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c_hh5f0rxdwjs5 | In mathematics, a translation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x'y'-Cartesian coordinate system in which the x' axis is parallel to the x axis and k units away, and the y' axis is parallel to the y axis and h units away. This means that the origin O' of the new coordin... | Translation of axes |
c_qb0i8j2wamg3 | For example, if the xy-system is translated a distance h to the right and a distance k upward, then P will appear to have been translated a distance h to the left and a distance k downward in the x'y'-system . A translation of axes in more than two dimensions is defined similarly. A translation of axes is a rigid trans... | Translation of axes |
c_8q9sw4xpqw0i | In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarguesian planes, and the vast majority of known non-Desarguesian planes ar... | Translation plane |
c_cosgjz9n6p0q | A line l in a projective plane Π is a translation line if the group of all elations with axis l acts transitively on the points of the affine plane obtained by removing l from the plane Π, Πl (the affine derivative of Π). A projective plane with a translation line is called a translation plane. The affine plane obtaine... | Translation plane |
c_ukq9658n6b2h | In mathematics, a transverse knot is a smooth embedding of a circle into a three-dimensional contact manifold such that the tangent vector at every point of the knot is transverse to the contact plane at that point. Any Legendrian knot can be C0-perturbed in a direction transverse to the contact planes to obtain a tran... | Transverse knot |
c_7f9up7w8rf8g | In mathematics, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. Any connected graph without simple cycles is a tree. A tree data structure simulates a hierarchical tree structure with a set of linked nodes. | XML tree |
c_lqocq6qqpf6e | A hierarchy consists of an order defined on a set. The term hierarchy is used to stress a hierarchical relation among the elements. The XML specification defines an XML document as a well-formed text if it satisfies a list of syntax rules defined in the specification. | XML tree |
c_dirxvclzr8us | This specification is long, however 2 key points relating to the tree structure of an XML document are: The begin, end, and empty-element tags that delimit the elements are correctly nested, with none missing and none overlapping A single "root" element contains all the other elementsThese features resemble those of tr... | XML tree |
c_df9mwqrvkri0 | The JavaScript (E4X) extension explicitly defines two specific objects (XML and XMLList), which support XML document nodes and XML node lists as distinct objects and use a dot-notation specifying parent-child relationships. These data structures represent XML documents as a tree structure. An XML Tree represented graph... | XML tree |
c_3myvzr9foxot | For instance, the XML document and the ASCII tree have the same structure. XML Trees do not show the content in an Instance document, only the structure of the document. In this example Product is the Root Element of the tree and the two child nodes of Product are Name and Details. Details contains two child nodes, Des... | XML tree |
c_gldq0mcwd08c | In mathematics, a tree of primitive Pythagorean triples is a data tree in which each node branches to three subsequent nodes with the infinite set of all nodes giving all (and only) primitive Pythagorean triples without duplication. A Pythagorean triple is a set of three positive integers a, b, and c having the propert... | Tree of primitive Pythagorean triples |
c_sh0omoq9cdcd | This was first discovered by B. Berggren in 1934.F. J. M. Barning showed that when any of the three matrices A = B = C = {\displaystyle {\begin{array}{lcr}A={\begin{bmatrix}1&-2&2\\2&-1&2\\2&-2&3\end{bmatrix}}&B={\begin{bmatrix}1&2&2\\2&1&2\\2&2&3\end{bmatrix}}&C={\begin{bmatrix}-1&2&2\\-2&1&2\\-2&2&3\end{bmatrix}}\... | Tree of primitive Pythagorean triples |
c_fdbx3e7wv0qq | Thus each primitive Pythagorean triple has three "children". All primitive Pythagorean triples are descended in this way from the triple (3, 4, 5), and no primitive triple appears more than once. The result may be graphically represented as an infinite ternary tree with (3, 4, 5) at the root node (see classic tree at r... | Tree of primitive Pythagorean triples |
c_zmln2s7i5pwh | In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane,... | Triangle group |
c_gl32vre0ga0k | In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero. Because matrix equations with triangular matr... | Triangular form |
c_wzpy2bikv8ex | In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abe... | Exact triangle |
