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wiki_2600_chunk_0 | Differential variational inequality | Differential variational inequalities were first formally introduced by Pang and Stewart, whose definition should not be confused with the differential variational inequality used in Aubin and Cellina (1984). Differential variational inequalities have the form to find u ( t ) ∈ K {\displaystyle u(t)\in K} such that ⟨ v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2601_chunk_0 | Digital manifold | In mathematics, a digital manifold is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space. A combinatorial manifold is a kind of manifold which is a discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2602_chunk_0 | Dihedral symmetry | In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2603_chunk_0 | Dihedral symmetry | In geometry, Dn or Dihn refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, D2n refers to this same dihedral group. This article uses the geometric convention, Dn. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2604_chunk_0 | Dijoin | A fractional weighted version of the conjecture, posed by Jack Edmonds and Rick Giles, was refuted by Alexander Schrijver.The Lucchesi–Younger theorem states that the minimum size of a dijoin, in any given directed graph, equals the maximum number of disjoint dicuts that can be found in the graph. The minimum weight di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2605_chunk_0 | Dilation theory | In mathematics, a dilation is a function f {\displaystyle f} from a metric space M {\displaystyle M} into itself that satisfies the identity d ( f ( x ) , f ( y ) ) = r d ( x , y ) {\displaystyle d(f(x),f(y))=rd(x,y)} for all points x , y ∈ M {\displaystyle x,y\in M} , where d ( x , y ) {\displaystyle d(x,y)} is the di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2606_chunk_0 | Direct limit | In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphism... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2607_chunk_0 | Direct limit | (This is a slight abuse of notation as it suppresses the system of homomorphisms that is crucial for the structure of the limit.) Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits, which are also a special case of limits in category theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2608_chunk_0 | Direct limit of groups | In mathematics, a direct limit of groups is the direct limit of a direct system of groups. These are central objects of study in algebraic topology, especially stable homotopy theory and homological algebra. They are sometimes called stable groups, though this term normally means something quite different in model theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2609_chunk_0 | Directed set | Likewise, lattices are directed sets both upward and downward. In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2610_chunk_0 | Discrete series representation | In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²(G). In the Plancherel measure, such representations have positive measure. The name comes from the fact that they a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2611_chunk_0 | Discrete valuation field | In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function: ν: K → Z ∪ { ∞ } {\displaystyle \nu :K\to \mathbb {Z} \cup \{\infty \}} satisfying the conditions: ν ( x ⋅ y ) = ν ( x ) + ν ( y ) {\displaystyle \nu (x\cdot y)=\nu (x)+\nu (y)} ν ( x + y ) ≥ min { ν ( x ) , ν ( y ) } {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2612_chunk_0 | Dispersive PDE | In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2613_chunk_0 | Dissipative operator | In mathematics, a dissipative operator is a linear operator A defined on a linear subspace D(A) of Banach space X, taking values in X such that for all λ > 0 and all x ∈ D(A) ‖ ( λ I − A ) x ‖ ≥ λ ‖ x ‖ . {\displaystyle \|(\lambda I-A)x\|\geq \lambda \|x\|.} A couple of equivalent definitions are given below. A dissipa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2614_chunk_0 | Ample line bundle | In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2615_chunk_0 | Distinguished limit | In mathematics, a distinguished limit is an appropriately chosen scale factor used in the method of matched asymptotic expansions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2616_chunk_0 | Summation method | In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2617_chunk_0 | Summation method | A counterexample is the harmonic series 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯ = ∑ n = 1 ∞ 1 n . {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n}}.} The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2618_chunk_0 | Summation method | In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A summability method or summation method is a partial function from the set of series to values. For example, Cesàro summation ass... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2619_chunk_0 | Summation method | Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums. Other methods involve analytic continuations of related series. In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2620_chunk_0 | Diversity (mathematics) | In mathematics, a diversity is a generalization of the concept of metric space. The concept was introduced in 2012 by Bryant and Tupper, who call diversities "a form of multi-way metric". The concept finds application in nonlinear analysis.Given a set X {\displaystyle X} , let ℘ fin ( X ) {\displaystyle \wp _{\mbox{fin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2621_chunk_0 | Divisor (number theory) | In mathematics, a divisor of an integer n {\displaystyle n} , also called a factor of n {\displaystyle n} , is an integer m {\displaystyle m} that may be multiplied by some integer to produce n {\displaystyle n} . In this case, one also says that n {\displaystyle n} is a multiple of m . {\displaystyle m.