specified either by giving both constituting numbers, written as "a to b" or "a:b", or by giving just the value of their quotient a/b. Equal quotients correspond to equal ratios. A statement expressing the equality of two ratios is called a proportion.
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https://en.wikipedia.org/wiki/Ratio_analysis
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Consequently, a ratio may be considered as an ordered pair of numbers, a fraction with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, are rational numbers, and may sometimes be natural numbers. A more specific definition adopted in physical sciences (especially in metrology) for ratio is the dimensionless quotient between two physical quantities measured with the same unit. A quotient of two quantities that are measured with different units may be called a rate.
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https://en.wikipedia.org/wiki/Ratio_analysis
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In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L. The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.
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https://en.wikipedia.org/wiki/Proper_rational_function
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In mathematics, a rational monoid is a monoid, an algebraic structure, for which each element can be represented in a "normal form" that can be computed by a finite transducer: multiplication in such a monoid is "easy", in the sense that it can be described by a rational function.
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https://en.wikipedia.org/wiki/Kleene_monoid
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In mathematics, a rational normal scroll is a ruled surface of degree n in projective space of dimension n + 1. Here "rational" means birational to projective space, "scroll" is an old term for ruled surface, and "normal" refers to projective normality (not normal schemes). A non-degenerate irreducible surface of degree m – 1 in Pm is either a rational normal scroll or the Veronese surface.
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https://en.wikipedia.org/wiki/Directrix_of_a_rational_normal_scroll
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In mathematics, a rational number is a number that can be expressed as the quotient or fraction p q {\displaystyle {\tfrac {p}{q}}} of two integers, a numerator p and a non-zero denominator q. For example, − 3 7 {\displaystyle {\tfrac {-3}{7}}} is a rational number, as is every integer (e.g., 5 = 5/1). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface Q, or blackboard bold Q . {\displaystyle \mathbb {Q} .}
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https://en.wikipedia.org/wiki/Field_of_rationals
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A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see Repeating decimal § Extension to other bases).
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https://en.wikipedia.org/wiki/Field_of_rationals
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A real number that is not rational is called irrational. Irrational numbers include the square root of 2 ( 2 {\displaystyle {\sqrt {2}}} ), π, e, and the golden ratio (φ). Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q) with q ≠ 0, using the equivalence relation defined as follows: ( p 1 , q 1 ) ∼ ( p 2 , q 2 ) ⟺ p 1 q 2 = p 2 q 1 .
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https://en.wikipedia.org/wiki/Field_of_rationals
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{\displaystyle (p_{1},q_{1})\sim (p_{2},q_{2})\iff p_{1}q_{2}=p_{2}q_{1}.} The fraction p q {\displaystyle {\tfrac {p}{q}}} then denotes the equivalence class of (p, q).Rational numbers together with addition and multiplication form a field which contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of Q {\displaystyle \mathbb {Q} } are called algebraic number fields, and the algebraic closure of Q {\displaystyle \mathbb {Q} } is the field of algebraic numbers.In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals (see Construction of the real numbers).
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https://en.wikipedia.org/wiki/Field_of_rationals
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In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to K ( U 1 , … , U d ) , {\displaystyle K(U_{1},\dots ,U_{d}),} the field of all rational functions for some set { U 1 , … , U d } {\displaystyle \{U_{1},\dots ,U_{d}\}} of indeterminates, where d is the dimension of the variety.
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https://en.wikipedia.org/wiki/Unirationality
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In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number x, the rational zeta series for x is given by x = ∑ n = 2 ∞ q n ζ ( n , m ) {\displaystyle x=\sum _{n=2}^{\infty }q_{n}\zeta (n,m)} where qn is a rational number, the value m is held fixed, and ζ(s, m) is the Hurwitz zeta function. It is not hard to show that any real number x can be expanded in this way.
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https://en.wikipedia.org/wiki/Rational_zeta_series
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In mathematics, a ray class field is an abelian extension of a global field associated with a ray class group of ideal classes or idele classes. Every finite abelian extension of a number field is contained in one of its ray class fields. The term "ray class group" is a translation of the German term "Strahlklassengruppe".