c_t77tgwpal178 | In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space X to complexes of sheaves, viewed as objects of the derived category of sheaves on X. More recently, triangulated categories have become objects of interest in their own right. Many equivalences between triangulated ca... | Exact triangle |
c_bjq6xu8fr3np | In mathematics, a tricategory is a kind of structure of category theory studied in higher-dimensional category theory. Whereas a weak 2-category is said to be a bicategory, a weak 3-category is said to be a tricategory (Gordon, Power & Street 1995; Baez & Dolan 1996; Leinster 1998).Tetracategories are the corresponding... | Tricategory |
c_b5fd598nlxxe | In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula: x y + a x 3 + b x 2 + c x = d {\displaystyle xy+ax^{3}+bx^{2}+cx=d} Trident curves are cubic plane curves with an ordinary double point in the real projective plane at x = 0, y ... | Trident curve |
c_kiqv5blt1uro | In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by ( a + b + c ) n = ∑ i , j , k i + j + k = n ( n i , j , k ) a i b j c k , {\displaystyle (a+b+c)^{n}=\sum _{{i,j,k} \atop {i+j+k=n}}{n \choose i,j,k}\,a^{i}\,b^{\;\!j}\;\!c^{k},} where n i... | Trinomial expansion |
c_t6t31gc03zd0 | k ! . {\displaystyle {n \choose i,j,k}={\frac {n!}{i!\,j!\,k!}}\,.} This formula is a special case of the multinomial formula for m = 3. The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron. | Trinomial expansion |
c_7wb1u81wokr4 | In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: 0 , 1 , {\displaystyle 0,1,} or e {\displaystyle e} dep... | Trivial group |
c_b76a5athnvz4 | {\displaystyle e\cdot e=e.} The similarly defined trivial monoid is also a group since its only element is its own inverse, and is hence the same as the trivial group. The trivial group is distinct from the empty set, which has no elements, hence lacks an identity element, and so cannot be a group. | Trivial group |
c_om3x7czsw6v1 | In mathematics, a trivial semigroup (a semigroup with one element) is a semigroup for which the cardinality of the underlying set is one. The number of distinct nonisomorphic semigroups with one element is one. If S = { a } is a semigroup with one element, then the Cayley table of S is The only element in S is the zero... | Trivial semigroup |
c_cyctwjdq26rv | It is the starting point for understanding the structure of semigroups. It serves as a counterexample in illuminating many situations. For example, the semigroup with one element is the only semigroup in which 0 = 1, that is, the zero element and the identity element are equal. Further, if S is a semigroup with one ele... | Trivial semigroup |
c_cafk151z3p88 | In mathematics, a tube domain is a generalization of the notion of a vertical strip (or half-plane) in the complex plane to several complex variables. A strip can be thought of as the collection of complex numbers whose real part lie in a given subset of the real line and whose imaginary part is unconstrained; likewise... | Tube domain |
c_bcz76w3wvet1 | Tubes over convex sets are domains of holomorphy. The Hardy spaces on tubes over convex cones have an especially rich structure, so that precise results are known concerning the boundary values of Hp functions. | Tube domain |
c_kl9tu58dfup8 | In mathematical physics, the future tube is the tube domain associated to the interior of the past null cone in Minkowski space, and has applications in relativity theory and quantum gravity. Certain tubes over cones support a Bergman metric in terms of which they become bounded symmetric domains. One of these is the S... | Tube domain |
c_4i3sohf3w4br | In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle. The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve draw a line perpend... | Tubular neighborhood |
c_f9b6chr7nqgo | Unless the curve is straight, these lines will intersect among themselves in a rather complicated fashion. However, if one looks only in a narrow band around the curve, the portions of the lines in that band will not intersect, and will cover the entire band without gaps. This band is a tubular neighborhood. | Tubular neighborhood |
c_8of227kugrfa | In general, let S be a submanifold of a manifold M, and let N be the normal bundle of S in M. Here S plays the role of the curve and M the role of the plane containing the curve. Consider the natural map i: N 0 → S {\displaystyle i:N_{0}\to S} which establishes a bijective correspondence between the zero section N 0 {\... | Tubular neighborhood |
c_t6t62x68oabq | In mathematics, a tuple is a finite sequence or ordered list of numbers or, more generally, mathematical objects, which are called the elements of the tuple. An n-tuple is a tuple of n elements, where n is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly ca... | Empty tuple |
