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Type of functions in algebra
algebra, a polynomial map or polynomial mapping P : V → W {\displaystyle P:V\to W} between vector spaces over an infinite field k is a polynomial in linear
Polynomial_mapping
Digital imaging technique
Polynomial texture mapping (PTM), also known as Reflectance Transformation Imaging (RTI), is a technique of imaging and interactively displaying objects
Polynomial_texture_mapping
Mathematical concept
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The
Degree_of_a_polynomial
This is a list of polynomial topics, by Wikipedia page. See also trigonometric polynomial, list of algebraic geometry topics. Degree: The maximum exponents
List_of_polynomial_topics
Pair of polynomial sequences
The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}
Chebyshev_polynomials
Type of mathematical expression
The mapping that associates the result of this substitution to the substituted value is a function, called a polynomial function; see § Polynomial functions
Polynomial
Method of defining surface detail on a computer-generated graphic or 3D model
Perspective correct texturing Time Texturing – texture mapping with bezier lines Polynomial Texture Mapping. Archived 2019-03-07 at the Wayback Machine – Interactive
Texture_mapping
Simple polynomial map exhibiting chaotic behavior
the quadratic difference equation It is a recurrence relation and a polynomial mapping of degree 2. It is often referred to as an archetypal example of how
Logistic_map
if a polynomial mapping over a characteristic-0 field has a constant nonzero Jacobian determinant, then it has a regular (i.e. with polynomial components)
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Function, homomorphism, or morphism
map or mapping is a function in its general sense.[vague] These terms may have originated as from the process of making a geographical map: mapping the Earth
Map_(mathematics)
Study of systems of inequalitites
equations with real-number coefficients, and mappings between them (in particular real polynomial mappings). Semialgebraic geometry is the study of semialgebraic
Real_algebraic_geometry
Form of interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through
Polynomial_interpolation
Point where function's value is zero
root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number
Zero_of_a_function
Statistics concept
In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable
Polynomial_regression
Machine learning kernel function
training/testing with a linear SVM, i.e. full computation of the mapping φ as in polynomial regression; basket mining (using a variant of the apriori algorithm)
Polynomial_kernel
Polynomial coprime with its derivative
In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct
Separable_polynomial
Properties of mathematical relationships
for two different properties: linearity of a function (or mapping); linearity of a polynomial. An example of a linear function is the function defined
Linearity
realized using the Pomeau–Manneville map. The Pomeau–Manneville map is a polynomial mapping (equivalently, recurrence relation), often referred to as an archetypal
Pomeau–Manneville_scenario
Universal construction of a complex Lie group from a real Lie group
exponential mapping is a polynomial mapping from the Lie algebra to the corresponding subgroup by nilpotence. The inverse is given by the logarithm mapping which
Complexification_(Lie_group)
Endofunctor on the category V of finite-dimensional vector spaces
{Hom} (X,Y)\to \operatorname {Hom} (F(X),F(Y))} is a polynomial mapping (i.e., a vector-valued polynomial in linear forms). Given linear maps f i : X → Y
Polynomial_functor
Fixed number that has received a name
between every period-doubling bifurcation. The logistic map is a polynomial mapping, often cited as an archetypal example of how chaotic behaviour can
Mathematical_constant
On invertibility of polynomial maps (mathematics)
fN(X1,...,XN)). Any map F: kN → kN arising in this way is called a polynomial mapping. The Jacobian determinant of F, denoted by JF, is defined as the determinant
Jacobian_conjecture
Polynomial sequence
In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike
Zernike_polynomials
Type of linear code
In coding theory, a polynomial code is a type of linear code whose set of valid code words consists of those polynomials (usually of some fixed length)
Polynomial_code
Orthogonal symmetric polynomial family
In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987
Macdonald_polynomials
Characteristic polynomial whose associated linear system is stable
In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either: all its
Stable_polynomial
Error-correcting codes
titled "Polynomial Codes over Certain Finite Fields". The original encoding scheme described in the Reed and Solomon article used a variable polynomial based
Reed–Solomon_error_correction
Polynomial with integer value for integer input
mathematics, an integer-valued polynomial (also known as a numerical polynomial) P ( t ) {\displaystyle P(t)} is a polynomial whose value P ( n ) {\displaystyle
Integer-valued_polynomial
Branch of mathematics
by iterating a complex analytic mapping. This article focuses on the case of algebraic dynamics, where a polynomial or rational function is iterated
Complex_dynamics
Number of times an object must be counted for making true a general formula
it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion
Multiplicity_(mathematics)
Mathematical theorem
In complex analysis, the Riemann mapping theorem states that if U {\displaystyle U} is a non-empty simply connected open subset of the complex number
Riemann_mapping_theorem
Field of algebraic geometry
lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the
Birational_geometry
Concepts from linear algebra
the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree n is the characteristic polynomial of some companion