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https://en.wikipedia.org/wiki/Ray_class_field
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Here "Strahl" is German for ray, and often means the positive real line, which appears in the positivity conditions defining ray class groups. Hasse (1926, p.6) uses "Strahl" to mean a certain group of ideals defined using positivity conditions, and uses "Strahlklasse" to mean a coset of this group. There are two slightly different notions of what a ray class field is, as authors differ in how the infinite primes are treated.
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https://en.wikipedia.org/wiki/Ray_class_field
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In mathematics, a read-once function is a special type of Boolean function that can be described by a Boolean expression in which each variable appears only once. More precisely, the expression is required to use only the operations of logical conjunction, logical disjunction, and negation. By applying De Morgan's laws, such an expression can be transformed into one in which negation is used only on individual variables (still with each variable appearing only once). By replacing each negated variable with a new positive variable representing its negation, such a function can be transformed into an equivalent positive read-once Boolean function, represented by a read-once expression without negations.
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https://en.wikipedia.org/wiki/Read-once_function
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In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.
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https://en.wikipedia.org/wiki/Real_closed_fields
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In mathematics, a real closed ring (RCR) is a commutative ring A that is a subring of a product of real closed fields, which is closed under continuous semi-algebraic functions defined over the integers.
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https://en.wikipedia.org/wiki/Real_closed_ring
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In mathematics, a real differential one-form ω on a surface is called a harmonic differential if ω and its conjugate one-form, written as ω∗, are both closed.
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https://en.wikipedia.org/wiki/Harmonic_differential
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In mathematics, a real function f {\displaystyle f} of real numbers is said to be uniformly continuous if there is a positive real number δ {\displaystyle \delta } such that function values over any function domain interval of the size δ {\displaystyle \delta } are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number ϵ {\displaystyle \epsilon } , then there is a positive real number δ {\displaystyle \delta } such that | f ( x ) − f ( y ) | < ϵ {\displaystyle |f(x)-f(y)|<\epsilon } at any x {\displaystyle x} and y {\displaystyle y} in any function interval of the size δ {\displaystyle \delta } . The difference between uniform continuity and (ordinary) continuity is that, in uniform continuity there is a globally applicable δ {\displaystyle \delta } (the size of a function domain interval over which function value differences are less than ϵ {\displaystyle \epsilon } ) that depends on only ε {\displaystyle \varepsilon } , while in (ordinary) continuity there is a locally applicable δ {\displaystyle \delta } that depends on the both ε {\displaystyle \varepsilon } and x {\displaystyle x} . So uniform continuity is a stronger continuity condition than continuity; a function that is uniformly continuous is continuous but a function that is continuous is not necessarily uniformly continuous.
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https://en.wikipedia.org/wiki/Uniform_continuity
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The concepts of uniform continuity and continuity can be expanded to functions defined between metric spaces. Continuous functions can fail to be uniformly continuous if they are unbounded on a bounded domain, such as f ( x ) = 1 x {\displaystyle f(x)={\tfrac {1}{x}}} on ( 0 , 1 ) {\displaystyle (0,1)} , or if their slopes become unbounded on an infinite domain, such as f ( x ) = x 2 {\displaystyle f(x)=x^{2}} on the real (number) line.
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https://en.wikipedia.org/wiki/Uniform_continuity
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However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map). Although continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space.
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https://en.wikipedia.org/wiki/Uniform_continuity
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In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.The set of real numbers is denoted R or R {\displaystyle \mathbb {R} } and is sometimes called "the reals". The adjective real, used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of −1.The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3.
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https://en.wikipedia.org/wiki/Real_number_system
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The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) are the root of a polynomial with integer coefficients, such as the square root √2 = 1.414...; these are called algebraic numbers. There are also real numbers which are not, such as π = 3.1415...; these are called transcendental numbers.Real numbers can be thought of as all points on a line called the number line or real line, where the points corresponding to integers (..., −2, −1, 0, 1, 2, ...) are equally spaced.
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https://en.wikipedia.org/wiki/Real_number_system
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Conversely, analytic geometry is the association of points on lines (especially axis lines) to real numbers such that geometric displacements are proportional to differences between corresponding numbers. The informal descriptions above of the real numbers are not sufficient for ensuring the correctness of proofs of theorems involving real numbers.
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https://en.wikipedia.org/wiki/Real_number_system
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The realization that a better definition was needed, and the elaboration of such a definition was a major development of 19th-century mathematics and is the foundation of real analysis, the study of real functions and real-valued sequences. A current axiomatic definition is that real numbers form the unique (up to an isomorphism) Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.
