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Polynomial function of degree two
In mathematics, a quadratic function of a single variable is a function of the form f ( x ) = a x 2 + b x + c {\displaystyle f(x)=ax^{2}+bx+c} with
Quadratic_function
Polynomial equation of degree two
solutions of the equation, and roots or zeros of the quadratic function on its left-hand side. A quadratic equation has at most two solutions. If there is
Quadratic_equation
Solving an optimization problem with a quadratic objective function
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks
Quadratic_programming
Formula that provides the solutions to a quadratic equation
algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations,
Quadratic_formula
Linear optimal control technique
by a quadratic function is called the LQ problem. One of the main results in the theory is that the solution is provided by the linear–quadratic regulator
Linear–quadratic_regulator
Mathematical relation assigning a probability event to a cost
regression theory, which is based on the quadratic loss function. The quadratic loss function is also used in linear-quadratic optimal control problems. In these
Loss_function
Continuous probability distribution
and statistics, the U-quadratic distribution is a continuous probability distribution defined by a unique convex quadratic function with lower limit a and
U-quadratic_distribution
Polynomial function of degree 4
quartic function is a cubic function. Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic function
Quartic_function
Independent parameter describing the state of a physical system
is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system
Degrees of freedom (physics and chemistry)
Degrees_of_freedom_(physics_and_chemistry)
Real function with secant line between points above the graph itself
number), a quadratic function c x 2 {\displaystyle cx^{2}} ( c {\displaystyle c} as a nonnegative real number) and an exponential function c e x {\displaystyle
Convex_function
Plane curve: conic section
another plane that is tangential to the conical surface. The graph of a quadratic function y = a x 2 + b x + c {\displaystyle y=ax^{2}+bx+c} (with a ≠ 0 {\displaystyle
Parabola
Method for solving quadratic equations
also used for graphing quadratic functions, deriving the quadratic formula, and more generally in computations involving quadratic polynomials, for example
Completing_the_square
Polynomial function of degree 3
solutions are called roots of the function. The derivative of a cubic function is a quadratic function. A cubic function with real coefficients has either
Cubic_function
Sum type in number theory
combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more
Quadratic_Gauss_sum
Association of one output to each input
include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain
Function_(mathematics)
Address collision resolution scheme
Quadratic probing is an open addressing scheme in computer programming for resolving hash collisions in hash tables. Quadratic probing operates by taking
Quadratic_probing
Function whose squared absolute value has finite integral
square-integrable function, also called a quadratically integrable function or L 2 {\displaystyle L^{2}} function or square-summable function, is a real- or
Square-integrable_function
Simple polynomial map exhibiting chaotic behavior
The logistic map is a discrete dynamical system defined by the quadratic difference equation It is a recurrence relation and a polynomial mapping of degree 2
Logistic_map
Optimization problem in mathematics
quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic functions
Quadratically constrained quadratic program
Quadratically_constrained_quadratic_program
Mathematical function
chemistry to form basis sets. Gaussian functions arise by composing the exponential function with a concave quadratic function: f ( x ) = exp ( α x 2 + β x
Gaussian_function
Study of mathematical algorithms for optimization problems
difficult than regular linear programming. Quadratic programming allows the objective function to have quadratic terms, while the feasible set must be specified
Mathematical_optimization
Topics referred to by the same term
for square. Quadratic function (or quadratic polynomial), a polynomial function that contains terms of at most second degree Complex quadratic polynomials
Quadratic
Circle associated with a quadratic equation
associated with a quadratic equation; it is named after Thomas Carlyle. The circle has the property that the solutions of the quadratic equation are the
Carlyle_circle
Concept in mathematics
generalizes the conjugate gradient method to nonlinear optimization. For a quadratic function f ( x ) {\displaystyle \displaystyle f(x)} f ( x ) = ‖ A x − b ‖ 2
Nonlinear conjugate gradient method
Nonlinear_conjugate_gradient_method
Concept in the analysis of dynamical systems
constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for
Lyapunov_function
Probability distribution