} An integer n ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2622_chunk_0 | Domino (mathematics) | In mathematics, a domino is a polyomino of order 2, that is, a polygon in the plane made of two equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there is only one free domino. Since it has reflection symmetry, it is also the only one-sided domino (with... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2623_chunk_0 | Double affine Hecke algebra | In mathematics, a double affine Hecke algebra, or Cherednik algebra, is an algebra containing the Hecke algebra of an affine Weyl group, given as the quotient of the group ring of a double affine braid group. They were introduced by Cherednik, who used them to prove Macdonald's constant term conjecture for Macdonald po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2624_chunk_0 | Double vector bundle | In mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent T E {\displaystyle TE} of a vector bundle E {\displaystyle E} and the double tangent bundle T 2 M {\displaystyle T^{2}M} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2625_chunk_0 | Doubly periodic function | In mathematics, a doubly periodic function is a function defined on the complex plane and having two "periods", which are complex numbers u and v that are linearly independent as vectors over the field of real numbers. That u and v are periods of a function ƒ means that f ( z + u ) = f ( z + v ) = f ( z ) {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2626_chunk_0 | Natural pairing | In mathematics, a dual system, dual pair, or duality over a field K {\displaystyle \mathbb {K} } is a triple ( X , Y , b ) {\displaystyle (X,Y,b)} consisting of two vector spaces X {\displaystyle X} and Y {\displaystyle Y} over K {\displaystyle \mathbb {K} } and a non-degenerate bilinear map b: X × Y → K {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2627_chunk_0 | Dual wavelet | In mathematics, a dual wavelet is the dual to a wavelet. In general, the wavelet series generated by a square-integrable function will have a dual series, in the sense of the Riesz representation theorem. However, the dual series is not itself in general representable by a square-integrable function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2628_chunk_0 | Duality (mathematics) | In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2629_chunk_0 | Duality (mathematics) | It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and anothe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2630_chunk_0 | Dyadic fraction | Dyadic rationals also have applications in weights and measures, musical time signatures, and early mathematics education. They can accurately approximate any real number. The sum, difference, or product of any two dyadic rational numbers is another dyadic rational number, given by a simple formula. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2631_chunk_0 | Dyadic fraction | In advanced mathematics, the dyadic rational numbers are central to the constructions of the dyadic solenoid, Minkowski's question-mark function, Daubechies wavelets, Thompson's group, Prüfer 2-group, surreal numbers, and fusible numbers. These numbers are order-isomorphic to the rational numbers; they form a subsystem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2632_chunk_0 | Global dynamical system | In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2633_chunk_0 | Global dynamical system | At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2634_chunk_0 | Global dynamical system | Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2635_chunk_0 | Global dynamical system | In physics, a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives". In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2636_chunk_0 | Geometry of natural structure | In mathematics, a dynamical system is chaotic if it is (highly) sensitive to initial conditions (the so-called "butterfly effect"), which requires the mathematical properties of topological mixing and dense periodic orbits.Alongside fractals, chaos theory ranks as an essentially universal influence on patterns in natur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2637_chunk_0 | Geometry of natural structure | Meanders are sinuous bends in rivers or other channels, which form as a fluid, most often water, flows around bends. As soon as the path is slightly curved, the size and curvature of each loop increases as helical flow drags material like sand and gravel across the river to the inside of the bend. The outside of the lo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2638_chunk_0 | Reversed process | In mathematics, a dynamical system is time-reversible if the forward evolution is one-to-one, so that for every state there exists a transformation (an involution) π which gives a one-to-one mapping between the time-reversed evolution of any one state and the forward-time evolution of another corresponding state, given... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2639_chunk_0 | Scientific fact | In mathematics, a fact is a statement (called a theorem) that can be proven by logical argument from certain axioms and definitions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2640_chunk_0 | Factor system | In mathematics, a factor system (sometimes called factor set) is a fundamental tool of Otto Schreier’s classical theory for group extension problem. It consists of a set of automorphisms and a binary function on a group satisfying certain condition (so-called cocycle condition). In fact, a factor system constitutes a r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2641_chunk_0 | Schützenberger theorem | In mathematics, a factorisation of a free monoid is a sequence of subsets of words with the property that every word in the free monoid can be written as a concatenation of elements drawn from the subsets. The Chen–Fox–Lyndon theorem states that the Lyndon words furnish a factorisation. The Schützenberger theorem relat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2642_chunk_0 | Indexed set | In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers, is a collection of real numbers, where a given function selects one real number for each integer (possibly the sam... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2643_chunk_0 | Zigzag poset | {\displaystyle 1,1,2,4,10,32,122,544,2770,15872,101042.} (sequence A001250 in the OEIS).The number of antichains in a fence is a Fibonacci number; the distributive lattice with this many elements, generated from a fence via Birkhoff's representation theorem, has as its graph the Fibonacci cube.A partially ordered set i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2644_chunk_0 | Fibrifold notation | In mathematics, a fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by John Horton Conway, Olaf Delgado Friedrichs, and Daniel H. Huson et al. (2001), who introduced a system of notation for 3-dimensional fibrifolds and used this to assign names to the 219 affine spac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2645_chunk_0 | Normalized valuation | Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2646_chunk_0 | Normalized valuation | Every local field is isomorphic (as a topological field) to one of the following: Archimedean local fields (characteristic zero): the real numbers R, and the complex numbers C. Non-Archimedean local fields of characteristic zero: finite extensions of the p-adic numbers Qp (where p is any prime number). Non-Archimedean ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2647_chunk_0 | Spherically complete field | In mathematics, a field K with an absolute value is called spherically complete if the intersection of every decreasing sequence of balls (in the sense of the metric induced by the absolute value) is nonempty: B 1 ⊇ B 2 ⊇ ⋯ ⇒ ⋂ n ∈ N B n ≠ ∅ . {\displaystyle B_{1}\supseteq B_{2}\supseteq \cdots \Rightarrow \bigcap _{n\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2648_chunk_0 | Field (algebra) | In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2649_chunk_0 | Field (algebra) | Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2650_chunk_0 | Field (algebra) | Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are, in general, algebraically unsolvable. Fields serve as foundational notions in several mathematical domains. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2651_chunk_0 | Field (algebra) | This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the stand... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2652_chunk_0 | Field of sets | In mathematics, a field of sets is a mathematical structure consisting of a pair ( X , F ) {\displaystyle (X,{\mathcal {F}})} consisting of a set X {\displaystyle X} and a family F {\displaystyle {\mathcal {F}}} of subsets of X {\displaystyle X} called an algebra over X {\displaystyle X} that contains the empty set as ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2653_chunk_0 | Symplectic filling | In mathematics, a filling of a manifold X is a cobordism W between X and the empty set. More to the point, the n-dimensional topological manifold X is the boundary of an (n + 1)-dimensional manifold W. Perhaps the most active area of current research is when n = 3, where one may consider certain types of fillings. Ther... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2654_chunk_0 | Filter quantifier | In mathematics, a filter on a set X {\displaystyle X} informally gives a notion of which subsets A ⊆ X {\displaystyle A\subseteq X} are "large". Filter quantifiers are a type of logical quantifier which, informally, say whether or not a statement is true for "most" elements of X . {\displaystyle X.} Such quantifiers ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2655_chunk_0 | Elementary filter | In mathematics, a filter on a set X {\displaystyle X} is a family B {\displaystyle {\mathcal {B}}} of subsets such that: X ∈ B {\displaystyle X\in {\mathcal {B}}} and ∅ ∉ B {\displaystyle \emptyset \notin {\mathcal {B}}} if A ∈ B {\displaystyle A\in {\mathcal {B}}} and B ∈ B {\displaystyle B\in {\mathcal {B}}} , then A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2656_chunk_0 | Filtered ring | In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory. A filtered algebra over the field k {\displaystyle k} is an algebra ( A , ⋅ ) {\displaystyle (A,\cdot )} over k {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2657_chunk_0 | Filtration (mathematics) | In mathematics, a filtration F {\displaystyle {\mathcal {F}}} is an indexed family ( S i ) i ∈ I {\displaystyle (S_{i})_{i\in I}} of subobjects of a given algebraic structure S {\displaystyle S} , with the index i {\displaystyle i} running over some totally ordered index set I {\displaystyle I} , subject to the conditi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2658_chunk_0 | Filtration (mathematics) | Again, it depends on the context how exactly the word "filtration" is to be understood. Descending filtrations are not to be confused with the dual notion of cofiltrations (which consist of quotient objects rather than subobjects). Filtrations are widely used in abstract algebra, homological algebra (where they are rel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2659_chunk_0 | Theory of relations | This is because there is only one 0-tuple, the empty tuple (). They are sometimes useful for constructing the base case of an induction argument. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2660_chunk_0 | Theory of relations | Unary relations can be viewed as a collection of members (such as the collection of Nobel laureates) having some property (such as that of having been awarded the Nobel prize). Binary relations are the most commonly studied form of finitary relations. When X1 = X2 it is called a homogeneous relation, for example: Equal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2661_chunk_0 | Galois fields | The order of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order pk, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including numbe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2662_chunk_0 | Combinatorial Riemann mapping theorem | In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over, they have slight variations in ea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2663_chunk_0 | Finite von Neumann algebra | In mathematics, a finite von Neumann algebra is a von Neumann algebra in which every isometry is a unitary. In other words, for an operator V in a finite von Neumann algebra if V ∗ V = I {\displaystyle V^{*}V=I} , then V V ∗ = I {\displaystyle VV^{*}=I} . In terms of the comparison theory of projections, the identity o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2664_chunk_0 | Finitely-generated algebra | In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra A over a field K where there exists a finite set of elements a1,...,an of A such that every element of A can be expressed as a polynomial in a1,...,an, with coefficients in K. Equivalently, there ex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2665_chunk_0 | Module of finite type | In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2666_chunk_0 | First-order partial differential equation | In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables. The equation takes the form F ( x 1 , … , x n , u , u x 1 , … u x n ) = 0. {\displaystyle F(x_{1},\ldots ,x_{n},u,u_{x_{1}},\ldots u_{x_{n}})=0.\,} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2667_chunk_0 | Stable fixed point | In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically for functions, a fixed point is an element that is mapped to itself by the function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2668_chunk_0 | Fixed point theory | In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2669_chunk_0 | Flow (mathematics) | Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2670_chunk_0 | Flow (mathematics) | Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and Anosov flows. Flows may also be defined for systems of random variables and stochastic processes, and occur in the study of ergodic dynamical systems. The most celebrated of these is perhaps t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2671_chunk_0 | Polynomial SOS | In mathematics, a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only if there exist forms g 1 ( x ) , … , g k ( x ) {\displaystyle g_{1}(x),\ldots ,g_{k}(x)} of degree m such that Every form that is SOS is also a positive polynomial, an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2672_chunk_0 | Formal distribution | In mathematics, a formal distribution is an infinite sum of powers of a formal variable, usually denoted z {\displaystyle z} in the theory of formal distributions. The coefficients of these infinite sums can be many different mathematical structures, such as vector spaces or rings, but in applications most often take v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2673_chunk_0 | Formal distribution | Rather, they are interpreted as distributions, that is, linear functionals on an appropriate space of test functions. They are closely related to formal Laurent series, but are not required to have finitely many negative powers. In particular, this means even if the coefficients are ring-valued, it is not necessarily p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2674_chunk_0 | Formal group law | In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by S. Bochner (1946). The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermedi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2675_chunk_0 | Formal power series ring | In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2676_chunk_0 | Formal power series ring | In a formal power series, the x n {\displaystyle x^{n}} are used only as position-holders for the coefficients, so that the coefficient of x 5 {\displaystyle x^{5}} is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2677_chunk_0 | Formal power series ring | More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary ring. Rings of formal power series are complete local rings, and this allows using calculus-like methods in the purely algebraic framework of algebraic geometry and commutativ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2678_chunk_0 | Formal sum | In mathematics, a formal sum, formal series, or formal linear combination may be: In group theory, an element of a free abelian group, a sum of finitely many elements from a given basis set multiplied by integer coefficients. In linear algebra, an element of a vector space, a sum of finitely many elements from a given ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2679_chunk_0 | Formula | In mathematics, a formula generally refers to an equation relating one mathematical expression to another, with the most important ones being mathematical theorems. For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion. H... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2680_chunk_0 | Formula | Having obtained this result, the volume of any sphere can be computed as long as its radius is known. Here, notice that the volume V and the radius r are expressed as single letters instead of words or phrases. This convention, while less important in a relatively simple formula, means that mathematicians can more quic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2681_chunk_0 | Formula | Mathematical formulas are often algebraic, analytical or in closed form.In a general context, formulas are often a manifestation of mathematical model to real world phenomena, and as such can be used to provide solution (or approximated solution) to real world problems, with some being more general than others. For exa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2682_chunk_0 | Formula | In all cases, however, formulas form the basis for calculations. Expressions are distinct from formulas in that they cannot contain an equals sign (=). Expressions can be likened to phrases the same way formulas can be likened to grammatical sentences. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2683_chunk_0 | Fractal dimension | In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. It is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2684_chunk_0 | Fractal geometry | In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2685_chunk_0 | Fractal geometry | One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radiu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2686_chunk_0 | Fractal geometry | However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension. This power is called the fractal dimension of the geometric object, to distinguish it from the conventional ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2687_chunk_0 | Fractal geometry | Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the wo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2688_chunk_0 | Fractal geometry | That's fractals." More formally, in 1982 Mandelbrot defined fractal as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2689_chunk_0 | Fractal geometry | Still later, Mandelbrot proposed "to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants".The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many examples hav... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2690_chunk_0 | Unitary frame bundle | In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex. The general linear group acts naturally on F(E) via a change of basis, giving the frame bundle the structure of a principal GL(k, R)-bun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2691_chunk_0 | Free Boolean algebra | In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that: Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean operations, and The generators are as independent as possible, in the sense that t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2692_chunk_0 | Free Lie algebra | In mathematics, a free Lie algebra over a field K is a Lie algebra generated by a set X, without any imposed relations other than the defining relations of alternating K-bilinearity and the Jacobi identity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2693_chunk_0 | Free abelian group | In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2694_chunk_0 | Free abelian group | Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free Z {\displaystyle \mathbb {Z} } -modules, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2695_chunk_0 | Free abelian group | In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors. The elements of a free abelian group with basis B {\displaystyle B} may be described in several equivalent ways. These include formal sums over B {\displaystyle B} , which are expressi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2696_chunk_0 | Free abelian group | Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of B {\displaystyle B} , with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a functio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2697_chunk_0 | Free boundary problem | In mathematics, a free boundary problem (FB problem) is a partial differential equation to be solved for both an unknown function u {\displaystyle u} and an unknown domain Ω {\displaystyle \Omega } . The segment Γ {\displaystyle \Gamma } of the boundary of Ω {\displaystyle \Omega } which is not known at the outset of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2698_chunk_0 | Free boundary problem | An interesting aspect of such a criticality is the so-called sandpile dynamic (or Internal DLA). The most classical example is the melting of ice: Given a block of ice, one can solve the heat equation given appropriate initial and boundary conditions to determine its temperature. But, if in any region the temperature i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_2699_chunk_0 | Free module | In mathematics, a free module is a module that has a basis, that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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