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In mathematics, a doctrine is simply a 2-category which is heuristically regarded as a system of theories. For example, algebraic theories, as invented by William Lawvere, is an example of a doctrine, as are multi-sorted theories, operads, categories, and toposes. The objects of the 2-category are called theories, the ... | https://en.wikipedia.org/wiki/Strict_2-category |
It is this vocabulary that makes the theory of doctrines worth while. For example, the 2-category Cat of categories, functors, and natural transformations is a doctrine. One sees immediately that all presheaf categories are categories of models. | https://en.wikipedia.org/wiki/Strict_2-category |
As another example, one may take the subcategory of Cat consisting only of categories with finite products as objects and product-preserving functors as 1-morphisms. This is the doctrine of multi-sorted algebraic theories. If one only wanted 1-sorted algebraic theories, one would restrict the objects to only those cate... | https://en.wikipedia.org/wiki/Strict_2-category |
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://en.wikipedia.org/wiki/Domino_(mathematics) |
In mathematics, a double Mersenne number is a Mersenne number of the form M M p = 2 2 p − 1 − 1 {\displaystyle M_{M_{p}}=2^{2^{p}-1}-1} where p is prime. | https://en.wikipedia.org/wiki/Double_Mersenne_number |
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://en.wikipedia.org/wiki/Double_affine_Hecke_algebra |
In mathematics, a double affine braid group is a group containing the braid group of an affine Weyl group. Their group rings have quotients called double affine Hecke algebras in the same way that the group rings of affine braid groups have quotients that are affine Hecke algebras. For affine An groups, the double affi... | https://en.wikipedia.org/wiki/Double_affine_braid_group |
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://en.wikipedia.org/wiki/Double_vector_bundle |
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://en.wikipedia.org/wiki/Doubly_periodic_function |
In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field K. | https://en.wikipedia.org/wiki/Poincare_line_bundle |
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://en.wikipedia.org/wiki/Natural_pairing |
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://en.wikipedia.org/wiki/Dual_wavelet |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Duality_(mathematics) |
In mathematics, a dyadic compactum is a Hausdorff topological space that is the continuous image of a product of discrete two-point spaces, and a dyadic space is a topological space with a compactification which is a dyadic compactum. However, many authors use the term dyadic space with the same meaning as dyadic compa... | https://en.wikipedia.org/wiki/Dyadic_space |
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in computer science because they are the only ones with finite binary representations... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Dyadic_fraction |
However, division of one dyadic rational number by another does not always produce a dyadic rational result. Mathematically, this means that the dyadic rational numbers form a ring, lying between the ring of integers and the field of rational numbers. This ring may be denoted Z {\displaystyle \mathbb {Z} } . | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Dyadic_fraction |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Global_dynamical_system |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Geometry_of_natural_structure |
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://en.wikipedia.org/wiki/Reversed_process |
In mathematics, a eutactic lattice (or eutactic form) is a lattice in Euclidean space whose minimal vectors form a eutactic star. This means they have a set of positive eutactic coefficients ci such that (v, v) = Σci(v, mi)2 where the sum is over the minimal vectors mi. "Eutactic" is derived from the Greek language, an... | https://en.wikipedia.org/wiki/Eutactic_lattice |
In mathematics, a fact is a statement (called a theorem) that can be proven by logical argument from certain axioms and definitions. | https://en.wikipedia.org/wiki/Scientific_fact |
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://en.wikipedia.org/wiki/Factor_system |
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://en.wikipedia.org/wiki/Schützenberger_theorem |
In mathematics, a fake 4-ball is a compact contractible topological 4-manifold. Michael Freedman proved that every three-dimensional homology sphere bounds a fake 4-ball. His construction involves the use of Casson handles and so does not work in the smooth category. | https://en.wikipedia.org/wiki/Fake_4-ball |
In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type. | https://en.wikipedia.org/wiki/Mumford_surface |
In mathematics, a fake projective space is a complex algebraic variety that has the same Betti numbers as some projective space, but is not isomorphic to it. There are exactly 50 fake projective planes. Prasad & Yeung (2006) found four examples of fake projective 4-folds, and showed that no arithmetic examples exist in... | https://en.wikipedia.org/wiki/Fake_projective_space |
In mathematics, a family F {\displaystyle {\mathcal {F}}} of sets is of finite character if for each A {\displaystyle A} , A {\displaystyle A} belongs to F {\displaystyle {\mathcal {F}}} if and only if every finite subset of A {\displaystyle A} belongs to F {\displaystyle {\mathcal {F}}} . That is, For each A ∈ F {\dis... | https://en.wikipedia.org/wiki/Finite_character |
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://en.wikipedia.org/wiki/Indexed_set |
In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set I {\displaystyle I} is called the index set of the family, and X {\displaystyle X} is the indexed set. Sequences are one type of families indexed by natural numbers. In general, the index set I {\displaystyle... | https://en.wikipedia.org/wiki/Indexed_set |
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... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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). | https://en.wikipedia.org/wiki/Zigzag_poset |