Eigenvalues_and_eigenvectors
Algebraic structure
mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]
Ring_of_polynomial_functions
Generalization of algebraic variety
sets such as X(k): define the Zariski topology on X(k), consider polynomial mappings between different sets of this type, and so on. But if k is not algebraically
Scheme_(mathematics)
Association of one output to each input
from the intersection of the domains of f and g. The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers
Function_(mathematics)
Graph drawing used to study Riemann surfaces
p {\displaystyle p} and q {\displaystyle q} are polynomials, transforms the Riemann sphere by mapping it to itself. Consider, for example, the rational
Dessin_d'enfant
Algorithms for zeros of functions
the function by a polynomial of low degree, which takes the same values at these approximate roots. Then the root of the polynomial is computed and used
Root-finding_algorithm
Mathematical concept in polynomial theory
resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root
Resultant
Polynomial function of degree two
function, is a quadratic polynomial, a polynomial of degree two. In elementary mathematics a polynomial and its associated polynomial function are rarely distinguished
Quadratic_function
Method for estimating new data within known data points
this interpolant with a polynomial of higher degree. Consider again the problem given above. The following sixth degree polynomial goes through all the seven
Interpolation
Arithmetic in a field with a finite number of elements
usual multiplication of polynomials, but with coefficients multiplied modulo p and polynomials multiplied modulo the polynomial m(x). This representation
Finite_field_arithmetic
Plane algebraic curve
exist various models. A related object is the classical modular polynomial, a polynomial in one variable defined as Φn(x, x). The classical modular curves
Classical_modular_curve
Mathematical function such that every output has at least one input
number y is the solution set of the cubic polynomial equation x3 − 3x − y = 0, and every cubic polynomial with real coefficients has at least one real
Surjective_function
Cubic function used for interpolation
cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives
Cubic_Hermite_spline
order polynomial is a polynomial studied in mathematics, in particular in algebraic graph theory and algebraic combinatorics. The order polynomial counts
Order_polynomial
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
spectral mapping theorem for the polynomial functional calculus: Let A be an n × n matrix with eigenvalues λ1, ..., λn, then for any polynomial p, p(A)
Jordan_normal_form
Cryptographic algorithm created by Adi Shamir
specifically that k {\displaystyle k} points on the polynomial uniquely determines a polynomial of degree less than or equal to k − 1 {\displaystyle
Shamir's_secret_sharing
Branch of functional analysis
Consequently, the mapping p ↦ p ( T ) {\displaystyle p\mapsto p(T)} is an isometry and a densely defined homomorphism on the ring of polynomial functions. Extending
Borel_functional_calculus
Method of interpolating functions on a 2D grid
is to write the solution to the interpolation problem as a multilinear polynomial f ( x , y ) ≈ a 00 + a 10 x + a 01 y + a 11 x y , {\displaystyle f(x,y)\approx
Bilinear_interpolation
In mathematics, the complex squaring map, a polynomial mapping of degree two, is a simple and accessible demonstration of chaos in dynamical systems. It
Complex_squaring_map
Subdivision of space into cells
where the mapping from the abstract to realized element is linear, and mesh edges are straight segments. Higher order polynomial mappings are common
Mesh_generation
Quadratic polynomial
complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers. Quadratic polynomials have the following
Complex_quadratic_polynomial
type II thin set is an image of an algebraic morphism (essentially a polynomial mapping) φ, applied to the K-points of some other d-dimensional algebraic
Thin_set_(Serre)
Type of Turing reduction
and computational complexity theory, a many-one reduction (also called mapping reduction) is a reduction that converts instances of one decision problem
Many-one_reduction
Branch of functional analysis
in A. It is known that the spectral mapping theorem holds for the polynomial functional calculus: for any polynomial p, σ(p(T)) = p(σ(T)). This can be extended
Holomorphic functional calculus
Holomorphic_functional_calculus
Concept in complex analysis
and a zero of order | n | {\displaystyle |n|} if n < 0. For example, a polynomial of degree n has a pole of degree n at infinity. The complex plane extended
Zeros_and_poles
Open convex self-dual cones
particular the exponential map is a polynomial mapping of n {\displaystyle {\mathfrak {n}}} onto N, with polynomial inverse given by the logarithm. Let
Symmetric_cone
Study of mathematical knots
theory. A knot polynomial is a knot invariant that is a polynomial. Well-known examples include the Jones polynomial, the Alexander polynomial, and the Kauffman
Knot_theory
Algebraic structure
"Galois field". In a finite field of order q {\displaystyle q} , the polynomial X q − X {\displaystyle X^{q}-X} has all q {\displaystyle q} elements of
Finite_field
Infinite sum that is considered independently from any notion of convergence
variable. Hence, formal power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from
Formal_power_series
Constants related to interpolation errors
interpolation maps the function f {\displaystyle f} to a polynomial p {\displaystyle p} . This defines a mapping X {\displaystyle X} from the space C ( [ a , b
Lebesgue_constant
Geometric transformation that preserves lines but not angles nor the origin
transformations Bent function Flat (geometry) Homography Multilinear polynomial Berger 1987, p. 38. Samuel 1988, p. 11. Snapper & Troyer 1989, p. 65.
