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Topics referred to by the same term
Function field may refer to: Function field of an algebraic variety Function field (scheme theory) Algebraic function field Function field sieve Function
Function_field
Finitely generated extension field of positive transcendence degree
function field (often abbreviated as function field) of n {\displaystyle n} variables over a field k {\displaystyle k} is a finitely generated field extension
Algebraic_function_field
Ratio of polynomial functions
over a field K is a field, the field of fractions of the ring of the polynomial functions over K. A function f is called a rational function if it can
Rational_function
Algorithm to solve the discrete logarithm problem
mathematics, the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has heuristic
Function_field_sieve
Distance from a point to the boundary of a set
In mathematics and its applications, the signed distance function or signed distance field (SDF) is the orthogonal distance of a given point x to the
Signed_distance_function
Algebraic structure with addition, multiplication, and division
such as fields of rational functions, algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly used and studied
Field_(mathematics)
Function that encodes the dependence of a coupling parameter on the energy scale
In theoretical physics, specifically quantum field theory, a beta function or Gell-Mann–Low function, β(g), encodes the dependence of a coupling parameter
Beta_function_(physics)
Algebraic variety
means such a function field has a single transcendental function as generator: for example the j-function generates the function field of X(1) = PSL(2
Modular_curve
Topics referred to by the same term
event Function Drinks, an American beverage company Function Health, an American health technology company Function field (disambiguation) Function hall
Function
Mathematical concept in algebraic geometry
algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic
Function field of an algebraic variety
Function_field_of_an_algebraic_variety
Assignment of numbers to points in space
In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The
Scalar_field
The sheaf of rational functions KX of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical
Function field (scheme theory)
Function_field_(scheme_theory)
Expectation value of time-ordered quantum operators
In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products
Correlation function (quantum field theory)
Correlation_function_(quantum_field_theory)
Mathematical concept
kinds of global fields: Algebraic number field: A finite extension of Q {\displaystyle \mathbb {Q} } Global function field: The function field of an irreducible
Global_field
Abelian group
{\displaystyle A|_{k(t)}} (the pullback of A {\displaystyle A} to the function field k ( t ) = k ( P 1 ) {\displaystyle k(t)=k(\mathbb {P} ^{1})} ) by a
Mordell–Weil_group
Index of lists with the same name
function of a number field Duursma zeta function of error-correcting codes Epstein zeta function of a quadratic form Goss zeta function of a function
List_of_zeta_functions
over the field of algebraic numbers is a global height function with local contributions coming from Fubini–Study metrics on the Archimedean fields and the
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Mathematical function
In mathematics, an algebraic function is a function that satisfies a polynomial equation. Thus an equation of the following form holds: a n ( x ) f ( x
Algebraic_function
Curve defined as zeros of polynomials
over a field F are categorically equivalent to algebraic function fields in one variable over F. Such an algebraic function field is a field extension
Algebraic_curve
Indian mathematician (born 1961)
University of Rochester in July 2013. Thakur wrote a research monograph Function Field Arithmetic. Thakur has been serving on the editorial boards of Journal
Dinesh_Thakur_(mathematician)
Algebraic curve
characteristic of the ground field is not 2, one can take h(x) = 0). A hyperelliptic function is an element of the function field of such a curve, or of the
Hyperelliptic_curve
Association of one output to each input
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
Function_(mathematics)
Algebraic structure with addition and multiplication
is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the
Ring_(mathematics)
Mathematical theory
correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from algebraic
Geometric Langlands correspondence
Geometric_Langlands_correspondence
Branch of number theory
algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring
Algebraic_number_theory
Generating function for quantum correlation functions
In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral
Partition function (quantum field theory)
Partition_function_(quantum_field_theory)