+Y_{m}^{2}\right)/m}}\sim F_{n,m}.} A quadratic form of a normal vector, i.e. a quadratic function q = ∑ x i 2 + ∑ x j + c {\textstyle q=\sum x_{i}^{2}+\sum
Normal_distribution
Function equal to the product of its values on coprime factors
known as quadratic functions or specially multiplicative functions. Euler's function φ ( n ) {\displaystyle \varphi (n)} is a totient function, and the
Multiplicative_function
Function with unusual fractal properties
question-mark function, denoted ?(x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers
Minkowski's question-mark function
Minkowski's_question-mark_function
Class of probability distributions
subset of NEF, called NEF with quadratic variance function (NEF-QVF) because the variance can be written as a quadratic function of the mean. NEF-QVF are discussed
Natural_exponential_family
straight line. Quadratic function: Second degree polynomial, graph is a parabola. Cubic function: Third degree polynomial. Quartic function: Fourth degree
List of mathematical functions
List_of_mathematical_functions
Gives conditions for the solvability of quadratic equations modulo prime numbers
theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime
Quadratic_reciprocity
Hash collision resolution technique
set to 1. Quadratic probing in which the interval between probes increases linearly (hence, the indices are described by a quadratic function). Double
Open_addressing
Variable used for specification
that determine which particular quadratic function is being considered. A parameter could be incorporated into the function name to indicate its dependence
Parameter
Optimization algorithm
a quadratic minimization problem. If the system matrix A {\displaystyle \mathbf {A} } is real symmetric and positive-definite, an objective function is
Gradient_descent
Mathematical operation
second derivative is related to the best quadratic approximation for a function f. This is the quadratic function whose first and second derivatives are
Second_derivative
Polynomial with all terms of degree two
In 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
Quadratic_form
Collective decision-making procedure
various issues. The number of votes to add is determined by a quadratic cost function, which means that the number of votes a person casts for a given
Quadratic_voting
Equation explaining wages via schooling and experience
logarithm of earnings is modelled as the sum of years of education and a quadratic function of "years of potential experience". ln w = f ( s , x ) = ln w
Mincer_earnings_function
Size of a two-dimensional surface
planimeter mechanical device. To find the bounded area between two quadratic functions, we first subtract one from the other, writing the difference as
Area
Measure of difference between two points
{\displaystyle t} , then f {\displaystyle f} is a quadratic function. Proof idea: For any quadratic function q : S → R {\displaystyle q:S\to \mathbb {R} }
Bregman_divergence
Product of a number by itself
function. The dot product of a Euclidean vector with itself is equal to the square of its length: v⋅v = v2. This is further generalised to quadratic forms
Square_(algebra)
Formulation of classical mechanics using momenta
{q}},{\boldsymbol {\dot {q}}})} T {\displaystyle T} is a homogeneous quadratic function in q ˙ {\displaystyle {\boldsymbol {\dot {q}}}} Regarding extensions
Hamiltonian_mechanics
Family of solutions to related differential equations
\alpha t\,dt.\end{aligned}}} Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for Re(ω) > 0): 2 K
Bessel_function
Mathematical concept
quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation
Quadratic_irrational_number
Polynomial equation of degree 4
discriminant of the quadratic function become zero. To explain this, first expand a perfect square so that it equals a quadratic function: ( s u + t ) 2 =
Quartic_equation
Combinatorial optimization method for a family of functions of discrete variables
of functions that can be optimised through graph cuts, such as submodular quadratic functions. Graph cut optimization can be extended to functions of
Graph_cut_optimization
Periodic motion of the atoms of a molecule
harmonic motion. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first
Molecular_vibration
Function defined by multiple sub-functions
mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned
Piecewise_function
Mathematical optimization algorithm
{\displaystyle \mathbf {x} _{*}} is also the unique minimizer of the following quadratic function f ( x ) = 1 2 x T A x − x T b , x ∈ R n . {\displaystyle f(\mathbf
Conjugate_gradient_method
Kind of probability distribution
(or generalized chi-square distribution) is the distribution of a quadratic function of a multinormal variable (normal vector), or a linear combination
Generalized chi-squared distribution
Generalized_chi-squared_distribution
Curve from a cone intersecting a plane
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola
Conic_section
Function used in Lagrangian mechanics
is more common. Moreover, the original theory is generalized from quadratic functions q ↦ R ( q ˙ ) = 1 2 q ˙ ⋅ V q ˙ {\displaystyle q\mapsto R({\dot {q}})={\frac