c_zdzuh5eltv2r | Tuple may be formally defined from ordered pairs by recurrence by starting from ordered pairs; indeed, a n-tuple can be identified with the ordered pair of its (n − 1) first elements and its nth element. Tuples are usually written by listing the elements within parentheses "( )", separated by a comma and a space; for e... | Empty tuple |
c_gj7bj8cpm14q | Braces "{ }" are used to specify arrays in some programming languages but not in mathematical expressions, as they are the standard notation for sets. The term tuple can often occur when discussing other mathematical objects, such as vectors. In computer science, tuples come in many forms. | Empty tuple |
c_id5nitt6axks | Most typed functional programming languages implement tuples directly as product types, tightly associated with algebraic data types, pattern matching, and destructuring assignment. Many programming languages offer an alternative to tuples, known as record types, featuring unordered elements accessed by label. A few pr... | Empty tuple |
c_4ap3mq54kb0y | In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (the twisted cubic, therefore). In algebraic geometry, the twisted cubic is a simple example of a projective vari... | Twisted cubic |
c_qu5hisigese9 | In mathematics, a twisted polynomial is a polynomial over a field of characteristic p {\displaystyle p} in the variable τ {\displaystyle \tau } representing the Frobenius map x ↦ x p {\displaystyle x\mapsto x^{p}} . In contrast to normal polynomials, multiplication of these polynomials is not commutative, but satisfies... | Noncommutative polynomials |
c_15tmf5gyxq61 | In mathematics, a twisted sheaf is a variant of a coherent sheaf. Precisely, it is specified by: an open covering in the étale topology Ui, coherent sheaves Fi over Ui, a Čech 2-cocycle θ on the covering Ui as well as the isomorphisms g i j: F j | U i j → ∼ F i | U i j {\displaystyle g_{ij}:F_{j}|_{U_{ij}}{\overset {\s... | Twisted sheaf |
c_t7xwzuekyluk | In mathematics, a two-graph is a set of (unordered) triples chosen from a finite vertex set X, such that every (unordered) quadruple from X contains an even number of triples of the two-graph. A regular two-graph has the property that every pair of vertices lies in the same number of triples of the two-graph. Two-graph... | Two-graph |
c_wbjhi3258thh | In mathematics, a unary function is a function that takes one argument. A unary operator belongs to a subset of unary functions, in that its range coincides with its domain. In contrast, a unary function's domain may or may not coincide with its range. | Unary function |
c_65npsacbeu14 | In mathematics, a unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function f: A → A, where A is a set. | Unary functional symbol |
c_vig3a2dwki3a | The function f is a unary operation on A. Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial n! ), functional notation (e.g. sin x or sin(x)), and superscripts (e.g. transpose AT). Other notations exist as well, for example, in the case of the square root, a horizontal bar extending the ... | Unary functional symbol |
c_05rlrvt51x60 | In mathematics, a unicoherent space is a topological space X {\displaystyle X} that is connected and in which the following property holds: For any closed, connected A , B ⊂ X {\displaystyle A,B\subset X} with X = A ∪ B {\displaystyle X=A\cup B} , the intersection A ∩ B {\displaystyle A\cap B} is connected. For example... | Unicoherent space |
c_d3jr3ypyzu77 | In mathematics, a uniform matroid is a matroid in which the independent sets are exactly the sets containing at most r elements, for some fixed integer r. An alternative definition is that every permutation of the elements is a symmetry. | Uniform matroid |
c_2ecoy1zrey66 | In mathematics, a uniform tree is a locally finite tree which is the universal cover of a finite graph. Equivalently, the full automorphism group G=Aut(X) of the tree, which is a locally compact topological group, is unimodular and G\X is finite. Also equivalent is the existence of a uniform X-lattice in G. | Uniform tree |
c_tbbzq8pwtsb0 | In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family. | Uniformly bounded |
c_abmu5lb4bd41 | In mathematics, a uniformly bounded representation T {\displaystyle T} of a locally compact group G {\displaystyle G} on a Hilbert space H {\displaystyle H} is a homomorphism into the bounded invertible operators which is continuous for the strong operator topology, and such that sup g ∈ G ‖ T g ‖ B ( H ) {\displaystyl... | Uniformly bounded representation |
