text stringlengths 559 401k | source stringlengths 13 121 |
|---|---|
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables.
For example,
{
3
x
+
... | Wikipedia/Systems_of_linear_equations |
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of pr... | Wikipedia/Projective_transformation |
In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to p... | Wikipedia/Direct_method_in_the_calculus_of_variations |
François Viète (French: [fʁɑ̃swa vjɛt]; 1540 – 23 February 1603), known in Latin as Franciscus Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy counci... | Wikipedia/New_Algebra |
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not... | Wikipedia/Rhetorical_algebra |
George Peacock FRS (9 April 1791 – 8 November 1858) was an English mathematician and Anglican cleric. He founded what has been called the British algebra of logic.
== Early life ==
Peacock was born on 9 April 1791 at Thornton Hall, Denton, near Darlington, County Durham. His father, Thomas Peacock, was a priest of th... | Wikipedia/Arithmetical_algebra |
George Peacock FRS (9 April 1791 – 8 November 1858) was an English mathematician and Anglican cleric. He founded what has been called the British algebra of logic.
== Early life ==
Peacock was born on 9 April 1791 at Thornton Hall, Denton, near Darlington, County Durham. His father, Thomas Peacock, was a priest of th... | Wikipedia/Symbolical_algebra |
Moderne Algebra is a two-volume German textbook on graduate abstract algebra by Bartel Leendert van der Waerden (1930, 1931), originally based on lectures given by Emil Artin in 1926 and by Emmy Noether (1929) from 1924 to 1928. The English translation of 1949–1950 had the title Modern algebra, though a later, extensiv... | Wikipedia/Moderne_Algebra |
In algebra, linear equations and systems of linear equations over a field are widely studied. "Over a field" means that the coefficients of the equations and the solutions that one is looking for belong to a given field, commonly the real or the complex numbers. This article is devoted to the same problems where "field... | Wikipedia/Linear_equation_over_a_ring |
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Co... | Wikipedia/Numerical_solution_of_linear_systems |
In mathematics, the solution set of a system of equations or inequality is the set of all its solutions, that is the values that satisfy all equations and inequalities. Also, the solution set or the truth set of a statement or a predicate is the set of all values that satisfy it.
If there is no solution, the solution ... | Wikipedia/Solution_set |
In mathematics, the Coates graph or Coates flow graph, named after C.L. Coates, is a graph associated with the Coates' method for the solution of a system of linear equations.
The Coates graph Gc(A) associated with an n × n matrix A is an n-node, weighted, labeled, directed graph. The nodes, labeled 1 through n, are e... | Wikipedia/Coates_graph |
In mathematics, the linear span (also called the linear hull or just span) of a set
S
{\displaystyle S}
of elements of a vector space
V
{\displaystyle V}
is the smallest linear subspace of
V
{... | Wikipedia/Span_(linear_algebra) |
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.
More generally, these terms may apply to an inequation or inequality; the right-hand side is everything on the r... | Wikipedia/Sides_of_an_equation |
In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it con... | Wikipedia/Jacobi_method |
A substitution is a syntactic transformation on formal expressions.
To apply a substitution to an expression means to consistently replace its variable, or placeholder, symbols with other expressions.
The resulting expression is called a substitution instance, or instance for short, of the original expression.
== Pr... | Wikipedia/Substitution_(algebra) |
A differential equation can be homogeneous in either of two respects.
A first order differential equation is said to be homogeneous if it may be written
f
(
x
,
y
)
d
y
=
g
(
x
,
y
... | Wikipedia/Homogeneous_differential_equation |
The Rybicki–Press algorithm is a fast algorithm for inverting a matrix whose entries are given by
A
(
i
,
j
)
=
exp
(
−
a
|
t
i
... | Wikipedia/Rybicki_Press_algorithm |
The Harrow–Hassidim–Lloyd (HHL) algorithm is a quantum algorithm for numerically solving a system of linear equations, designed by Aram Harrow, Avinatan Hassidim, and Seth Lloyd. The algorithm estimates the result of a scalar measurement on the solution vector to a given linear system of equations.