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https://en.wikipedia.org/wiki/Real_number_system
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In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density b−n. Intuitively, a number being simply normal means that no digit occurs more frequently than any other. If a number is normal, no finite combination of digits of a given length occurs more frequently than any other combination of the same length.
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https://en.wikipedia.org/wiki/Normal_number
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A normal number can be thought of as an infinite sequence of coin flips (binary) or rolls of a die (base 6). Even though there will be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally many of any other sequence of equal length. No digit or sequence is "favored".
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https://en.wikipedia.org/wiki/Normal_number
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A number is said to be normal (sometimes called absolutely normal) if it is normal in all integer bases greater than or equal to 2. While a general proof can be given that almost all real numbers are normal (meaning that the set of non-normal numbers has Lebesgue measure zero), this proof is not constructive, and only a few specific numbers have been shown to be normal. For example, any Chaitin's constant is normal (and uncomputable). It is widely believed that the (computable) numbers √2, π, and e are normal, but a proof remains elusive.
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https://en.wikipedia.org/wiki/Normal_number
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In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α > 0, such that | f ( x ) − f ( y ) | ≤ C ‖ x − y ‖ α {\displaystyle |f(x)-f(y)|\leq C\|x-y\|^{\alpha }} for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. If α = 1, then the function satisfies a Lipschitz condition.
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https://en.wikipedia.org/wiki/Hölder_exponent
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For any α > 0, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder. We have the following chain of strict inclusions for functions defined on a closed and bounded interval of the real line with a < b: Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous ⊂ uniformly continuous ⊂ continuous,where 0 < α ≤ 1.
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https://en.wikipedia.org/wiki/Hölder_exponent
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In mathematics, a real plane curve is usually a real algebraic curve defined in the real projective plane.
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https://en.wikipedia.org/wiki/Real_plane_curve
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In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation map σ: C → C {\displaystyle \sigma :{\mathbb {C} }\to {\mathbb {C} }\,} , with σ ( z ) = z ¯ {\displaystyle \sigma (z)={\bar {z}}} , giving the "canonical" real structure on C {\displaystyle {\mathbb {C} }\,} , that is C = R ⊕ i R {\displaystyle {\mathbb {C} }={\mathbb {R} }\oplus i{\mathbb {R} }\,} . The conjugation map is antilinear: σ ( λ z ) = λ ¯ σ ( z ) {\displaystyle \sigma (\lambda z)={\bar {\lambda }}\sigma (z)\,} and σ ( z 1 + z 2 ) = σ ( z 1 ) + σ ( z 2 ) {\displaystyle \sigma (z_{1}+z_{2})=\sigma (z_{1})+\sigma (z_{2})\,} .
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https://en.wikipedia.org/wiki/Real_structure
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In mathematics, a real-valued function K(x,y) is said to fulfill Mercer's condition if for all square-integrable functions g(x) one has ∬ g ( x ) K ( x , y ) g ( y ) d x d y ≥ 0. {\displaystyle \iint g(x)K(x,y)g(y)\,dx\,dy\geq 0.}
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https://en.wikipedia.org/wiki/Mercer's_theorem
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In mathematics, a real-valued function f on the interval is said to be singular if it has the following properties: f is continuous on . (**) there exists a set N of measure 0 such that for all x outside of N the derivative f ′(x) exists and is zero, that is, the derivative of f vanishes almost everywhere. f is non-constant on .A standard example of a singular function is the Cantor function, which is sometimes called the devil's staircase (a term also used for singular functions in general). There are, however, other functions that have been given that name.
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https://en.wikipedia.org/wiki/Singular_function
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One is defined in terms of the circle map. If f(x) = 0 for all x ≤ a and f(x) = 1 for all x ≥ b, then the function can be taken to represent a cumulative distribution function for a random variable which is neither a discrete random variable (since the probability is zero for each point) nor an absolutely continuous random variable (since the probability density is zero everywhere it exists). Singular functions occur, for instance, as sequences of spatially modulated phases or structures in solids and magnets, described in a prototypical fashion by the Frenkel–Kontorova model and by the ANNNI model, as well as in some dynamical systems. Most famously, perhaps, they lie at the center of the fractional quantum Hall effect.
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https://en.wikipedia.org/wiki/Singular_function
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In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions.
