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Bimodal function
In mathematics, a positive-definite function is, depending on the context, either of two types of function. Let R {\displaystyle \mathbb {R} } be the set
Positive-definite_function
Index of articles associated with the same name
positive-definite. See, in particular: Positive-definite bilinear form Positive-definite function Positive-definite function on a group Positive-definite functional
Positive_definiteness
Generalization of a positive-definite matrix
branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced
Positive-definite_kernel
Property of a mathematical matrix
with real entries is positive-definite if the real number x T M x {\displaystyle \mathbf {x} ^{\mathsf {T}}M\mathbf {x} } is positive for every nonzero real
Definite_matrix
and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and
Positive-definite function on a group
Positive-definite_function_on_a_group
Type of homogeneous polynomial of degree 2
that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the
Definite_quadratic_form
Mathematical theorem
representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in
Mercer's_theorem
In functional analysis, a Hilbert space
symmetric and positive definiteness follows from the properties of inner product in F {\displaystyle F} . Conversely, every positive definite function and corresponding
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Topics referred to by the same term
mathematics, positive semidefinite may refer to: Positive semidefinite function Positive semidefinite matrix Positive semidefinite operator Positive semidefinite
Positive_semidefinite
_{r\rightarrow \infty }\alpha (r)=\infty } . A nondecreasing positive definite function β {\displaystyle \beta } satisfying all conditions of class K
Class_kappa_function
Theorem of Fourier transforms of Borel measures
Fourier transform a continuous positive-definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual
Bochner's_theorem
Concept in the analysis of dynamical systems
Lyapunov-candidate-function V {\displaystyle V} is locally positive definite, and the time derivative of the Lyapunov-candidate-function is locally negative definite: V
Lyapunov_function
Positive-real functions, often abbreviated to PR function or PRF, are a kind of mathematical function that first arose in electrical network synthesis
Positive-real_function
Type of mathematical function
These radial basis functions are from C ∞ ( R ) {\displaystyle C^{\infty }(\mathbb {R} )} and are strictly positive-definite functions that require tuning
Radial_basis_function
Extension of the factorial function
every positive integer n {\displaystyle n} . The gamma function can be defined via a convergent improper integral for complex numbers with positive real
Gamma_function
Operation in mathematical calculus
of calculus relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative
Integral
Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients. A positive function f on the unit disk with f(0) = 1 is harmonic
Positive_harmonic_function
Method in approximation theory
non-singular is exactly the definition of a strictly positive definite function. Such functions, including the Gaussian, inverse quadratic, and inverse
Radial basis function interpolation
Radial_basis_function_interpolation
Probability distribution
}}\end{cases}}} The probability density function of the symmetric generalized normal distribution is a positive-definite function for β ∈ ( 0 , 2 ] {\displaystyle
Generalized normal distribution
Generalized_normal_distribution
Mathematical function
= C {\displaystyle C^{\mathsf {T}}=C} , and positive-definite. The following integrals with this function can be calculated with the same technique: ∫
Gaussian_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
for sigmoid functions not evident or intuitive M1: Inverse of singularity functions M2: Sigmoid functions of embedded positive functions M3: Rising a
Sigmoid_function
Negative of a convex function
nonnegative-definite matrix B, is concave. Rays bending in the computation of radiowave attenuation in the atmosphere involve concave functions. In expected
Concave_function
Function with a repeating pattern
{\displaystyle f(x)} is an integrable periodic function with period P {\displaystyle P} , then its definite integral over any interval of length P {\displaystyle
Periodic_function
Matrix of second derivatives
a polynomial of degree 3. The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test whether a critical
Hessian_matrix
Real function with secant line between points above the graph itself
means that ∇ 2 f ( x ) − m I {\displaystyle \nabla ^{2}f(x)-mI} is positive semi-definite. This is equivalent to requiring that the minimum eigenvalue of
Convex_function
enveloping algebra of G. Thus a normalised positive definite K-biinvariant function f on G is a zonal spherical function if and only if for each D in D(G/K)
Zonal_spherical_function
Function that is continuous everywhere but differentiable nowhere
Differentialquotienten" (Mr. Weierstrass read [a paper] about continuous functions without definite [i.e., well-defined] derivatives [to members of the Academy])
Weierstrass_function
Types of special mathematical functions
Scott, Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions, AAECC (Applicable Algebra in
Incomplete_gamma_function
Linear combination of indicator functions of real intervals
{\displaystyle x\in A_{i}.} The definite integral of a step function is a piecewise linear function. The Lebesgue integral of a step function f = ∑ i = 0 n α i χ
Step_function
Function returning minus 1, zero or plus 1
of a given real number is positive or negative, or the given number is itself zero. In mathematical notation the sign function is often represented as sgn
Sign_function
Special mathematical function defined as sin(x)/x
causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value
Sinc_function
Number whose square is a given number
terms of nth roots. If A is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator B with B2 =
Square_root
Linear mathematical operator which translates a function
of almost periodic functions, positive-definite functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear
Shift_operator
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
Distance from zero to a number
generalization of this notion to other domains: Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition.