The algorithm is one... | Wikipedia/Quantum_algorithm_for_linear_systems_of_equations |
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,... | Wikipedia/Dynamical_systems |
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in th... | Wikipedia/Springer_Science_&_Business_Media |
In computer science, function composition is an act or mechanism to combine simple functions to build more complicated ones. Like the usual composition of functions in mathematics, the result of each function is passed as the argument of the next, and the result of the last one is the result of the whole.
Programmers f... | Wikipedia/Function_composition_(computer_science) |
In universal algebra, a clone is a set C of finitary operations on a set A such that
C contains all the projections πkn: An → A, defined by πkn(x1, …, xn) = xk,
C is closed under (finitary multiple) composition (or "superposition"): if f, g1, …, gm are members of C such that f is m-ary, and gj is n-ary for all j, then... | Wikipedia/Clone_(algebra) |
In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations (functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a perm... | Wikipedia/Transformation_monoid |
In engineering, functional decomposition is the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts.
This process of decomposition may be undertaken to gain insight into the identity of the constituen... | Wikipedia/Functional_decomposition |
In set theory, a projection is one of two closely related types of functions or operations, namely:
A set-theoretic operation typified by the
j
{\displaystyle j}
th projection map, written
p
r
o
... | Wikipedia/Projection_function |
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually ... | Wikipedia/Infinite_compositions_of_analytic_functions |
In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X)... | Wikipedia/Transformation_group |
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x1 ≠ x2 implies f(x1) ≠ f(x2) (equivalently by contraposition, f(x1) = f(x2) implies x1 = x2). In other words, every element ... | Wikipedia/One-to-one_function |
In mathematics, a transformation, transform, or self-map is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X.
Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and ... | Wikipedia/Transformation_(function) |
In mathematics, a surjective function (also known as surjection, or onto function ) is a function f such that, for every element y of the function's codomain, there exists at least one element x in the function's domain such that f(x) = y. In other words, for a function f : X → Y, the codomain Y is the image of the fun... | Wikipedia/Onto_function |
Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function h, find the function Ψ such that
Schröder's equation is an eigenvalue equation for the composition operator Ch that sends a function f to f(h(.)).
If a is a fixed point of h, meaning h(a) = a, t... | Wikipedia/Schröder's_equation |
In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again into the ... | Wikipedia/Iterated_function |
In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations (functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a perm... | Wikipedia/Full_transformation_semigroup |
Mathematics, Form and Function, a book published in 1986 by Springer-Verlag, is a survey of the whole of mathematics, including its origins and deep structure, by the American mathematician Saunders Mac Lane.
== Mathematics and human activities ==
Throughout his book, and especially in chapter I.11, Mac Lane informal... | Wikipedia/Mathematics,_Form_and_Function |
In mathematics, if
A
{\displaystyle A}
is a subset of
B
,
{\displaystyle B,}
then the inclusion map is the function
ι
{\displaystyle \iota }
that sends each element
... | Wikipedia/Inclusion_function |
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates seve... | Wikipedia/Functional_equation |
In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, th... | Wikipedia/Multivariate_function |
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.
The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it opera... | Wikipedia/Simplex_algorithm |
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple e... | Wikipedia/Object_(category_theory) |
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two mo... | Wikipedia/Dual_(category_theory) |
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which... | Wikipedia/Coproduct_(category_theory) |
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability... | Wikipedia/Computability_theory_(computer_science) |
In mathematics, specifically in category theory, an
F
{\displaystyle F}
-coalgebra is a structure defined according to a functor
F
{\displaystyle F}
, with specific properties as defined below. For both algebras and coalgebras, a func... | Wikipedia/F-coalgebra |
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total co... | Wikipedia/Ackermann_function |
In computer programming, especially functional programming and type theory, an algebraic data type (ADT) is a kind of composite data type, i.e., a data type formed by combining other types.