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https://en.wikipedia.org/wiki/Real-valued_function
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In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include a linear function f ( x ) = c x {\displaystyle f(x)=cx} (where c {\displaystyle c} is a real number), a quadratic function c x 2 {\displaystyle cx^{2}} ( c {\displaystyle c} as a nonnegative real number) and a exponential function c e x {\displaystyle ce^{x}} ( c {\displaystyle c} as a nonnegative real number).
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https://en.wikipedia.org/wiki/Convex_functions
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In simple terms, a convex function refers to a function whose graph is shaped like a cup ∪ {\displaystyle \cup } (or a straight line like a linear function), while a concave function's graph is shaped like a cap ∩ {\displaystyle \cap } . Convex functions play an important role in many areas of mathematics.
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https://en.wikipedia.org/wiki/Convex_functions
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They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and as a result, they are the most well-understood functionals in the calculus of variations. In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable. This result, known as Jensen's inequality, can be used to deduce inequalities such as the arithmetic–geometric mean inequality and Hölder's inequality.
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https://en.wikipedia.org/wiki/Convex_functions
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In mathematics, a real-valued function u ( x , y ) {\displaystyle u(x,y)} defined on a connected open set Ω ⊂ R 2 {\displaystyle \Omega \subset \mathbb {R} ^{2}} is said to have a conjugate (function) v ( x , y ) {\displaystyle v(x,y)} if and only if they are respectively the real and imaginary parts of a holomorphic function f ( z ) {\displaystyle f(z)} of the complex variable z := x + i y ∈ Ω . {\displaystyle z:=x+iy\in \Omega .} That is, v {\displaystyle v} is conjugate to u {\displaystyle u} if f ( z ) := u ( x , y ) + i v ( x , y ) {\displaystyle f(z):=u(x,y)+iv(x,y)} is holomorphic on Ω .
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https://en.wikipedia.org/wiki/Conjugate_harmonic_functions
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{\displaystyle \Omega .} As a first consequence of the definition, they are both harmonic real-valued functions on Ω {\displaystyle \Omega } . Moreover, the conjugate of u , {\displaystyle u,} if it exists, is unique up to an additive constant. Also, u {\displaystyle u} is conjugate to v {\displaystyle v} if and only if v {\displaystyle v} is conjugate to − u {\displaystyle -u} .
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https://en.wikipedia.org/wiki/Conjugate_harmonic_functions
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In mathematics, a reality structure on a complex vector space V is a decomposition of V into two real subspaces, called the real and imaginary parts of V: V = V R ⊕ i V R . {\displaystyle V=V_{\mathbb {R} }\oplus iV_{\mathbb {R} }.} Here VR is a real subspace of V, i.e. a subspace of V considered as a vector space over the real numbers. If V has complex dimension n (real dimension 2n), then VR must have real dimension n. The standard reality structure on the vector space C n {\displaystyle \mathbb {C} ^{n}} is the decomposition C n = R n ⊕ i R n .
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https://en.wikipedia.org/wiki/Reality_structure
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{\displaystyle \mathbb {C} ^{n}=\mathbb {R} ^{n}\oplus i\,\mathbb {R} ^{n}.} In the presence of a reality structure, every vector in V has a real part and an imaginary part, each of which is a vector in VR: v = Re { v } + i Im { v } {\displaystyle v=\operatorname {Re} \{v\}+i\,\operatorname {Im} \{v\}} In this case, the complex conjugate of a vector v is defined as follows: v ¯ = Re { v } − i Im { v } {\displaystyle {\overline {v}}=\operatorname {Re} \{v\}-i\,\operatorname {Im} \{v\}} This map v ↦ v ¯ {\displaystyle v\mapsto {\overline {v}}} is an antilinear involution, i.e. v ¯ ¯ = v , v + w ¯ = v ¯ + w ¯ , and α v ¯ = α ¯ v ¯ . {\displaystyle {\overline {\overline {v}}}=v,\quad {\overline {v+w}}={\overline {v}}+{\overline {w}},\quad {\text{and}}\quad {\overline {\alpha v}}={\overline {\alpha }}\,{\overline {v}}.}
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https://en.wikipedia.org/wiki/Reality_structure
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Conversely, given an antilinear involution v ↦ c ( v ) {\displaystyle v\mapsto c(v)} on a complex vector space V, it is possible to define a reality structure on V as follows. Let Re { v } = 1 2 ( v + c ( v ) ) , {\displaystyle \operatorname {Re} \{v\}={\frac {1}{2}}\left(v+c(v)\right),} and define V R = { Re { v } ∣ v ∈ V } . {\displaystyle V_{\mathbb {R} }=\left\{\operatorname {Re} \{v\}\mid v\in V\right\}.}
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https://en.wikipedia.org/wiki/Reality_structure
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Then V = V R ⊕ i V R . {\displaystyle V=V_{\mathbb {R} }\oplus iV_{\mathbb {R} }.} This is actually the decomposition of V as the eigenspaces of the real linear operator c. The eigenvalues of c are +1 and −1, with eigenspaces VR and i {\displaystyle i} VR, respectively. Typically, the operator c itself, rather than the eigenspace decomposition it entails, is referred to as the reality structure on V.