c_fm49bqa2391e | The result on unitarizability of uniformly bounded representations was extended in 1950 by Dixmier, Day and Nakamura-Takeda to all locally compact amenable groups, following essentially the method of proof of Sz-Nagy. The result is known to fail for non-amenable groups such as SL(2,R) and the free group on two generato... | Uniformly bounded representation |
c_nr46bes8zyef | In mathematics, a uniformly disconnected space is a metric space ( X , d ) {\displaystyle (X,d)} for which there exists λ > 0 {\displaystyle \lambda >0} such that no pair of distinct points x , y ∈ X {\displaystyle x,y\in X} can be connected by a λ {\displaystyle \lambda } -chain. A λ {\displaystyle \lambda } -chain be... | Uniformly disconnected space |
c_fvr5wtaocpx0 | In mathematics, a uniformly smooth space is a normed vector space X {\displaystyle X} satisfying the property that for every ϵ > 0 {\displaystyle \epsilon >0} there exists δ > 0 {\displaystyle \delta >0} such that if x , y ∈ X {\displaystyle x,y\in X} with ‖ x ‖ = 1 {\displaystyle \|x\|=1} and ‖ y ‖ ≤ δ {\displaystyle ... | Uniformly smooth space |
c_g873tknf84eg | In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N that is its inverse (these are equivalent under Cramer's rule). Thus every equation Mx = b, where M and b both have integ... | Totally unimodular |
c_30qp4dl3a9fh | In mathematics, a unimodular polynomial matrix is a square polynomial matrix whose inverse exists and is itself a polynomial matrix. Equivalently, a polynomial matrix A is unimodular if its determinant det(A) is a nonzero constant. | Unimodular polynomial matrix |
c_h52bdcbrgglv | In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n. In particular, a square matrix M is a unipotent matrix if and only if its characteristic polynomial P(t) is a power of t − 1. Thus all the eigenvalues of a unipotent matrix are 1... | Unipotent group |
c_gi0zu0aryri7 | In mathematics, a unipotent representation of a reductive group is a representation that has some similarities with unipotent conjugacy classes of groups. Informally, Langlands philosophy suggests that there should be a correspondence between representations of a reductive group and conjugacy classes of a Langlands dua... | Unipotent representation |
c_bnx2pqe6hzft | In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any... | Unique factorisation |
c_3l18kea7d3yp | In mathematics, a unique sink orientation is an orientation of the edges of a polytope such that, in every face of the polytope (including the whole polytope as one of the faces), there is exactly one vertex for which all adjoining edges are oriented inward (i.e. towards that vertex). If a polytope is given together wi... | Unique sink orientation |
c_xtfio46y1w5r | In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems include: Alexandrov's uniqueness theorem of three-dimensional polyhedra... | Uniqueness theorem |
c_wb7gfb8shwlc | Fundamental theorem of arithmetic, the uniqueness of prime factorization. Holmgren's uniqueness theorem for linear partial differential equations with real analytic coefficients. Picard–Lindelöf theorem, the uniqueness of solutions to first-order differential equations. Thompson uniqueness theorem in finite group theor... | Uniqueness theorem |
c_r08qqnivq8cg | In mathematics, a unistochastic matrix (also called unitary-stochastic) is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix. A square matrix B of size n is doubly stochastic (or bistochastic) if all its entries are non-negative real numbers and each o... | Unistochastic matrix |
c_tbfilc3c4k4b | Since all orthogonal matrices are necessarily unitary matrices, all orthostochastic matrices are also unistochastic. The converse, however, is not true. First, all 2-by-2 doubly stochastic matrices are both unistochastic and orthostochastic, but for larger n this is not the case. | Unistochastic matrix |
c_asudnzc1s51a | For example, take n = 3 {\displaystyle n=3} and consider the following doubly stochastic matrix: B = 1 2 . {\displaystyle B={\frac {1}{2}}{\begin{bmatrix}1&1&0\\0&1&1\\1&0&1\end{bmatrix}}.} This matrix is not unistochastic, since any two vectors with moduli equal to the square root of the entries of two columns (or ro... | Unistochastic matrix |
c_71mfmtgt8v6n | In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S1 because it is a one-dimension... | Unit circle |
c_eboyrqn44qut | Thus, by the Pythagorean theorem, x and y satisfy the equation Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant. The interior of ... | Unit circle |
c_yqz7v2kphmb3 | In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ball is the closed set of points of distance less than or equal to 1 from a f... | Unit ball |
c_3yi0cv8wz8tl | Special cases are the unit circle and the unit disk. The importance of the unit sphere is that any sphere can be transformed to a unit sphere by a combination of translation and scaling. In this way the properties of spheres in general can be reduced to the study of the unit sphere. | Unit ball |
