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relation (mathematics) | For example, "is less than" is irreflexive, asymmetric, and transitive, but neither reflexive nor symmetric. "is sister of" is transitive, but neither reflexive (e.g. Pierre Curie is not a sister of himself), nor symmetric, nor asymmetric; while being irreflexive or not may be a matter of definition (is every woman a s... | wikipedia |
relation (mathematics) | Mathematical theorems are known about combinations of relation properties, such as "A transitive relation is irreflexive if, and only if, it is asymmetric". Of particular importance are relations that satisfy certain combinations of properties. A partial order is a relation that is reflexive, antisymmetric, and transit... | wikipedia |
biquaternion algebra | In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field. The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dual quaternions do not form biquaternion algebras in this sense. | wikipedia |
bitopological space | In mathematics, a bitopological space is a set endowed with two topologies. Typically, if the set is X {\displaystyle X} and the topologies are σ {\displaystyle \sigma } and τ {\displaystyle \tau } then the bitopological space is referred to as ( X , σ , τ ) {\displaystyle (X,\sigma ,\tau )} . The notion was introduced... | wikipedia |
boxcar function | In mathematics, a boxcar function is any function which is zero over the entire real line except for a single interval where it is equal to a constant, A. The function is named after its graph's resemblance to a boxcar, a type of railroad car. The boxcar function can be expressed in terms of the uniform distribution as... | wikipedia |
bracket algebra | In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra with a symbolic representation of projective invariants. Given that L is a proper signed alphabet and Super is the supersymmetric algebra, the bracket algebra Bracket of dimension n over the field K is the quoti... | wikipedia |
braided hopf algebra | In mathematics, a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf algebra H, particularly the Nichols algebra of a braided vector space in that category. The notion should not be confused with quasitriangula... | wikipedia |
building (mathematics) | In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Buildings were initially introduced by Jacques Tits as a means to understand... | wikipedia |
bundle (mathematics) | In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π: E→ B... | wikipedia |
character (topology) | In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. | wikipedia |
category (mathematics) | 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 ex... | wikipedia |
category (mathematics) | Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundat... | wikipedia |
category (mathematics) | In addition to formalizing mathematics, category theory is also used to formalize many other systems in computer science, such as the semantics of programming languages. Two categories are the same if they have the same collection of objects, the same collection of arrows, and the same associative method of composing a... | wikipedia |
category (mathematics) | Well-known categories are denoted by a short capitalized word or abbreviation in bold or italics: examples include Set, the category of sets and set functions; Ring, the category of rings and ring homomorphisms; and Top, the category of topological spaces and continuous maps. All of the preceding categories have the id... | wikipedia |
category (mathematics) | Other references are given in the References below. The basic definitions in this article are contained within the first few chapters of any of these books. Any monoid can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and so can an... | wikipedia |
character (mathematics) | In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified. | wikipedia |
characterization theorem | In mathematics, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it. To say that "Property P characterizes object X" is to say that not only does X have property P, but that X is the only thing that has property P (i.e., P is a de... | wikipedia |
characterization theorem | Common mathematical expressions for a characterization of X in terms of P include "P is necessary and sufficient for X", and "X holds if and only if P". It is also common to find statements such as "Property Q characterizes Y up to isomorphism". The first type of statement says in different words that the extension of ... | wikipedia |
characterization theorem | A reference on mathematical terminology notes that characteristic originates from the Greek term kharax, "a pointed stake":From Greek kharax came kharakhter, an instrument used to mark or engrave an object. Once an object was marked, it became distinctive, so the character of something came to mean its distinctive natu... | wikipedia |
characterization theorem | Characterization is not unique to mathematics, but since the science is abstract, much of the activity can be described as "characterization". For instance, in Mathematical Reviews, as of 2018, more than 24,000 articles contain the word in the article title, and 93,600 somewhere in the review. In an arbitrary context o... | wikipedia |
chiral algebra | In mathematics, a chiral algebra is an algebraic structure introduced by Beilinson & Drinfeld (2004) as a rigorous version of the rather vague concept of a chiral algebra in physics. In Chiral Algebras, Beilinson and Drinfeld introduced the notion of chiral algebra, which based on the pseudo-tensor category of D-module... | wikipedia |
chord diagram (mathematics) | In mathematics, a chord diagram consists of a cyclic order on a set of objects, together with a one-to-one pairing (perfect matching) of those objects. Chord diagrams are conventionally visualized by arranging the objects in their order around a circle, and drawing the pairs of the matching as chords of the circle. The... | wikipedia |
chord diagram (mathematics) | {\displaystyle (2n-1)!!} . There is a Catalan number of chord diagrams on a given ordered set in which no two chords cross each other. | wikipedia |
chord diagram (mathematics) | The crossing pattern of chords in a chord diagram may be described by a circle graph, the intersection graph of the chords: it has a vertex for each chord and an edge for each two chords that cross.In knot theory, a chord diagram can be used to describe the sequence of crossings along the planar projection of a knot, w... | wikipedia |