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://en.wikipedia.org/wiki/Fibrifold_notation |
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. | https://en.wikipedia.org/wiki/Algebraically_closed_field |
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 ... | https://en.wikipedia.org/wiki/Quasi-algebraically_closed |
In mathematics, a field K is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation v and if its residue field k is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, ... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Normalized_valuation |
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://en.wikipedia.org/wiki/Spherically_complete_field |
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. | https://en.wikipedia.org/wiki/Pseudo_algebraically_closed |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Field_(algebra) |
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://en.wikipedia.org/wiki/Field_of_sets |
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... | https://en.wikipedia.org/wiki/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. | https://en.wikipedia.org/wiki/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... | https://en.wikipedia.org/wiki/Chirality |
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://en.wikipedia.org/wiki/Symplectic_filling |
Mathematicians call this orientation the outward normal first convention.All the following cobordisms are oriented, with the orientation on W given by a symplectic structure. Let ξ denote the kernel of the contact form α. A weak symplectic filling of a contact manifold (X,ξ) is a symplectic manifold (W,ω) with ∂ W = X ... | https://en.wikipedia.org/wiki/Symplectic_filling |
A Stein filling of a contact manifold (X,ξ) is a Stein manifold W which has X as its strictly pseudoconvex boundary and ξ is the set of complex tangencies to X – that is, those tangent planes to X that are complex with respect to the complex structure on W. The canonical example of this is the 3-sphere where the comple... | https://en.wikipedia.org/wiki/Symplectic_filling |
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://en.wikipedia.org/wiki/Filter_quantifier |
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://en.wikipedia.org/wiki/Elementary_filter |
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, ... | https://en.wikipedia.org/wiki/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 ... | https://en.wikipedia.org/wiki/Filter_(mathematics) |
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://en.wikipedia.org/wiki/Filtered_ring |
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://en.wikipedia.org/wiki/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... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Filtration_(mathematics) |
In mathematics, a finitary relation over sets X1, ..., Xn is a subset of the Cartesian product X1 × ⋯ × Xn; that is, it is a set of n-tuples (x1, ..., xn) consisting of elements xi in Xi. Typically, the relation describes a possible connection between the elements of an n-tuple. For example, the relation "x is divisibl... | https://en.wikipedia.org/wiki/Theory_of_relations |
A relation with n "places" is variously called an n-ary relation, an n-adic relation or a relation of degree n. Relations with a finite number of places are called finitary relations (or simply relations if the context is clear). It is also possible to generalize the concept to infinitary relations with infinite sequen... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Theory_of_relations |
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most commo... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Galois_fields |
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://en.wikipedia.org/wiki/Combinatorial_Riemann_mapping_theorem |
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... | https://en.wikipedia.org/wiki/Finite_topological_space |
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://en.wikipedia.org/wiki/Finite_von_Neumann_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://en.wikipedia.org/wiki/Finitely-generated_algebra |
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://en.wikipedia.org/wiki/Module_of_finite_type |
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://en.wikipedia.org/wiki/First-order_partial_differential_equation |
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://en.wikipedia.org/wiki/Stable_fixed_point |
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://en.wikipedia.org/wiki/Fixed_point_theory |
In mathematics, a flip graph is a graph whose vertices are combinatorial or geometric objects, and whose edges link two of these objects when they can be obtained from one another by an elementary operation called a flip. Flip graphs are special cases of geometric graphs. Among noticeable flip graphs, one finds the 1-s... | https://en.wikipedia.org/wiki/Flip_graph |
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. | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Flow_(mathematics) |
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://en.wikipedia.org/wiki/Polynomial_SOS |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Formal_distribution |
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://en.wikipedia.org/wiki/Formal_group_law |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Formal_power_series_ring |
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://en.wikipedia.org/wiki/Formal_sum |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Formula |
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://en.wikipedia.org/wiki/Fractal_dimension |
In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail wi... | https://en.wikipedia.org/wiki/Fractal_dimension |
After several iterations over years, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants. "One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, bu... | https://en.wikipedia.org/wiki/Fractal_dimension |
No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively by thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional. Therefore its dime... | https://en.wikipedia.org/wiki/Fractal_dimension |
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