Two common classes of algebraic types are product types (i.e., tuples, and records) and sum types (i.e., tagged or disjoint unions... | Wikipedia/Algebraic_data_type |
In category theory, a branch of mathematics, a monad is a triple
(
T
,
η
,
μ
)
{\displaystyle (T,\eta ,\mu )}
consisting of a functor T from a category to itself and two natural transformations
η
... | Wikipedia/Monad_(category_theory) |
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family ... | Wikipedia/Product_(category_theory) |
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms i... | Wikipedia/Algebras_over_a_field |
In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework for induction and recursion.
== Examples ==
=== Functor 1 + (−) ===
Consider the endofunctor 1 + (−), i.e. F : Set → Set sending X to 1 + X, where 1 is a one... | Wikipedia/Initial_algebra |
In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon ... | Wikipedia/Nil_algebra |
In mathematical genetics, a genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. Some variations of these algebras are called train algebras, special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras (also called weigh... | Wikipedia/Genetic_algebra |
In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon ... | Wikipedia/Nilpotent_algebra |
In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there are three such Jordan algebras up to isomorphism. One of them, which was... | Wikipedia/Albert_algebra |
In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras a... | Wikipedia/Derivation_algebra |
In mathematics, a derivation is a function on an algebra that generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D : A → A that satisfies Leibniz's law:
D
(
a
b
)
... | Wikipedia/Derivation_(abstract_algebra) |
In mathematics, the Cayley–Dickson construction, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. It is named after Arthur Cayley and Leonard Eugene Dickson. The algebras ... | Wikipedia/Cayley–Dickson_algebra |
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.
== History ==
In the nineteenth century, number systems ca... | Wikipedia/Hypercomplex_algebra |
In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most important class of graded quadratic algebras is Koszul algebras.
== Defi... | Wikipedia/Quadratic_algebra |
In mathematics, specifically category theory, a family of generators (or family of separators) of a category
C
{\displaystyle {\mathcal {C}}}
is a collection
G
... | Wikipedia/Generator_(category_theory) |
In mathematics, an even function is a real function such that
f
(
−
x
)
=
f
(
x
)
{\displaystyle f(-x)=f(x)}
for every
x
{\displaystyle x}
in its domain. Simil... | Wikipedia/Even_and_odd_functions |
In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations (functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a perm... | Wikipedia/Full_transformation_monoid |
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be a... | Wikipedia/Continuous_function_(topology) |
In computer programming, an anonymous function (function literal, expression or block) is a function definition that is not bound to an identifier. Anonymous functions are often arguments being passed to higher-order functions or used for constructing the result of a higher-order function that needs to return a functio... | Wikipedia/Function_constant |
In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts.
== History and motivation ==
Mereotopology begins in philoso... | Wikipedia/Mereotopology |
In general topology and related areas of mathematics, the final topology (or coinduced, weak, colimit, or inductive topology) on a set
X
,
{\displaystyle X,}
with respect to a family of functions from topological spaces into
X
,
... | Wikipedia/Final_topology |
In mathematics, a topological algebra
A
{\displaystyle A}
is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
== Definition ==
A topological algebra
A
... | Wikipedia/Topological_algebra |
The cocountable topology, also known as the countable complement topology, is a topology that can be defined on any infinite set
X
{\displaystyle X}
. In this topology, a set is open if its complement in
X
{\displaystyle X}
is either... | Wikipedia/Cocountable_topology |
In mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If X is a totally ordered set, the order topology on X is generated by the subbase of "open rays"
... | Wikipedia/Order_topology |
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
== Definition ==
A topology on a set may be defined as the collection of subsets which are considered to be "open". (An alter... | Wikipedia/Coarsest_topology |
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product... | Wikipedia/Product_topology |
In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. That is, a topological space X is a Moore space if the following conditions hold:
Any two distinct points can be separated by neighbourhoods, and any closed set and any point in its complement can be separated... | Wikipedia/Moore_space_(topology) |
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in... | Wikipedia/Neighborhood_(topology) |
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be a... | Wikipedia/Continuous_(topology) |
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open i... | Wikipedia/Discrete_topology |
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on
R
{\displaystyle \mathbb {R} }
, the set of real numbers; it is different from the standard topology on
R
... | Wikipedia/Lower_limit_topology |
In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.