cohomological invariant | In mathematics, a cohomological invariant of an algebraic group G over a field is an invariant of forms of G taking values in a Galois cohomology group. | wikipedia |
collapsing algebra | In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used to generate collapsing algebras were introduced by Azriel Lévy in 1963.The collapsing algebra of λω is a complete Boolean algebra with at least λ elements but generat... | wikipedia |
compact quantum group | In mathematics, a compact quantum group is an abstract structure on a unital separable C*-algebra axiomatized from those that exist on the commutative C*-algebra of "continuous complex-valued functions" on a compact quantum group. The basic motivation for this theory comes from the following analogy. The space of compl... | wikipedia |
compact quantum group | On the other hand, by the Gelfand Theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism. S. L. Woronowicz introduced the important concept o... | wikipedia |
complete boolean algebra | In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra A has an essentially unique completion, which is a complete Boolea... | wikipedia |
complex lie algebra | In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate g ¯ {\displaystyle {\overline {\mathfrak {g}}}} is a complex Lie algebra with the same underlying real vector space but with i = − 1 {\displaystyle i={\sqrt {-1}... | wikipedia |
concave function | In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex. | wikipedia |
configuration space (mathematics) | In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space. In mathematics, they are used to describe assignments of a collection of points to positio... | wikipedia |
confluent hypergeometric series | In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The term confluent refers to the merging of singular points of ... | wikipedia |
confluent hypergeometric series | There is a different and unrelated Kummer's function bearing the same name. Tricomi's (confluent hypergeometric) function U(a, b, z) introduced by Francesco Tricomi (1947), sometimes denoted by Ψ(a; b; z), is another solution to Kummer's equation. | wikipedia |
confluent hypergeometric series | This is also known as the confluent hypergeometric function of the second kind. Whittaker functions (for Edmund Taylor Whittaker) are solutions to Whittaker's equation. Coulomb wave functions are solutions to the Coulomb wave equation.The Kummer functions, Whittaker functions, and Coulomb wave functions are essentially... | wikipedia |
mathematical conjecture | In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of... | wikipedia |
connector (mathematics) | In mathematics, a connector is a map which can be defined for a linear connection and used to define the covariant derivative on a vector bundle from the linear connection. | wikipedia |
constant map | In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function y(x) = 4 is a constant function because the value of y(x) is 4 regardless of the input value x (see image). | wikipedia |
non-binding constraint | In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set. | wikipedia |
continuous (topology) | In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if ar... | wikipedia |
continuous (topology) | A discontinuous function is a function that is not continuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. | wikipedia |
continuous (topology) | Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definiti... | wikipedia |
continuous (topology) | A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity. As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M(t) denoting the amount of... | wikipedia |
continuum structure function | In mathematics, a continuum structure function (CSF) is defined by Laurence Baxter as a nondecreasing mapping from the unit hypercube to the unit interval. It is used by Baxter to help in the Mathematical modelling of the level of performance of a system in terms of the performance levels of its components. | wikipedia |
degeneracy (math) | In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.The definitions of many classes of composite or structured objects often impli... | wikipedia |
degeneracy (math) | Equivalently, it becomes a "line segment".Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its di... | wikipedia |
degeneracy (math) | As another example, the solution set of a system of equations that depends on parameters generally has a fixed cardinality and dimension, but cardinality and/or dimension may be different for some exceptional values, called degenerate cases. In such a degenerate case, the solution set is said to be degenerate. For some... | wikipedia |
degeneracy (math) | In particular, the class of objects may often be defined or characterized by systems of equations. In most scenarios, a given class of objects may be defined by several different systems of equations, and these different systems of equations may lead to different degenerate cases, while characterizing the same non-dege... | wikipedia |
degeneracy (math) | A degenerate case thus has special features which makes it non-generic, or a special case. However, not all non-generic or special cases are degenerate. | wikipedia |
degeneracy (math) | For example, right triangles, isosceles triangles and equilateral triangles are non-generic and non-degenerate. In fact, degenerate cases often correspond to singularities, either in the object or in some configuration space. For example, a conic section is degenerate if and only if it has singular points (e.g., point,... | wikipedia |
dendrite (mathematics) | In mathematics, a dendrite is a certain type of topological space that may be characterized either as a locally connected dendroid or equivalently as a locally connected continuum that contains no simple closed curves. | wikipedia |
homogeneous derivation | In mathematics, a derivation is a function on an algebra which 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 ) = a D ( b ) + D ( a ) b . {\displaystyle D(ab)=aD(b)+D(a)b.... | wikipedia |
homogeneous derivation | The collection of all K-derivations of A to itself is denoted by DerK(A). The collection of K-derivations of A into an A-module M is denoted by DerK(A, M). Derivations occur in many different contexts in diverse areas of mathematics. | wikipedia |