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https://en.wikipedia.org/wiki/Reality_structure
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In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f ( x ) {\displaystyle f(x)} with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial f ( x ) = x 2 + a x + b {\displaystyle f(x)=x^{2}+ax+b} splits into linear terms when reduced mod p {\displaystyle p} . That is, it determines for which prime numbers the relation f ( x ) ≡ f p ( x ) = ( x − n p ) ( x − m p ) ( mod p ) {\displaystyle f(x)\equiv f_{p}(x)=(x-n_{p})(x-m_{p}){\text{ }}({\text{mod }}p)} holds. For a general reciprocity lawpg 3, it is defined as the rule determining which primes p {\displaystyle p} the polynomial f p {\displaystyle f_{p}} splits into linear factors, denoted Spl { f ( x ) } {\displaystyle {\text{Spl}}\{f(x)\}} .
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https://en.wikipedia.org/wiki/Reciprocity_law
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There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert norm residue symbols (a,b/p), taking values in roots of unity, is equal to 1.
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https://en.wikipedia.org/wiki/Reciprocity_law
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Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see. The name reciprocity law was coined by Legendre in his 1785 publication Recherches d'analyse indéterminée, because odd primes reciprocate or not in the sense of quadratic reciprocity stated below according to their residue classes mod 4 {\displaystyle {\bmod {4}}} . This reciprocating behavior does not generalize well, the equivalent splitting behavior does. The name reciprocity law is still used in the more general context of splittings.
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https://en.wikipedia.org/wiki/Reciprocity_law
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In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set. As such, it has many of the desirable properties of smooth manifolds, including tangent spaces that are defined almost everywhere. Rectifiable sets are the underlying object of study in geometric measure theory.
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https://en.wikipedia.org/wiki/Rectifiable_set
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In mathematics, a recurrence relation is an equation according to which the n {\displaystyle n} th term of a sequence of numbers is equal to some combination of the previous terms. Often, only k {\displaystyle k} previous terms of the sequence appear in the equation, for a parameter k {\displaystyle k} that is independent of n {\displaystyle n} ; this number k {\displaystyle k} is called the order of the relation. If the values of the first k {\displaystyle k} numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In linear recurrences, the nth term is equated to a linear function of the k {\displaystyle k} previous terms.
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https://en.wikipedia.org/wiki/Difference_equation
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A famous example is the recurrence for the Fibonacci numbers, where the order k {\displaystyle k} is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n {\displaystyle n} . For these recurrences, one can express the general term of the sequence as a closed-form expression of n {\displaystyle n} .
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https://en.wikipedia.org/wiki/Difference_equation
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As well, linear recurrences with polynomial coefficients depending on n {\displaystyle n} are also important, because many common elementary and special functions have a Taylor series whose coefficients satisfy such a recurrence relation (see holonomic function). Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n {\displaystyle n} . The concept of a recurrence relation can be extended to multidimensional arrays, that is, indexed families that are indexed by tuples of natural numbers.
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https://en.wikipedia.org/wiki/Difference_equation
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In mathematics, a recurrent point for a function f is a point that is in its own limit set by f. Any neighborhood containing the recurrent point will also contain (a countable number of) iterates of it as well.