Conjecture on zeros of the zeta function
hypothesis for curves over finite fields, which was proved by André Weil. The Riemann zeta function ζ is a function whose argument may be any complex
Riemann_hypothesis
Concept in mathematics
over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex
Drinfeld_module
Assignment of a vector to each point in a subset of Euclidean space
fields are one kind of tensor field. Given a subset S of Rn, a vector field is represented by a vector-valued function V: S → Rn in standard Cartesian
Vector_field
Special fields Over a finite field, d = 1; over the reals, d = 1 or 2; over a p-adic field or a number field, or any local or global function field, d is
List of irreducible Tits indices
List_of_irreducible_Tits_indices
Abstract algebra concept
] {\displaystyle k[t]} is the rational function field k ( t ) {\displaystyle k(t)} . For any field k, the field of fractions of the formal power series
Field_of_fractions
number field or a global function field). It is used to encode ramification data for abelian extensions of a global field. Let K be a global field with
Modulus (algebraic number theory)
Modulus_(algebraic_number_theory)
Finite extension of the rationals
function fields, the local fields are completions of the local rings at all points of the curve for function fields. Many results valid for function fields
Algebraic_number_field
Concept in abstract algebra
\mathbb {R} [x],\,g(0)\neq 0\},} considered as a subring of the field of rational functions R ( x ) {\displaystyle \mathbb {R} (x)} . R {\displaystyle R}
Discrete_valuation_ring
Vector function in optics
leaving a four-dimensional function variously termed the photic field, the 4D light field or lumigraph. Formally, the field is defined as radiance along
Light_field
Method of solution to differential equations
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with
Green's_function
Algebraic variety
over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic
Rational_variety
Function with a multiplicative scaling behaviour
to functions whose domain and codomain are vector spaces over a field F: a function f : V → W {\displaystyle f:V\to W} between two F-vector spaces is
Homogeneous_function
Field extension that is not algebraic
transcendence degree of its function field. Also, global function fields are transcendental extensions of degree one of a finite field, and play in number theory
Transcendental_extension
Theoretical object in mathematics
fields starts with a curve C over a finite field k, which comes equipped with a function field F, which is a field extension of k. Each such function
Field_with_one_element
Concept in number theory
curves. Let K {\displaystyle K} be a global field, meaning either a number field or a global function field. Let v {\displaystyle v} run over the places
Adele_ring
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_function
Branch of mathematics
such as the field of rational numbers, number fields, finite fields, function fields, and p-adic fields. A large part of singularity theory is devoted
Algebraic_geometry
Kind of partial function between algebraic varieties
equivalent function fields. That is, every rational function f : X → P 1 {\displaystyle f:X\to \mathbb {P} ^{1}} can be restricted to a rational function U →
Rational_mapping
Extension of the factorial function
and other formulas in the fields of probability, statistics, analytic number theory, and combinatorics. The gamma function can be seen as a solution to
Gamma_function
function fields". Compositio Mathematica. 55 (2): 209–239. Rosen, Michael (2002), "15. The Brumer-Stark conjecture", Number theory in function fields
Brumer–Stark_conjecture
One-dimensional complex manifold
meromorphic function on T {\displaystyle T} . This function and its derivative ℘ τ ′ ( z ) {\displaystyle \wp '_{\tau }(z)} generate the function field of T
Riemann_surface
Construction of a larger algebraic field by "adding elements" to a smaller field
function defined on M. More generally, given an algebraic variety V over some field K, the function field K(V), consisting of the rational functions defined
Field_extension
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is the logistic function. Other sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial
Sigmoid_function
Mathematical conjecture about zeros of L-functions
the algebraic function field case (not the number field case). Global L-functions can be associated to elliptic curves, number fields (in which case
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Conjectures connecting number theory and geometry
fields (with subcases corresponding to number fields or function fields). Analogues for finite fields. More general fields, such as function fields over
Langlands_program
larger class of L-functions (Artin-Hecke L-functions); and to the global function field case. Here the inclusion of Artin L-functions, in particular, implicates