Affine_transformation
In mathematics, invariant of square matrices
more efficient. Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry, the
Determinant
Matrix of geometric progressions
a=V^{-1}y} . That is, the map from coefficients to values of polynomials is a bijective linear mapping with matrix V, and the interpolation problem has a unique
Vandermonde_matrix
Hamiltonian cycle polynomial of a matrix received from its weighted adjacency matrix via subjecting its rows and columns to any permutation mapping i to 1 and
Hamiltonian_cycle_polynomial
Methods of error detection and correction in communications
after division in the ring of polynomials over GF(2) (the finite field of integers modulo 2). That is, the set of polynomials where each coefficient is either
Mathematics of cyclic redundancy checks
Mathematics_of_cyclic_redundancy_checks
Strong form of uniform continuity
as a consequence of Weierstrass approximation theorem, because every polynomial is locally Lipschitz continuous). Every Lipschitz continuous map is uniformly
Lipschitz_continuity
Mathematical space
{O}}_{\mathbf {R} ^{n}}} of semialgebraic functions. (For example, any polynomial mapping between semialgebraic sets is a semialgebraic function, as is the
Semialgebraic_space
Natural number
oblong, kite, rhombus, and square. Four is the highest degree general polynomial equation for which there is a solution in radicals. Four is the only square
4
Operation on mathematical functions
→ X X → ℂ ℂ → X ℂn → X Classes/properties Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective
Function_composition
A fully polynomial-time approximation scheme (FPTAS) is an algorithm for finding approximate solutions to function problems, especially optimization problems
Fully polynomial-time approximation scheme
Fully_polynomial-time_approximation_scheme
Mapping between functions in the quantum phase space
invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödinger picture. Often the mapping from functions
Wigner–Weyl_transform
Typically linear operator defined in terms of differentiation of functions
of differentiation) because of the symmetry of second derivatives. The polynomial p obtained by replacing partials ∂ ∂ x i {\displaystyle {\frac {\partial
Differential_operator
Nonlinear differential operator used to study conformal mappings
plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces. It is named after the German mathematician Hermann
Schwarzian_derivative
Scientific discipline
come about in qualitatively very similar systems. Logistic maps are polynomial mappings, and are often cited as providing archetypal examples of how chaotic
Theoretical_ecology
Method of representing curves and surfaces in computer graphics
mathematicians started studying the spline shape, and derived the piecewise polynomial formula known as the spline curve or spline function. I. J. Schoenberg
Non-uniform_rational_B-spline
Algebraic structure in mathematics
example: The continuous mappings in a topological group. The polynomial functions on a ring with identity under addition and polynomial composition. The affine
Near-ring
Mathematical idealization of the trace left by a moving point
the zero set of a polynomial in two indeterminates. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies
Curve
Counting polynomial real roots based on coefficients
described by René Descartes in his La Géométrie, counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive
Descartes'_rule_of_signs
Type of function in mathematics
analytic functions are The following elementary functions: All polynomials: if a polynomial has degree n {\displaystyle n} , any terms of degree larger
Analytic_function
Result in modular arithmetic
Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number p, then this root can be lifted
Hensel's_lemma
Ratio of polynomial functions
such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in
Rational_function
Map in projective geometry
categorical product (in the category of projective varieties and homogeneous polynomial maps) of P n {\displaystyle \mathbb {P^{n}} } and P m {\displaystyle \mathbb
Segre_embedding
Mathematical representation
polynomial, consider H1(Cn) as a module over the group-ring of covering transformations Z[Z], which is isomorphic to the ring of Laurent polynomials Z[t
Burau_representation