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https://en.wikipedia.org/wiki/Recurrent_point
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In mathematics, a recurrent word or sequence is an infinite word over a finite alphabet in which every factor occurs infinitely many times. An infinite word is recurrent if and only if it is a sesquipower.A uniformly recurrent word is a recurrent word in which for any given factor X in the sequence, there is some length nX (often much longer than the length of X) such that X appears in every block of length nX. The terms minimal sequence and almost periodic sequence (Muchnik, Semenov, Ushakov 2003) are also used.
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https://en.wikipedia.org/wiki/Uniformly_recurrent_word
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In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive.
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https://en.wikipedia.org/wiki/Reductive_group
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Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a number field, the classification is well understood.
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https://en.wikipedia.org/wiki/Reductive_group
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The classification of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic group G over a finite field k, or as minor variants of that construction. Reductive groups have a rich representation theory in various contexts. First, one can study the representations of a reductive group G over a field k as an algebraic group, which are actions of G on k-vector spaces. But also, one can study the complex representations of the group G(k) when k is a finite field, or the infinite-dimensional unitary representations of a real reductive group, or the automorphic representations of an adelic algebraic group. The structure theory of reductive groups is used in all these areas.
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https://en.wikipedia.org/wiki/Reductive_group
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In mathematics, a refinement monoid is a commutative monoid M such that for any elements a0, a1, b0, b1 of M such that a0+a1=b0+b1, there are elements c00, c01, c10, c11 of M such that a0=c00+c01, a1=c10+c11, b0=c00+c10, and b1=c01+c11. A commutative monoid M is said to be conical if x+y=0 implies that x=y=0, for any elements x,y of M.
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https://en.wikipedia.org/wiki/Refinement_monoid
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In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis (a vertical reflection) would look like q. Its image by reflection in a horizontal axis (a horizontal reflection) would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term reflection is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions.
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https://en.wikipedia.org/wiki/Horizontal_reflection
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Such isometries have a set of fixed points (the "mirror") that is an affine subspace, but is possibly smaller than a hyperplane. For instance a reflection through a point is an involutive isometry with just one fixed point; the image of the letter p under it would look like a d. This operation is also known as a central inversion (Coxeter 1969, §7.2), and exhibits Euclidean space as a symmetric space.
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https://en.wikipedia.org/wiki/Horizontal_reflection
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In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane. Some mathematicians use "flip" as a synonym for "reflection".
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https://en.wikipedia.org/wiki/Horizontal_reflection
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In mathematics, a reflection formula or reflection relation for a function f is a relationship between f(a − x) and f(x). It is a special case of a functional equation, and it is very common in the literature to use the term "functional equation" when "reflection formula" is meant. Reflection formulas are useful for numerical computation of special functions. In effect, an approximation that has greater accuracy or only converges on one side of a reflection point (typically in the positive half of the complex plane) can be employed for all arguments.
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https://en.wikipedia.org/wiki/Reflection_formula
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In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star regular 4-polytopes, giving a total of sixteen.
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https://en.wikipedia.org/wiki/Regular_star_4-polytope
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In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in a complex semisimple Lie algebra, an element X ∈ g {\displaystyle X\in {\mathfrak {g}}} is regular if its centralizer in g {\displaystyle {\mathfrak {g}}} has dimension equal to the rank of g {\displaystyle {\mathfrak {g}}} , which in turn equals the dimension of some Cartan subalgebra h {\displaystyle {\mathfrak {h}}} (note that in earlier papers, an element of a complex semisimple Lie algebra was termed regular if it is semisimple and the kernel of its adjoint representation is a Cartan subalgebra). An element g ∈ G {\displaystyle g\in G} a Lie group is regular if its centralizer has dimension equal to the rank of G {\displaystyle G} .
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https://en.wikipedia.org/wiki/Regular_element_of_a_Lie_algebra
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In mathematics, a regular map is a symmetric tessellation of a closed surface. More precisely, a regular map is a decomposition of a two-dimensional manifold (such as a sphere, torus, or real projective plane) into topological disks such that every flag (an incident vertex-edge-face triple) can be transformed into any other flag by a symmetry of the decomposition. Regular maps are, in a sense, topological generalizations of Platonic solids. The theory of maps and their classification is related to the theory of Riemann surfaces, hyperbolic geometry, and Galois theory. Regular maps are classified according to either: the genus and orientability of the supporting surface, the underlying graph, or the automorphism group.
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https://en.wikipedia.org/wiki/Regular_map_(graph_theory)
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