Weil's_criterion
S-shaped curve
networks. There are various generalizations, depending on the field. The logistic function was introduced in a series of three papers by Pierre François
Logistic_function
mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1)
Tsen's_theorem
Mathematical linear code
to algebraic function fields, the definitions of the codes are often given in the language of algebraic function fields over finite fields. Nevertheless
Algebraic_geometry_code
Generalization of the Riemann zeta function for algebraic number fields
Dedekind zeta function of an algebraic number field K, usually denoted ζ K ( s ) {\displaystyle \zeta _{K}(s)} , is an analytic function that represents
Dedekind_zeta_function
Romanian-American mathematician
University of California, San Diego. Popescu formulated and proved function field versions of the Gras conjectures and Rubin's integral refinement of
Cristian_Dumitru_Popescu
in the Taylor expansion of an Artin L-function associated with a Galois extension K/k of algebraic number fields. The conjectures generalize the analytic
Stark_conjectures
Type of energy
close to the solid to be influenced by ambient electric fields in the vacuum. The work function is not a characteristic of a bulk material, but rather
Work_function
over any field K {\displaystyle K} that has a Weil height function, the Bogomolov conjecture admits a natural extension to more general fields. When K
Bogomolov_conjecture
Best results achieved to date
variant of the medium-sized base field function field sieve, for binary fields, to compute a discrete logarithm in a field of 21971 elements. In order to
Discrete_logarithm_records
Theoretical framework in physics
classical field is a function of spatial and time coordinates. Examples include the gravitational field g(x, t) in Newtonian gravity and the electric field E(x
Quantum_field_theory
Euclidean Wightman distributions
In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to
Schwinger_function
Theorem in algebraic geometry
mathematics, Lüroth's theorem asserts that every field that lies between a field K and the rational function field K(X) must be generated as an extension of
Lüroth's_theorem
Concept in algebraic geometry
nonsingular model for the function field of a variety X, in other words a complete non-singular variety X′ with the same function field. In practice it is more
Resolution_of_singularities
Definite integral of a scalar or vector field along a path
plane. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points
Line_integral
Symmetric holomorphic function
{\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )} , and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular
Modular_lambda_function
German mathematician
the theory of algebraic function fields and in particular for his definition of a zeta function for algebraic function fields and his proof of the generalized
Friedrich_Karl_Schmidt
Function in quantum field theory showing probability amplitudes of moving particles
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one
Propagator
Concept in class field theory
topology is "locally profinite".) For global fields of characteristic p > 0 {\displaystyle p>0} (function fields), the Weil group is the subgroup of the absolute
Weil_group
defined over the field K and K(C) denotes the function field of C, then the gonality is the minimum value taken by the degrees of field extensions K(C)/K(f)
Gonality of an algebraic curve
Gonality_of_an_algebraic_curve
Mathematical function, used to describe magnetization
Langevin function is derived using statistical mechanics and describes how magnetic dipoles are aligned by an applied field. The Brillouin function was developed
Brillouin and Langevin functions
Brillouin_and_Langevin_functions
American mathematician
Berkeley, and Brandeis University. He worked on function fields and introduced the Goss zeta function. In 2012, he became a fellow of the American Mathematical
David_Goss
Shreeram Abhyankar posed in 1957, on the Galois groups of algebraic function fields of characteristic p. The soluble case was solved by Serre in 1990 and
Abhyankar's_conjecture
means that at the level of function fields it is possible to construct J by taking linearly disjoint copies of the function field of C, and within their compositum
Symmetric product of an algebraic curve
Symmetric_product_of_an_algebraic_curve
Mathematical concept
Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Theorem about polynomials, analogous to the abc conjecture for integers
of polynomials is replaced by a one-dimensional function field. Let k be an algebraically closed field of characteristic 0, let C/k be a smooth projective
Mason–Stothers_theorem
Estimates the number of points on an elliptic curve over a finite field