Rayleigh_dissipation_function
Conjecture on zeros of the zeta function
discriminant of an imaginary quadratic number field K. Assume the generalized Riemann hypothesis for L-functions of all imaginary quadratic Dirichlet characters
Riemann_hypothesis
Method for solving continuous operator problems (such as differential equations)
of a quadratic function representing the system energy and the approximate solution is a linear combination of the given set of the basis functions. Bubnov–Galerkin
Galerkin_method
Function in number theory
the Legendre symbol is a function of a {\displaystyle a} and p {\displaystyle p} defined as ( a p ) = { 1 if a is a quadratic residue modulo p and
Legendre_symbol
the family of quadratic functions having the general form y = a x 2 + b x + c , {\displaystyle y=ax^{2}+bx+c\,,} the simplest function is y = x 2 {\displaystyle
Parent_function
Mathematical proportionality to a square
mathematics, a function or sequence is said to exhibit quadratic growth when its values are proportional to the square of the function argument or sequence
Quadratic_growth
Combinatorial optimization problem
Quadratic unconstrained binary optimization (QUBO), also known as unconstrained binary quadratic programming (UBQP), is a combinatorial optimization problem
Quadratic unconstrained binary optimization
Quadratic_unconstrained_binary_optimization
Loss function used in robust regression
\left(|a|-{\frac {1}{2}}\delta \right),&{\text{otherwise.}}\end{cases}}} This function is quadratic for small values of a, and linear for large values, with equal values
Huber_loss
Quadratic homogeneous polynomial in two variables
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables q ( x , y ) = a x 2 + b x y + c y 2 , {\displaystyle q(x
Binary_quadratic_form
In statistics, the rational quadratic covariance function is used in spatial statistics, geostatistics, machine learning, image analysis, and other fields
Rational quadratic covariance function
Rational_quadratic_covariance_function
Field (mathematics) generated by the square root of an integer
theory, a quadratic field is an algebraic number field of degree two over Q {\displaystyle \mathbf {Q} } , the rational numbers. Every such quadratic field
Quadratic_field
Combinatorial optimization problem
except that the cost function is expressed in terms of quadratic inequalities, hence the name. The formal definition of the quadratic assignment problem
Quadratic_assignment_problem
Least squares approximation of linear functions to data
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems
Linear_least_squares
Function or value which does not change
depend on the main variable(s) in question. For example, a general quadratic function is commonly written as a x 2 + b x + c {\displaystyle ax^{2}+bx+c}
Constant_(mathematics)
Method of calculating an investment's rate of return
cash flows; the NPV is a quadratic function of 1/(1 + r), where r is the rate of return, or put differently, a quadratic function of the discount rate r/(1 + r);
Internal_rate_of_return
Concept in mathematical invariant theory
the biquadratic function of x, y was first brought into notice as an invariant by Mr Boole; and the discriminant of the quadratic function of x, y is identical
Catalecticant
Method for finding stationary points of a function
x_{k+1}=x_{k}+t} . If the second derivative is positive, the quadratic approximation is a convex function of t {\displaystyle t} , and its minimum can be found
Newton's method in optimization
Newton's_method_in_optimization
Statistical method
; Mirkes, E.M.; Zinovyev, A. (2016) "Piece-wise quadratic approximations of arbitrary error functions for fast and robust machine learning." Neural Networks
Lasso_(statistics)
Root of a quadratic polynomial with a unit leading coefficient
number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it
Quadratic_integer
Property of a mathematical matrix
positive definite if and only if its quadratic form is a strictly convex function. More generally, any quadratic function from R n {\displaystyle \mathbb {R}
Definite_matrix
Quantum search algorithm
require exponentially many steps, and Grover's algorithm provides at most a quadratic speedup over the classical solution for unstructured search, this suggests
Grover's_algorithm
Branch of pure mathematics
century. Gauss proved in this work the law of quadratic reciprocity and developed the theory of quadratic forms. He also introduced some basic notation
Number_theory
Subfield of mathematical optimization
the constraints are all linear, but the objective may be a convex quadratic function. Second order cone programming are more general. Semidefinite programming
Convex_optimization
Family of continuous probability distributions
values in the probability mass function of the hypergeometric distribution (which yields the linear-divided-by-quadratic structure). In equation (1), the
Pearson_distribution
Mathematical functions
quartic analog of the (quadratic) π = {\displaystyle \pi =} 3.141592..., ratio of perimeter to diameter of a circle. As complex functions, sl and cl have a