Absolute_value
Special function defined by an integral
is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument
Exponential_integral
Mathematical function
§ Relationship to the gamma function. The beta function is also closely related to binomial coefficients. When m (or n, by symmetry) is a positive integer, it follows
Beta_function
Family of solutions to related differential equations
amplitudes chosen for the functions originate from the early work in which the functions appeared as solutions to definite integrals rather than solutions
Bessel_function
Mathematical function, denoted exp(x) or e^x
for definiting the complex exponential function, and the proof of their equivalence is the same as in the real case. The complex exponential function can
Exponential_function
Generalized function whose value is zero everywhere except at zero
acting on functions. The graph of the Dirac delta is usually thought of as following the whole x {\displaystyle x} -axis and the positive y {\displaystyle
Dirac_delta_function
Bounded operators with sub-unit norm
operator-valued positive-definite function arises in this way. Recall that every (continuous) scalar-valued positive-definite function on a topological
Contraction_(operator_theory)
Special function defined by an integral
integral has an integral representation defined for all positive real numbers x ≠ 1 by the definite integral li ( x ) = ∫ 0 x d t ln t . {\displaystyle
Logarithmic_integral_function
Matrix decomposition method
(pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate
Cholesky_decomposition
Vector space with generalized dot product
form in question need not be positive definite (need not be an inner product). Bilinear form – Scalar-valued bilinear function Biorthogonal system – Pair
Inner_product_space
Integral of the Gaussian function, equal to sqrt(π)
}}\right)}^{N}} for any positive-definite symmetric matrix A {\displaystyle A} . Suppose A is a symmetric positive-definite (hence invertible) n × n
Gaussian_integral
Topics referred to by the same term
The term positive map may refer to: Positive-definite functions in classical analysis Choi's theorem on completely positive maps between C*-algebras (pronounced
Positive_map
Mathematical function, inverse of an exponential function
This function is written as f(x) = b x. When b is positive and unequal to 1, we show below that f is invertible when considered as a function from the
Logarithm
Mathematical functions having established names and notations
Analogues of several special functions have been defined on the space of positive definite matrices, among them the power function which goes back to Atle
Special_functions
Stability notion for nonlinear control systems with external inputs
{\displaystyle j\neq i} , χ i i := 0 {\displaystyle \chi _{ii}:=0} and a positive-definite function α i {\displaystyle \alpha _{i}} , such that: ψ i 1 ( | x i | )
Input-to-state_stability
Indefinite integral
definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is
Antiderivative
Smooth and compactly supported function
serve as a bump function. When b < c the plateau has positive length; when b = c it degenerates to a single point, but the function is still smooth.