== Scope ==
The central object of study in topological dynamics is a topological dynamical system, i.e. a topological... | Wikipedia/Topological_dynamics |
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a numbe... | Wikipedia/Dimension_theory |
In mathematics, pointless topology, also called point-free topology (or pointfree topology) or topology without points and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this approach it becomes possible to construct topologi... | Wikipedia/Pointless_topology |
In general topology and related areas of mathematics, the initial topology (or induced topology or strong topology or limit topology or projective topology) on a set
X
,
{\displaystyle X,}
with respect to a family of functions on
X
,
... | Wikipedia/Initial_topology |
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonic... | Wikipedia/Quotient_space_(topology) |
In mathematics, a cofinite subset of a set
X
{\displaystyle X}
is a subset
A
{\displaystyle A}
whose complement in
X
{\displaystyle X}
is a finite set. In other words,
... | Wikipedia/Cofinite_topology |
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be di... | Wikipedia/Trivial_topology |
In topology, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought ... | Wikipedia/Closure_(topology) |
In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua.
== Definitions ==
A continuum that contains more than o... | Wikipedia/Continuum_theory |
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (an... | Wikipedia/Separated_sets |
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be di... | Wikipedia/Indiscrete_topology |
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonic... | Wikipedia/Quotient_topology |
In mathematics, a base (or basis; pl.: bases) for the topology τ of a topological space (X, τ) is a family
B
{\displaystyle {\mathcal {B}}}
of open subsets of X such that every open set of the topology is equal to the union of some ... | Wikipedia/Base_(topology) |
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of water and other liquids in motion... | Wikipedia/Fluid_dynamics |
Complex dynamics, or holomorphic dynamics, is the study of dynamical systems obtained by iterating a complex analytic mapping. This article focuses on the case of algebraic dynamics, where a polynomial or rational function is iterated. In geometric terms, that amounts to iterating a mapping from some algebraic variety ... | Wikipedia/Complex_dynamics |
In the mathematical field of graph theory, a path graph (or linear graph) is a graph whose vertices can be listed in the order v1, v2, ..., vn such that the edges are {vi, vi+1} where i = 1, 2, ..., n − 1. Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices of degree 1)... | Wikipedia/Linear_graph |
In topology and related areas of mathematics, a subspace of a topological space (X, 𝜏) is a subset S of X which is equipped with a topology induced from that of 𝜏 called the subspace topology (or the relative topology, or the induced topology, or the trace topology).
== Definition ==
Given a topological space
... | Wikipedia/Subspace_topology |
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.
In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of topological properties. It is often useful in the s... | Wikipedia/Counterexamples_in_Topology |
In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain sense "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither Lindelöf nor separable). Therefore, it serves as an important counterexam... | Wikipedia/Long_line_(topology) |
In set theory, a projection is one of two closely related types of functions or operations, namely:
A set-theoretic operation typified by the
j
{\displaystyle j}
th projection map, written
p
r
o
... | Wikipedia/Projection_(set_theory) |
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M.
Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, becau... | Wikipedia/Completeness_(topology) |
In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational nu... | Wikipedia/Dense_(topology) |
In topology and related areas of mathematics, a subspace of a topological space (X, 𝜏) is a subset S of X which is equipped with a topology induced from that of 𝜏 called the subspace topology (or the relative topology, or the induced topology, or the trace topology).
== Definition ==
Given a topological space
... | Wikipedia/Subspace_(topology) |
In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that the topology in question arises naturally or canonically (see mathematica... | Wikipedia/Natural_topology |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.