c_5f0p2oxeuexn | In mathematics, a unit square is a square whose sides have length 1. Often, the unit square refers specifically to the square in the Cartesian plane with corners at the four points (0, 0), (1, 0), (0, 1), and (1, 1). | Unit square |
c_k45eowca995n | In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in v ^ {\displaystyle {\hat {\mathbf {v} }}} (pronounced "v-hat"). | Hat operator |
c_hr0oku9z0uld | In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in v ^ {\displaystyle {\hat {\mathbf {v} }}} (pronounced "v-hat"). The term direction vector, commonly denoted as d, is used to de... | Unit vector |
c_6ynotehbyd6h | 2D spatial directions are numerically equivalent to points on the unit circle and spatial directions in 3D are equivalent to a point on the unit sphere. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., u ^ = u ‖ u ‖ {\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{\... | Unit vector |
c_qgma3457sdj1 | In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in the case that G is a locally compact (Hausdorff) topological group and the representations are strongly conti... | Unitary representation |
c_du8z1dmak70b | In mathematics, a unitary spider diagram adds existential points to an Euler or a Venn diagram. The points indicate the existence of an attribute described by the intersection of contours in the Euler diagram. These points may be joined forming a shape like a spider. Joined points represent an "or" condition, also know... | Spider diagram |
c_9m45md4j6bax | In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. | Unitary transformation |
c_4zh42vtdytsc | In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate cases is fundamental; for example, the fundamental theorem of algebra and E... | Univariate |
c_hjpypyx4o1eb | In statistics, a univariate distribution characterizes one variable, although it can be applied in other ways as well. For example, univariate data are composed of a single scalar component. In time series analysis, the whole time series is the "variable": a univariate time series is the series of values over time of a... | Univariate |
c_7j1dkml3teay | Correspondingly, a "multivariate time series" characterizes the changing values over time of several quantities. In some cases, the terminology is ambiguous, since the values within a univariate time series may be treated using certain types of multivariate statistical analyses and may be represented using multivariate... | Univariate |
c_1v5ndx8ipw0m | In mathematics, a univariate polynomial of degree n with real or complex coefficients has n complex roots, if counted with their multiplicities. They form a multiset of n points in the complex plane. This article concerns the geometry of these points, that is the information about their localization in the complex plan... | Properties of polynomial roots |
c_kmu4et4hez92 | Such bounds are widely used for root-finding algorithms for polynomials, either for tuning them, or for computing their computational complexity. Some other properties are probabilistic, such as the expected number of real roots of a random polynomial of degree n with real coefficients, which is less than 1 + 2 π ln ... | Properties of polynomial roots |
c_lblnr4rgai8b | In mathematics, a universal C*-algebra is a C*-algebra described in terms of generators and relations. In contrast to rings or algebras, where one can consider quotients by free rings to construct universal objects, C*-algebras must be realizable as algebras of bounded operators on a Hilbert space by the Gelfand-Naimar... | Universal C*-algebra |
c_7tnly9jyprlz | In mathematics, a universal graph is an infinite graph that contains every finite (or at-most-countable) graph as an induced subgraph. A universal graph of this type was first constructed by Richard Rado and is now called the Rado graph or random graph. More recent work has focused on universal graphs for a graph famil... | Universal graph |
c_7gs1td0iw0wd | However it is not the smallest such graph: it is known that there is a universal graph for n-vertex trees, with only n vertices and O(n log n) edges, and that this is optimal. A construction based on the planar separator theorem can be used to show that n-vertex planar graphs have universal graphs with O(n3/2) edges, a... | Universal graph |
c_zxdrc129k3l0 | It is also possible to construct universal graphs for planar graphs that have n1+o(1) vertices.Sumner's conjecture states that tournaments are universal for polytrees, in the sense that every tournament with 2n − 2 vertices contains every polytree with n vertices as a subgraph.A family F of graphs has a universal graph... | Universal graph |