Bump_function
Inverse functions of sin, cos, tan, etc.
at one point gives an expression for the inverse trigonometric function as a definite integral: arcsin ( x ) = ∫ 0 x 1 1 − z 2 d z , | x | ≤ 1 arccos
Inverse trigonometric functions
Inverse_trigonometric_functions
Tool in linear algebra and matrix analysis
}\succ 0.} If A is positive definite, then X is positive semi-definite if and only if the complement X/A is positive semi-definite: If A ≻ 0 , then
Schur_complement
Function in probability theory
function Correlation function Covariance matrix Covariance operator – Operator in probability theory Kriging Positive-definite kernel Random field Stochastic
Covariance_function
theta function of a lattice is a function whose coefficients give the number of vectors of a given norm. One can associate to any (positive-definite) lattice
Theta_function_of_a_lattice
Integral of sin(x)/x from 0 to infinity
integral, an antiderivative of the sinc function, is not an elementary function. In this case, the improper definite integral can be determined in several
Dirichlet_integral
Mathematical description of quantum state
numbers) labeling different solutions, the strictly positive function w is called a weight function, and δmn is the Kronecker delta. The integration is
Wave_function
Multivalued function in mathematics
{x}{\ddots }}}}}}.} There are several useful definite integral formulas involving the principal branch of the W function, including the following: ∫ 0 π W 0 (
Lambert_W_function
Solving an optimization problem with a quadratic objective function
(component-wise inequality). As a special case when Q is symmetric positive-definite, the cost function reduces to least squares: where Q = RTR follows from the
Quadratic_programming
Mathematical function
in functional analysis, a seminorm is like a norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm
Seminorm
Special functions of several complex variables
representation of the Heisenberg group. If F is a positive-definite quadratic form in n variables, then the theta function associated with F is θ F ( z ) = ∑ m ∈
Theta_function
Function related to statistics and probability theory
_{r}}}\ {\frac {\partial \log f}{\partial \theta _{s}}}\ f\,dz} is positive definite and | I ( θ ) | {\textstyle \,\left|\mathbf {I} (\theta )\right|\
Likelihood_function
Methods of calculating definite integrals
solution to a definite integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} to a given degree of accuracy. If f(x) is a smooth function integrated
Numerical_integration
matrices with the positive elements being the positive-definite matrices. The trace function defined on this C*-algebra is a positive functional, as the
Positive_linear_functional
Mathematical function with multiple real-number arguments
analytic function and equals its Taylor series about any point in the domain, the notation Cω denotes this differentiability class. Definite integration
Function of several real variables
Function_of_several_real_variables
Optimization algorithm
large problems. When f {\displaystyle f} is a convex quadratic function with positive-definite Hessian B {\displaystyle B} , one would expect the matrices
Quasi-Newton_method
equivalent statement, to the effect that a certain generalized function is positive definite. Weil's idea was formulated first in a 1952 paper. It is based
Weil's_criterion
Study of mathematical algorithms for optimization problems
function global minimum. Further, critical points can be classified using the definiteness of the Hessian matrix: If the Hessian is positive definite
Mathematical_optimization
About mathematical functions
"AXIOM III. (Axiom of separation). Whenever the propositional function Φ(x) is definite for all elements of a set M, M possesses a subset MΦ containing
History of the function concept
History_of_the_function_concept
Probability distribution
distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior
Inverse-Wishart_distribution
Branch of mathematics
upper-case letters for a function and its indefinite integral is common in calculus.) The definite integral inputs a function and outputs a number, which
Calculus
Generalization of gamma distribution to multiple dimensions
density function as follows: Let X be a p × p symmetric matrix of random variables that is positive semi-definite. Let V be a (fixed) symmetric positive definite
Wishart_distribution
Generalization of definite integrals to functions of multiple variables
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis
Multiple_integral
Partial order on matrices
of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar functions to monotone
Loewner_order
Polynomial sequence
constant. Gegenbauer polynomials also appear in the theory of positive-definite functions. The Askey–Gasper inequality reads ∑ j = 0 n C j α ( x ) ( 2
Gegenbauer_polynomials
Class of algorithms for pattern analysis
… , c n ) {\displaystyle (c_{1},\dots ,c_{n})} (cf. positive definite kernel), then the function k {\displaystyle k} satisfies Mercer's condition. Some
Kernel_method
associated to a positive definite lattice, generalizing the 1-variable theta function of a lattice. Suppose that L is a positive definite lattice. The Siegel
Siegel_theta_series
algebraic function over an algebraic domain is known as a period. The following is a list of some of the most common or interesting definite integrals
List_of_definite_integrals
Conjecture on zeros of the zeta function
the unlikely event that one could show the existence of a suitable positive definite inner product on this space, the Riemann hypothesis would follow.