Lemniscate_elliptic_functions
elsewhere. Linear and convex quadratic functions are self-concordant, since their third derivative is zero. Any function f ( x ) = − log ( − g ( x )
Self-concordant_function
Artificial neural network node function
activation functions. Usually the sinusoid is used, as any periodic function is decomposable into sinusoids by the Fourier transform. Quadratic activation
Activation_function
Combinatorial optimization method for pseudo-Boolean functions
Quadratic pseudo-Boolean optimisation (QPBO) is a combinatorial optimization method for minimizing quadratic pseudo-Boolean functions in the form f (
Quadratic pseudo-Boolean optimization
Quadratic_pseudo-Boolean_optimization
Measure of inequality of a statistical distribution
with a quadratic function across pairs of intervals or building an appropriately smooth approximation to the underlying distribution function that matches
Gini_coefficient
The quadratic knapsack problem (QKP), first introduced in 19th century, is an extension of knapsack problem that allows for quadratic terms in the objective
Quadratic_knapsack_problem
Set of methods for supervised statistical learning
problem is a quadratic function of the c i {\displaystyle c_{i}} subject to linear constraints, it is efficiently solvable by quadratic programming algorithms
Support_vector_machine
Nearest integers from a number
Floor and ceiling functions In mathematics, the floor function is the function that takes a real number x as input and returns the greatest integer less
Floor_and_ceiling_functions
Branch of differential geometry
general relativity. On the other hand, replacing the quadratic form by a more general non-quadratic function leads to Finsler geometry. There exists a close
Riemannian_geometry
Theorem for proving more complex theorems
contexts. Often, a theorem is broken into multiple cases (for example, a quadratic function may have no real roots, one double root, or two distinct roots), and
Lemma_(mathematics)
Algorithms for zeros of functions
last two computed points. Three values define a parabolic curve: a quadratic function. This is the basis of Muller's method. Although all root-finding algorithms
Root-finding_algorithm
Algorithms for solving convex optimization problems
\\\end{aligned}}} We assume that the constraint functions belong to some family (e.g. quadratic functions), so that the program can be represented by a
Interior-point_method
Mathematical formula expressing equality
letters at the beginning, a, b, c, d, ... . For example, the general quadratic equation is usually written ax2 + bx + c = 0. The process of finding the
Equation
Multivalued function in mathematics
a quadratic polynomial in x: where r1 and r2 are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function which
Lambert_W_function
scheme – in a 1979 paper. In order to find the cell face value a quadratic function passing through two bracketing or surrounding nodes and one node on
QUICK_scheme
Principle and applications of MINFLUX microscopy
position r → {\displaystyle {\vec {r}}} can be approximated by a quadratic function. Therefore, the recorded number of emission photons is: n ( r → ,
Minflux
Algorithm for finding zeros of functions
and that f is a smooth function. So, even before any computation, it is known that any convergent Newton iteration has a quadratic rate of convergence.
Newton's_method
Number divisible only by 1 and itself
modulo large primes. As well as in the hash function, prime numbers are used for the hash table size in quadratic probing based hash tables to ensure that
Prime_number
Special functions of several complex variables
including abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions in two dimensions are functions of two complex arguments. In one choice
Theta_function
Method for estimating new data within known data points
Online tools for linear Archived 2016-09-18 at the Wayback Machine, quadratic Archived 2016-09-18 at the Wayback Machine, cubic spline Archived 2016-08-20
Interpolation
Operation in calculus
the function yield better approximations to the integral, can be carried further: Simpson's rule approximates the integrand by a piecewise quadratic function
Integral
Function defined by a hypergeometric series
then there is a quadratic transformation of the hypergeometric function, connecting it to a different value of z related by a quadratic equation. The first
Hypergeometric_function
1993 Canadian TV series or program
program uses computer animation to demonstrate quadratic equations and their corresponding functions in the Cartesian coordinate system. Each program
Quadratics
Function used in signal processing
processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside
Window_function
Number of integers coprime to and less than n
( x ) {\displaystyle \log _{e}(x)} . In number theory, Euler's totient function counts the positive integers up to a given integer n {\displaystyle n}
Euler's_totient_function
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