local zeta-function of E {\displaystyle E} . In this form it can be seen to be the analogue of the Riemann hypothesis for the function field associated
Hasse's theorem on elliptic curves
Hasse's_theorem_on_elliptic_curves
Unsolved problem in mathematics
conjecture comes from Srinivasa Ramanujan, who proposed it for Ramanujan tau function, and Hans Petersson, who generalized it for coefficients of modular forms
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Mathematical description of quantum state
In quantum mechanics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common
Wave_function
Type of Dirichlet series associated to number field extensions
In mathematics, Artin L-functions are a type of Dirichlet series defined for finite extensions of number fields, encoding informations about linear representations
Artin_L-function
Class of mathematical function
an algebraic point of view, if the function's domain is connected, then the set of meromorphic functions is the field of fractions of the integral domain
Meromorphic_function
Type of zeta function
mathematics, the Goss zeta function, named after David Goss, is an analogue of the Riemann zeta function for function fields. Sheats proved that it satisfies
Goss_zeta_function
Multivariate derivative (mathematics)
gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) ∇f whose value at a point p gives the
Gradient
Group-like structure appearing in global fields
infrastructure of a real quadratic function field to the divisor class group. This connection can be generalized to arbitrary function fields and, combining with R
Infrastructure (number theory)
Infrastructure_(number_theory)
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Completes the Langlands program for general linear groups over algebraic function fields
completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic forms on these groups
Lafforgue's_theorem
finite field is quasi-algebraically closed by the Chevalley–Warning theorem. Algebraic function fields of dimension 1 over algebraically closed fields are
Quasi-algebraically closed field
Quasi-algebraically_closed_field
Type of mathematical function
elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are polynomial
Elementary_function
Concept in mathematics
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
Meromorphic function on the complex plane
number theory and related fields. L-functions share fundamental properties and characteristics with the Riemann zeta function, which serves as the prototypical
L-function
Indian mathematician (born 1966)
1007/BF01231440 1998: "Isotropy of quadratic forms over function fields in one variable over p-adic fields", Publ. de I.H.E.S. 88, 129–150 (with Raman Parimala)
Suresh_Venapally
Analytic function that does not satisfy a polynomial equation
mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable
Transcendental_function
Type of algebraic curve
\deg(G)=\deg({\overline {G}})} The function field K(C) of C over K is the field of fractions of K[C], and the function field K ¯ ( C ) {\displaystyle {\overline
Imaginary_hyperelliptic_curve
Mathematics award
geometric interpretations for the higher derivatives of L-functions in the function field case." Wei Zhang – "For deep work on the global Gan-Gross-Prasad
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group G that is constant
Class_function
Linear map or polynomial function of degree one
the term linear function refers to two distinct but related notions: In calculus and related areas, a linear function is a function whose graph is a
Linear_function
Commutative ring with a Euclidean division
⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields Let R be an integral domain. A Euclidean function on R is a function f from R \ {0} to the
Euclidean_domain
Curves of genus > 1 over the rationals have only finitely many rational points
conjectures have been put forth by Paul Vojta. The Mordell conjecture for function fields was proved by Yuri Ivanovich Manin and by Hans Grauert. In 1990, Robert
Faltings'_theorem
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FUNCTION FIELD
FUNCTION FIELD
Surname or Lastname
English
English : variant of Field, from the dative plural of Old English feld ‘open country’.
Surname or Lastname
English
English : habitational name from any of various places, such as Merryfield in Devon and Cornwall or Mirfield in West Yorkshire, all named with the Old English elements myrige ‘pleasant’ + feld ‘pasture’, ‘open country’ (see Field).
Biblical
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Surname or Lastname
English (chiefly Gloucestershire and Worcestershire)
English (chiefly Gloucestershire and Worcestershire) : variant of Millward.French (northern) : from a Germanic personal name composed of the elements mil ‘good’, ‘gracious’ + hard ‘hardy’, ‘brave’, ‘strong’.Southern French : from a variant spelling of Occitan milhar ‘millet field’ (from mil ‘millet’).