Riemann_hypothesis
Length in a vector space
Although this article defined "positive" to be a synonym of "positive definite", some authors instead define "positive" to be a synonym of "non-negative";
Norm_(mathematics)
Algebras arising in harmonic analysis
linear span of the set P ( G ) {\displaystyle P(G)} of continuous positive-definite functions on G {\displaystyle G} . Since L 1 ( G ^ ) {\displaystyle L_{1}({\hat
Fourier_algebra
x\rangle } where A = A T ≥ 0 {\displaystyle A=A^{T}\geq 0} is a positive semi-definite symmetric matrix, the logarithmic barrier f ( x ) = − log ϕ (
Self-concordant_function
Topics referred to by the same term
discount factor used in mathematical finance Positive-definite kernel, a generalization of a positive-definite matrix Kernel trick, in statistics Reproducing
Kernel
Generalization of the concept from statistical mechanics
derivatives with regard to the Lagrange multipliers gives rise to a positive semi-definite covariance matrix g i j ( β ) = ∂ 2 ∂ β i ∂ β j ( − log Z ( β
Partition function (mathematics)
Partition_function_(mathematics)
Function in control theory
control-Lyapunov function (CLF) is a function V : D → R {\displaystyle V:D\to \mathbb {R} } that is continuously differentiable, positive-definite (that is,
Control-Lyapunov_function
Special mathematical functions defined on the surface of a sphere
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving
Spherical_harmonics
, the condition is not verified even though the second function is globally positive definite. Terrell, William J. (2009), Stability and stabilization
Radially_unbounded_function
Uniform restraint of the change in functions
In mathematics, a real function f {\displaystyle f} of real numbers is said to be uniformly continuous if there is a positive real number δ {\displaystyle
Uniform_continuity
Function that takes two inputs
appropriate size and M is a matrix. If M is a positive definite matrix, this yields an inner product. Functions whose domain is a subset of R 2 {\displaystyle
Binary_function
{\displaystyle {\mathcal {H}}_{k}} is a Hilbert space of functions defined by a symmetric, positive-definite function k : X × X → R {\displaystyle k:{\mathcal {X}}\times
Bayesian interpretation of kernel regularization
Bayesian_interpretation_of_kernel_regularization
Type of mathematical function
dom f and 0 < θ < 1. If f is strictly positive, this is equivalent to saying that the logarithm of the function, log ∘ f, is concave; that is, log f
Logarithmically concave function
Logarithmically_concave_function
Mathematics term
dual of G with Fell topology. (2) Any sequence of continuous positive definite functions on G converging to 1 uniformly on compact subsets, converges
Kazhdan's_property_(T)
English words "the", "a(n)", and sometimes "some"
and [ ] are used here, see this page. The articles in English are the definite article the and the indefinite article a (which takes the alternate form
English_articles
(and positive eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly
Totally_positive_matrix
Determiners in the English language
meanings to the noun phrase, such as definiteness, proximity, number, and the like. While the determinative function is typically realized by determiner
English_determiners
Multivariate generalization of the gamma function
as the following integral over the p × p {\displaystyle p\times p} positive-definite real matrices: Γ p ( a ) = ∫ S > 0 exp ( − t r ( S ) ) | S | a −
Multivariate_gamma_function
Concept in mathematical analysis
mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral
Improper_integral
{\displaystyle f} and whose difference A − B {\displaystyle A-B} is a positive semi-definite matrix, then necessarily f ( A ) − f ( B ) ≥ 0 {\displaystyle f(A)-f(B)\geq
Operator_monotone_function
r>0\right\}\end{aligned}}} Functions of class P {\displaystyle {\mathcal {P}}} are also called positive-definite functions. One of the most important
Comparison_function
Transcendental single-variable function
the sine function inside the absolute value sign remains strictly positive, so the absolute value signs may be omitted. The Clausen function also has