Springer-Verlag, ISBN 0-387-98428-3 Narkiewicz, Władysław (1995), Polynomial mappings, Lecture Notes in Mathematics, vol. 1600, Berlin: Springer-Verlag
Prüfer_domain
Mathematical technique used in environmental sciences
such as geology and soil science. The method involves using low-order polynomials of spatial coordinates to estimate a regular grid of points from scattered
Trend_surface_analysis
Study of abstract algebraic structures
commutative algebras, namely the polynomial algebras. In this particularly simple and important case of the polynomial algebra F [ T 1 , … , T k ] {\displaystyle
Algebra_representation
System where changes of output are not proportional to changes of input
equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words
Nonlinear_system
Index of articles associated with the same name
characteristic function of a cooperative game in game theory. The characteristic polynomial in linear algebra. The characteristic state function in statistical mechanics
Characteristic_function
Fractal named after mathematician Benoit Mandelbrot
parameters c {\displaystyle c} for which the Julia set of the corresponding polynomial forms a connected set. In the same way, the boundary of the Mandelbrot
Mandelbrot_set
Audio process
_{n=0}^{N}a_{n}x^{n}} Polynomial functions are convenient as shaping functions because, when given a single sinusoid as input, a polynomial of degree N will
Waveshaper
Differential equation that is linear with respect to the unknown function
any. The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is
Linear_differential_equation
Mathematical term; type of polynomial transformation
type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. Simply, it is a method for transforming a polynomial equation
Tschirnhaus_transformation
Sum of elements on the main diagonal
trace. As a consequence, one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing
Trace_(linear_algebra)
polynomials, which implies that every symmetric polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression
Symmetry_in_mathematics
Continuous generalization of cellular automata
( α − α 4 r ( 1 − r ) ) , exponential , α = 4 ( 4 r ( 1 − r ) ) α , polynomial , α = 4 1 [ 1 4 , 3 4 ] ( r ) , rectangular … , etc. {\displaystyle
Lenia
Iterative algorithm on numbers
K_{b}(n)=\alpha -\beta } is the Kaprekar mapping. Some numbers map to themselves; these are the fixed points of the Kaprekar mapping, and are called Kaprekar’s constants
Kaprekar's_routine
Polynomial with all terms of degree two
mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, 4 x 2 + 2 x y
Quadratic_form
Function used as a performance test problem for optimization algorithms
function of x {\displaystyle x} . For small N {\displaystyle N} the polynomials can be determined exactly and Sturm's theorem can be used to determine
Rosenbrock_function
POLYNOMIAL MAPPING
POLYNOMIAL MAPPING
POLYNOMIAL MAPPING
POLYNOMIAL MAPPING
Girl/Female
Arabic, Australian, French, German
The Weaning; The Abstaining
Girl/Female
Indian
Sweet Fruit
Boy/Male
Arabic, Muslim
Servant of the Praiseworthy; Ever-praised
Female
Croatian
, from Hadria.
Girl/Female
Hindu
Wisdom, Knowledge, Learning, Goddess Durga
Boy/Male
American, Australian, British, Celtic, Christian, English, Gaelic, Irish, Jamaican
Exalted; Wise; High Longing; Wolf; Lover; Hound; King; Ulster; Hound Lover; Lover of Wolves
Boy/Male
Hindu
Plenty
Female/Male/Unisex
Korean
Korean name SHIN means "faith, trust." Compare with another form of Shin.
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Beautiful Image
Girl/Female
British, English
Warrior Maid
POLYNOMIAL MAPPING
POLYNOMIAL MAPPING
POLYNOMIAL MAPPING
POLYNOMIAL MAPPING
POLYNOMIAL MAPPING
n.
The art of describing or delineating the stars; a description or mapping of the heavens.
n.
the mapping or description of a region or district.
p. pr. & vb. n.
of Map
n.
An expression composed of two or more terms, connected by the signs plus or minus; as, a2 - 2ab + b2.
a.
Possessing the same number of factors of a given kind; as, a homogeneous polynomial.
a.
Consisting of two or more words; having names consisting of two or more words; as, a polynomial name; polynomial nomenclature.
n.
A polynomial of four terms connected by the signs plus or minus.
n.
A polynomial name or term.
a.
Containing many names or terms; multinominal; as, the polynomial theorem.
n. & a.
Same as Polynomial.