c_43t6u88iflkt | In mathematics, a universal quadratic form is a quadratic form over a ring that represents every element of the ring. A non-singular form over a field which represents zero non-trivially is universal. | Universal quadratic form |
c_751tb02a3jtz | In mathematics, a universal space is a certain metric space that contains all metric spaces whose dimension is bounded by some fixed constant. A similar definition exists in topological dynamics. | Universal space (topology) |
c_rvcuydozudj8 | In mathematics, a vampire number or true vampire number is a composite natural number v, with an even number of digits n, that can be factored into two integers x and y each with n/2 digits and not both with trailing zeroes, where v contains all the digits from x and from y, in any order. x and y are called the fangs. ... | Wonders of Numbers |
c_w2dpfvy18yom | Similarly, 136,948 is a vampire because 136,948 = 146 × 938. Vampire numbers first appeared in a 1994 post by Clifford A. Pickover to the Usenet group sci.math, and the article he later wrote was published in chapter 30 of his book Keys to Infinity.In addition to "Vampire numbers", a term Pickover actually coined, he h... | Wonders of Numbers |
c_evsjahkx8ey1 | In 1990, he asked "Is There a Double Smoothly Undulating Integer? ", and he computed "All Known Replicating Fibonacci Digits Less than One Billion". With his colleague John R. Hendricks, he was the first to compute the smallest perfect (nasik) magic tesseract. The "Pickover sequence" dealing with e and pi was named aft... | Wonders of Numbers |
c_hxcvd5szj38t | In mathematics, a variable (from Latin variabilis, "changeable") is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set.Algebraic computations with variables as if they were explicit numbers solve a ... | Variable (logics) |
c_1sw09jnlxkl7 | In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities was initially developed to deal with equilibrium problems, precisely the Signorini ... | Variational inequalities |
c_ma83jl70lyqd | In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X {\displaystyle X} (for example X {\displaystyle X} could be a topological space, a manifold, or an algebraic variety): to every point x {\displaystyle x} of the space X... | Vector bundle morphism |
c_28pzduvqublu | Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial. | Vector bundle morphism |
c_n183uhl2lrns | Vector bundles are almost always required to be locally trivial, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can b... | Vector bundle morphism |
c_40mnx53tc3wa | In mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection. | Flat vector bundle |
c_f05tjgqiwzbe | In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only. | Lyapunov's theorem |
c_u45o929oy6wa | In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-value... | Vector-valued differential form |
c_i1t3wk4c76v2 | In mathematics, a versor is a quaternion of norm one (a unit quaternion). Each versor has the form q = exp ( a r ) = cos a + r sin a , r 2 = − 1 , a ∈ , {\displaystyle q=\exp(a\mathbf {r} )=\cos a+\mathbf {r} \sin a,\quad \mathbf {r} ^{2}=-1,\quad a\in ,} where the r2 = −1 condition means that r is a unit-length... | Unit quaternion |
c_qkksygjxa3am | In mathematics, a vertex cycle cover (commonly called simply cycle cover) of a graph G is a set of cycles which are subgraphs of G and contain all vertices of G. If the cycles of the cover have no vertices in common, the cover is called vertex-disjoint or sometimes simply disjoint cycle cover. This is sometimes known a... | Vertex cycle cover |
c_1jno6irlxb8n | Similar definitions exist for digraphs, in terms of directed cycles. Finding a vertex-disjoint cycle cover of a directed graph can also be performed in polynomial time by a similar reduction to perfect matching. However, adding the condition that each cycle should have length at least 3 makes the problem NP-hard. | Vertex cycle cover |
c_uy0ut0kxnsjp | In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geomet... | Vertex algebra |
c_0m2rf5i66jn7 | Borcherds formulated the notion of vertex algebra by axiomatizing the relations between the lattice vertex operators, producing an algebraic structure that allows one to construct new Lie algebras by following Frenkel's method. The notion of vertex operator algebra was introduced as a modification of the notion of vert... | Vertex algebra |
c_ki6qb9s88f76 | Motivated by this observation, they added the Virasoro action and bounded-below property as axioms. We now have post-hoc motivation for these notions from physics, together with several interpretations of the axioms that were not initially known. | Vertex algebra |
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