homogeneous derivation | The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on Rn. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor... | wikipedia |
homogeneous derivation | The Pincherle derivative is an example of a derivation in abstract algebra. If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K. That is, = G + F {\displaystyle =G+F} where {\displaystyle } is ... | wikipedia |
continuously differentiable | In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well... | wikipedia |
continuously differentiable | In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). f is said to be differentiable on U if it is differentiable at every point of U. f is said to be continuously differentiable if its derivative is also a continuous function over the domain of the function f {\displaystyle f} . Gene... | wikipedia |
differential algebraic group | In mathematics, a differential algebraic group is a differential algebraic variety with a compatible group structure. Differential algebraic groups were introduced by Cassidy (1972). | wikipedia |
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... | wikipedia |
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. | wikipedia |
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... | wikipedia |
diversity (mathematics) | Bryant and Tupper observe that these axioms imply monotonicity; that is, if A ⊆ B {\displaystyle A\subseteq B} , then δ ( A ) ≤ δ ( B ) {\displaystyle \delta (A)\leq \delta (B)} . They state that the term "diversity" comes from the appearance of a special case of their definition in work on phylogenetic and ecological ... | wikipedia |
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... | wikipedia |
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... | wikipedia |
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... | wikipedia |
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... | wikipedia |
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... | wikipedia |
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... | wikipedia |
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... | wikipedia |
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... | wikipedia |
scientific fact | In mathematics, a fact is a statement (called a theorem) that can be proven by logical argument from certain axioms and definitions. | wikipedia |
zigzag poset | In mathematics, a fence, also called a zigzag poset, is a partially ordered set (poset) in which the order relations form a path with alternating orientations: a < b > c < d > e < f > h < i ⋯ {\displaystyle a c e h b < c > d < e > f < h > i ⋯ {\displaystyle a>b d f i\cdots } A fence may be finite, or it may be formed b... | wikipedia |
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... | wikipedia |
zigzag poset | For instance, Q(2,9) has the elements and relations a > b > c < d > e > f < g > h > i . {\displaystyle a>b>c e>f h>i.} In this notation, a fence is a partially ordered set of the form Q(1,n). | wikipedia |
algebraically closed field | In mathematics, a field F is algebraically closed if every non-constant polynomial in F (the univariate polynomial ring with coefficients in F) has a root in F. | wikipedia |
quasi-algebraically closed | In mathematics, a field F is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a ... | wikipedia |
pseudo algebraically closed | In mathematics, a field K {\displaystyle K} is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967. | wikipedia |
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... | wikipedia |
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. | wikipedia |
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. | wikipedia |
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... | wikipedia |
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 ... | wikipedia |
chirality | In mathematics, a figure is chiral (and said to have chirality) if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from anticlockwise. See for a full mathematical definition. A chiral object and its mirror i... | wikipedia |
chirality | The word enantiomorph stems from the Greek ἐναντίος (enantios) 'opposite' + μορφή (morphe) 'form'. A non-chiral figure is called achiral or amphichiral. The helix (and by extension a spun string, a screw, a propeller, etc.) and Möbius strip are chiral two-dimensional objects in three-dimensional ambient space. | wikipedia |
chirality | The J, L, S and Z-shaped tetrominoes of the popular video game Tetris also exhibit chirality, but only in a two-dimensional space. Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves, glasses (sometimes), and shoes. A similar notion of chirality is considered in knot theory, a... | wikipedia |
filter (mathematics) | In mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal. Special cases of filters include ultrafilters, ... | wikipedia |
filter (mathematics) | Filters on sets were introduced by Henri Cartan in 1937. Nicolas Bourbaki, in their book Topologie Générale, popularized filters as an alternative to E. H. Moore and Herman L. Smith's 1922 notion of a net; order filters generalize this notion from the specific case of a power set under inclusion to arbitrary partially ... | wikipedia |
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... | wikipedia |
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... | wikipedia |
filtration (mathematics) | This article does not impose this requirement. There is also the notion of a descending filtration, which is required to satisfy S i ⊇ S j {\displaystyle S_{i}\supseteq S_{j}} in lieu of S i ⊆ S j {\displaystyle S_{i}\subseteq S_{j}} (and, occasionally, ⋂ i ∈ I S i = 0 {\displaystyle \bigcap _{i\in I}S_{i}=0} instead o... | wikipedia |
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... | wikipedia |
finite topological space | In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conject... | wikipedia |
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... | wikipedia |
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... | wikipedia |
flow (mathematics) | In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. | wikipedia |
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. | wikipedia |
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... | wikipedia |
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... | wikipedia |
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... | wikipedia |
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... | wikipedia |
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