Clausen_function
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
Boy/Male
Hindu, Indian, Japanese
Yonit; Good; Positive
Girl/Female
Tamil
Positive energy, Horseless
Boy/Male
Hindu, Indian, Tamil
Positive Energy
Boy/Male
Tamil
Anirved | அநீரà¯à®µà¯‡à®¤
Positive, Courageous, Resilient, Independent
Anirved | அநீரà¯à®µà¯‡à®¤
Boy/Male
Indian
Positive Power
Boy/Male
Arabic, Australian, Chinese, German, Muslim
Definite; Decisive
Boy/Male
Tamil
Positive, Suitable
Boy/Male
Hindu, Indian
Positive Thinking
Boy/Male
Hindu
Positive, Suitable
Boy/Male
Hindu, Indian
Positive Thinking Person
Boy/Male
Hindu
Positive energy, Horseless
Girl/Female
Arabic, Australian, Muslim
Inspiring; Positive Attitude
Boy/Male
Tamil
Positive, Suitable
Boy/Male
Indian, Tamil
Positive; The Lord Ganesh
Boy/Male
Hindu
Positive, Suitable
Boy/Male
Tamil
Positive energy, Horseless
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Mythological, Sanskrit, Tamil, Telugu, Traditional
Powerful; Positive Thinker; Self Confidence; Positive; Frank; Powerful Character of Mahabharat
Boy/Male
Hindu, Indian
Positive
Girl/Female
Hindu
Positive energy, Horseless
Boy/Male
Hindu
Positive, Courageous, Resilient, Independent
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
Girl/Female
Hindu, Indian
Momentary; Twinkling of Eye
Girl/Female
Hindu
Desh ki Bhoomi mitii
Boy/Male
Tamil
To make melodic sounds, Chanting
Girl/Female
Indian, Sanskrit, Telugu
Blessing; Fortune; Luck
Boy/Male
Hindu
Famous, Scholar
Girl/Female
Hebrew
a village near Jerusalem where Jesus visited Mary; Martha and Lazarus.
Boy/Male
Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Wealth
Boy/Male
Sikh
One who delights in devotion to God
Male
Italian
Italian, Portuguese and Spanish form of Latin Eusebius, EUSEBIO means "pious."
Male
English
English form of Latin Adolphus, ADOLPH means "noble wolf."
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
adv.
In a definite manner; with precision; precisely; determinately.
n.
The positive degree or form.
a.
Having no determined or certain limits; large and unmeasured, though not infinite; unlimited; as indefinite space; the indefinite extension of a straight line.
a.
Serving to define or restrict; limiting; determining; as, the definite article.
n.
An infinite quantity or magnitude.
a.
Having certain limits in signification; determinate; certain; precise; fixed; exact; clear; as, a definite word, term, or expression.
a.
Electro-positive.
a.
Definitely laid down; explicitly stated; clearly expressed; -- opposed to implied; as, a positive declaration or promise.
a.
Having certain or distinct; determinate in extent or greatness; limited; fixed; as, definite dimensions; a definite measure; a definite period or interval.
p. pr. & vb. n.
of Define
a.
Corresponding with the original in respect to the position of lights and shades, instead of having the lights and shades reversed; as, a positive picture.
a.
Without limit in power, capacity, knowledge, or excellence; boundless; immeasurably or inconceivably great; perfect; as, the infinite wisdom and goodness of God; -- opposed to finite.
n.
The positive plate of a voltaic or electrolytic cell.
a.
Not definite; not limited, defined, or specified; not explicit; not determined or fixed upon; not precise; uncertain; vague; confused; obscure; as, an indefinite time, plan, etc.
a.
Having the power of direct action or influence; as, a positive voice in legislation.
a.
Determinate; positive; final; conclusive; unconditional; express.
a.
Having a limit; limited in quantity, degree, or capacity; bounded; -- opposed to infinite; as, finite number; finite existence; a finite being; a finite mind; finite duration.
a.
Boundless; infinite.
a.
Hence: Not admitting of any doubt, condition, qualification, or discretion; not dependent on circumstances or probabilities; not speculative; compelling assent or obedience; peremptory; indisputable; decisive; as, positive instructions; positive truth; positive proof.
n.
A word used to define or limit the extent of the signification of a common noun, such as the definite article, and some pronouns.