id stringlengths 14 14 | text stringlengths 9 3.55k | source stringlengths 1 250 |
|---|---|---|
c_a923r4im7eb0 | 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. | Quasifield |
c_mhinqljp45fq | 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... | Quasiperfect number |
c_moyxtmpv560g | 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" fun... | Quasiperiodic function |
c_2y2vqp4w2ag4 | 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 quasiper... | Quasiperiodic function |
c_gkxs6wirlh59 | 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 gro... | Quasirandom group |
c_1yxpowhwin1b | 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... | Quasisimple group |
c_zoxohd743hjo | 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 set... | Quasisymmetric map |
c_pg683z68hzkr | 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 group... | Quasithin group |
c_pr75vh70hk0y | 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 ... | Quasitoric manifold |
c_jm8p8pyg12yx | 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. | Quasivariety |
c_15923yuso3wk | 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. | Sylvester pentahedron |
c_g5d4r12y7cax | 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... | Quaternion algebra |
c_oe9i1g23vhgr | 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 ... | Quaternionic discrete series representation |
c_qt3v5c02nv9a | 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 eleme... | Quaternionic structure |
c_ya44ade5uljl | }}\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 quaternio... | Quaternionic structure |
c_54780j4akhx2 | 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. I... | Queen's graph |
c_szyiwiplh2u8 | 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\ti... | Queen's graph |
c_3c5rnrx89p8i | 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 ... | Quintic equation |
c_zl8h3kdcag3i | 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 al... | Quintic equation |
c_ebm7jh36gfb4 | 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 ... | Quintic threefold |
c_z5p2o6e6r06f | 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. | Quippian |
c_59tbn3fvqox5 | 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... | Maltsev conditions |
c_8nuzbsh452tk | 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. | Quotient category |
c_hpphl2qr24ad | 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 ... | Radial function |
c_n4r2by0pjxw9 | 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 rad... | Radial function |
c_8leu22l5r0yk | 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}... | Radial function |
c_dms07m6rgccl | 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 a... | Radially unbounded function |
c_r4b8k774qqeu | 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. | Random compact set |
c_1esg52e8de2m | 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 f... | Random minimal spanning tree |
c_ct9fj92wi3ki | 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. | Random polytope |
c_iueo1zdaqn2n | 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, ... | Random walk with drift |
c_zzs123x82nom | 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. | Rank ring |
c_j69fljpyd3gj | 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 defini... | Ranked poset |
c_0e3bvop2nlpn | 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 var... | Temporal rate of change |
c_itdyhjmqackm | 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 met... | Temporal rate of change |
c_mirabx2q0c2k | 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 ... | Ratio analysis |
c_9b7ungbkygvb | 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. | Ratio analysis |
c_4dezwu42xfyt | 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 ... | Ratio analysis |
c_ss2sk73goq64 | 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... | Proper rational function |
c_ynst3r010jnu | 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. | Kleene monoid |
c_cczr0ql4fw4q | 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 degre... | Directrix of a rational normal scroll |
c_v362o4z8qe2c | 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 r... | Field of rationals |
c_jikraj5wxapo | 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 on... | Field of rationals |
c_c9blhbbh40v7 | 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 ... | Field of rationals |
c_t344ikx3yh1e | {\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 integ... | Field of rationals |
c_4topdw6urqil | 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... | Unirationality |
c_o5wncp7e90j5 | 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 ) {\display... | Rational zeta series |
c_pzwpo1zplnck | 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 "Strahlklassengrup... | Ray class field |
c_w2sex0qwron1 | 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 sligh... | Ray class field |
c_mstxbn50jbs3 | 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... | Read-once function |
c_g6m1etanzc6v | 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. | Real closed fields |
c_vuzeoksgg9f4 | 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. | Real closed ring |
c_xn8btwv25ktc | 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. | Harmonic differential |
c_0ntd3t54cphk | 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... | Uniform continuity |
c_b2ih3bpkp38b | 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 sl... | Uniform continuity |
c_cn41emkac84b | 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 neighbour... | Uniform continuity |
c_ix9mb95kg2f5 | 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 expansi... | Real number system |
c_3tb5mcs7sdf7 | 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 ar... | Real number system |
c_oj7i6u2dhhkg | 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 th... | Real number system |
c_slwuqsmzfhc4 | 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 i... | Real number system |
c_0oywx7s18i85 | 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 digit... | Normal number |
c_z0ml6vqmh995 | 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 ... | Normal number |
c_2n9ud6rl7hv8 | 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 ... | Normal number |
c_1j8itfijc3nh | 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. M... | Hölder exponent |
c_7rzmeu1e6atm | 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 continu... | Hölder exponent |
c_2kqit7yozael | In mathematics, a real plane curve is usually a real algebraic curve defined in the real projective plane. | Real plane curve |
c_we5dz7th17qv | 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 {\displa... | Real structure |
c_fsys4n4frrya | 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.} | Mercer's theorem |
c_j9kmrynledkj | 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-co... | Singular function |
c_bl7tlkd9667i | 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 ra... | Singular function |
c_q7ke7rd20e5z | 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 ... | Real-valued function |
c_pbm5v25nehhe | 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-diff... | Convex functions |
c_8l032893mfv7 | 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. | Convex functions |
c_gdi9d6yagdn2 | 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 continu... | Convex functions |
c_garr84ib6ogw | 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 f... | Conjugate harmonic functions |
c_df2f0ngfymw6 | {\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 i... | Conjugate harmonic functions |
c_c39i55cnwmsn | 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... | Reality structure |
c_cys46eeay0zl | {\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... | Reality structure |
c_dt9xpyu5dfj0 | 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 }... | Reality structure |
c_ijhijglq8piy | 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... | Reality structure |
c_r9glbqy2qzqy | 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 {\displ... | Reciprocity law |
c_cfoz6ikxa5yt | 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 re... | Reciprocity law |
c_fxo5otr04bwg | 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 thei... | Reciprocity law |
c_ka0z3oglfolw | 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... | Rectifiable set |
c_gxdvlgtg7o6w | 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 independ... | Difference equation |
c_eby4ig1t6a9v | 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 ... | Difference equation |
c_hx8ejwo434t8 | 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-f... | Difference equation |
c_3gnz2dj3v2lk | 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. | Recurrent point |
c_0kq3z5tu3mci | 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 lengt... | Uniformly recurrent word |
c_7daddok3to4k | 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 m... | Reductive group |
c_p4e89avv0l3i | 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 harde... | Reductive group |
c_lp7pjw90868r | 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 rep... | Reductive group |
c_rfucwrszhcby | 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 e... | Refinement monoid |
c_nwr4r7n9oan3 | 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 ... | Horizontal reflection |
c_3mw5f62tap2m | 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 centra... | Horizontal reflection |
c_nvsv5gv1lysu | 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 syno... | Horizontal reflection |
c_jnny2ykbr2k8 | 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 nu... | Reflection formula |
c_pm2fb94frnhc | 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. | Regular star 4-polytope |
c_y7yfcluvcwqj | 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 equ... | Regular element of a Lie algebra |
c_e7brgx4x7l0d | 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 ... | Regular map (graph theory) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.