Girl/Female
Tamil
Ankshika | அஂகà¯à®·à¯€à®•à®¾
It’s derived from the root word - anksh that means a fraction. Ankshika means the fraction of the cosmos
Ankshika | அஂகà¯à®·à¯€à®•à®¾
Boy/Male
English
In the field.
Boy/Male
Indian
Friction
Girl/Female
Indian
It’s derived from the root word - anksh that means a fraction. Ankshika means the fraction of the cosmos
Girl/Female
Afghan, Arabic, Australian, Indian, Muslim
Fiction; Romance; Story
Surname or Lastname
English (chiefly West Midlands and northern England)
English (chiefly West Midlands and northern England) : topographic name for someone who lived in a house (Middle English hous) in open pasture land (see Field). Reaney draws attention to the form de Felhouse (Staffordshire 1332), and suggests that this may have become Fellows.
Girl/Female
Bengali, Indian
Fraction of Time
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Surname or Lastname
South German
South German : occupational name for an official in charge of the legal auction of property confiscated in default of a fine; such a sale was known in Middle High German as a gant (from Italian incanto, a derivative of Late Latin inquantare ‘to auction’, from the phrase In quantum? ‘To how much (is the price raised)?’).German : metonymic occupational name for a cooper, from Middle High German ganter, kanter ‘barrel rack’.German : variant of Gander 3.English : occupational name for a glover, from Old French gantier, an agent derivative of gant ‘glove’ (see Gant).
Boy/Male
Australian, British, English
A Field
Boy/Male
French Greek
Cyrano de Bergerac was a seventeenth-century soldier and science-fiction writer.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Boy/Male
American, British, English
Lives in the Field
Surname or Lastname
English
English : topographic name for someone who lived by a watercourse or road junction, Old English gelǣt, or a habitational name from Leat in Devon, or The Leete in Essex, named with this element.
Girl/Female
Hindu, Indian
Fraction of the Cosmos
FUNCTION FIELD
FUNCTION FIELD
FUNCTION FIELD
FUNCTION FIELD
FUNCTION FIELD
FUNCTION FIELD
FUNCTION FIELD
n.
The act of feigning, inventing, or imagining; as, by a mere fiction of the mind.
v. t.
To sell by auction.
v. t.
To give sanction to; to ratify; to confirm; to approve.
n.
A derived function; a function obtained from a given function by a certain algebraic process.
v. i.
Alt. of Functionate
n.
The course of action which peculiarly pertains to any public officer in church or state; the activity appropriate to any business or profession.
a.
Pertaining to the function of an organ or part, or to the functions in general.
n.
The act of anointing, smearing, or rubbing with an unguent, oil, or ointment, especially for medical purposes, or as a symbol of consecration; as, mercurial unction.
n.
The act of joining, or the state of being joined; union; combination; coalition; as, the junction of two armies or detachments; the junction of paths.
n.
The act of anointing, or the state of being anointed; unction; specifically (Med.), the rubbing of ointments into the pores of the skin, by which medicinal agents contained in them, such as mercury, iodide of potash, etc., are absorbed.
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
n.
The natural or assigned action of any power or faculty, as of the soul, or of the intellect; the exertion of an energy of some determinate kind.
v. t.
To separate by means of, or to subject to, fractional distillation or crystallization; to fractionate; -- frequently used with out; as, to fraction out a certain grade of oil from pretroleum.
v. t.
To supply with an organ or organs having a special function or functions.
a.
Pertaining to, or connected with, a function or duty; official.
n.
The things sold by auction or put up to auction.
n.
The place or point of union, meeting, or junction; specifically, the place where two or more lines of railway meet or cross.
n.
The office, duties, or functions of a minister, servant, or agent; ecclesiastical, executive, or ambassadorial function or profession.
v. t.
The act of uniting, or the state of being united; junction.
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
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