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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
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
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
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
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
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
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
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
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
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
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
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
Operation in calculus
of calculus relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative
Integral
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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)
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
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
Function in probability theory
function Correlation function Covariance matrix Covariance operator – Operator in probability theory Kriging Positive-definite kernel Random field Stochastic
Covariance_function
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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, inverse of an exponential function
{\displaystyle f} is invertible when considered as a function from the reals to the positive reals. Let b be a positive real number not equal to 1 {\displaystyle
Logarithm
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
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
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
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
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
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
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
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
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
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
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
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
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)
Probability distribution on complex matrices
variables. It has support for p × p {\displaystyle p\times p} Hermitian positive definite matrices. The complex Wishart distribution is also encountered in
Complex_Wishart_distribution
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
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
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
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
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
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
(and positive eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly
Totally_positive_matrix
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
Mathematical function
Slepian functions are solutions to either of these types of equations with positive-definite kernels, that is, they are bandlimited functions G = L F
Slepian_function
{\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
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
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
Indian statistician (1936–2023)
JSTOR 25051432. Bhatia, Rajendra; Parthasarathy, K. R. (2000). "Positive Definite Functions and Operator Inequalities". Bulletin of the London Mathematical
K. R. Parthasarathy (probabilist)
K._R._Parthasarathy_(probabilist)
theorem Plancherel's theorem Convolution Convolution theorem Positive-definite function Poisson summation formula Paley–Wiener theorem Sobolev space Time–frequency
List of harmonic analysis topics
List_of_harmonic_analysis_topics
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
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
, the condition is not verified even though the second function is globally positive definite. Terrell, William J. (2009), Stability and stabilization
Radially_unbounded_function
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
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
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
Mathematical function often applied to matrices
the induced operator norm. It quantifies key notions such as positive/negative definiteness in matrix theory, uniformly coercive or monotone vector fields
Logarithmic_norm
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
Romanian-American mathematician (1903–1990)
data by analytic functions, Quart. Appl. Math., vol. 4, pp. 45–99 and 112–141, 1946. Schoenberg, I. J. (1942). "Positive definite functions on spheres". Duke
Isaac_Jacob_Schoenberg
Lattice in 8-dimensional space with special properties
is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact
E8_lattice
On when an integer positive definite quadratic form represents all positive integers
Q(x)=x^{t}Qx=\sum _{i,j}x_{i}Q_{ij}x_{j}} This function is called a quadratic form. We say Q {\displaystyle Q} is positive definite if Q ( x ) > 0 {\displaystyle Q(x)>0}
15_and_290_theorems
Matrix B such that B² equals a given matrix A
semidefinite matrix is real. The principal square root of a positive definite matrix is positive definite; more generally, the rank of the principal square root
Square_root_of_a_matrix
{\displaystyle C_{0}} : there is a sequence of normalized continuous positive-definite functions ϕ n {\displaystyle \phi _{n}} which vanish at infinity on G {\displaystyle
Haagerup_property
{\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
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
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
Positive, Suitable
Girl/Female
Tamil
Positive energy, Horseless
Boy/Male
Tamil
Positive, Suitable
Boy/Male
Hindu, Indian
Positive Thinking Person
Boy/Male
Hindu, Indian
Positive
Boy/Male
Indian, Tamil
Positive; The Lord Ganesh
Boy/Male
Hindu, Indian, Japanese
Yonit; Good; Positive
Boy/Male
Hindu
Positive, Courageous, Resilient, Independent
Boy/Male
Hindu, Indian, Tamil
Positive Energy
Boy/Male
Hindu
Positive energy, Horseless
Boy/Male
Tamil
Positive, Suitable
Boy/Male
Indian
Positive Power
Girl/Female
Arabic, Australian, Muslim
Inspiring; Positive Attitude
Boy/Male
Arabic, Australian, Chinese, German, Muslim
Definite; Decisive
Boy/Male
Hindu, Indian
Positive Thinking
Boy/Male
Hindu
Positive, Suitable
Girl/Female
Hindu
Positive energy, Horseless
Boy/Male
Tamil
Positive energy, Horseless
Boy/Male
Tamil
Anirved | அநீரà¯à®µà¯‡à®¤
Positive, Courageous, Resilient, Independent
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
POSITIVE DEFINITE-FUNCTION
a.
Having certain limits in signification; determinate; certain; precise; fixed; exact; clear; as, a definite word, term, or expression.
n.
The positive plate of a voltaic or electrolytic cell.
a.
Having certain or distinct; determinate in extent or greatness; limited; fixed; as, definite dimensions; a definite measure; a definite period or interval.
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.
adv.
In a definite manner; with precision; precisely; determinately.
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.
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.
Electro-positive.
a.
Definitely laid down; explicitly stated; clearly expressed; -- opposed to implied; as, a positive declaration or promise.
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.
Determinate; positive; final; conclusive; unconditional; express.
a.
Having the power of direct action or influence; as, a positive voice in legislation.
p. pr. & vb. n.
of Define
n.
An infinite quantity or magnitude.
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.
n.
The positive degree or form.
a.
Boundless; infinite.
a.
Serving to define or restrict; limiting; determining; as, the definite article.
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.
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.