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Hessian matrix (3,539 words) [view diff] no match in snippet view article find links to article

In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function

continuously (for a real-valued function of a real variable) or discretely (for a sequence of real numbers, i.e., real-valued function of an integer or natural

approximate value for the box-counting dimension of the graph of a real-valued function or time series. This value is obtained via an algorithmic approximation

phase or simply phase) of a complex-valued function s(t), is the real-valued function: φ ( t ) = arg ⁡ { s ( t ) } , {\displaystyle \varphi (t)=\arg\{s(t)\}

other by the Hilbert transform. The analytic representation of a real-valued function is an analytic signal, comprising the original function and its Hilbert

that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum

In mathematics, the support of a real-valued function f {\displaystyle f} is the subset of the function domain of elements that are not mapped to zero

function is said to be bounded (from) below by B {\displaystyle B} . A real-valued function is bounded if and only if it is bounded from above and below.[additional

sometimes refers to entrywise ordering, e.g. the real-valued function f majorizes the real-valued function g when f ( x ) ≥ g ( x ) {\displaystyle f(x)\geq

set in multidimensional space is defined by a single continuous real-valued function f ( X ) {\displaystyle f(X)} of point coordinates X [ x 1 , x 2

notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifically

A kernel smoother is a statistical technique to estimate a real valued function f : R p → R {\displaystyle f:\mathbb {R} ^{p}\to \mathbb {R} } as the weighted

a Banach functional, on a vector space X {\displaystyle X} is a real-valued function with only some of the properties of a seminorm. Unlike seminorms

In mathematics, the positive part of a real or extended real-valued function is defined by the formula f + ( x ) = max ( f ( x ) , 0 ) = { f ( x )  if 

In mathematics, a real-valued function u ( x , y ) {\displaystyle u(x,y)} defined on a connected open set Ω ⊂ R 2 {\displaystyle \Omega \subset \mathbb

a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of

\varphi (x)=0\right\rbrace } where φ {\displaystyle \varphi } is a real-valued function. The set O {\displaystyle {\mathcal {O}}} is said to be positively

manifold, the lower bound for the number of critical points that a real-valued function on X {\displaystyle X} could possess (this should be compared with

In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse

problem is as follows: given a closed, smooth manifold M and a smooth, real-valued function ƒ on M, construct a Riemannian metric on M whose scalar curvature

analysis and variational analysis, a point at which some given extended real-valued function is minimized is typically sought, where such a point is called a

finite-dimensional component and an infinite-dimensional component (often a real-valued function defined on the real line). Thus, Θ ⊆ R k × V {\displaystyle \Theta

differentiable, and no derivatives are taken. The function must be a real-valued function of a fixed number of real-valued inputs. The caller passes in the

minimization problem once again has an immediate answer. Extended real-valued function for which the minimization problem is not solved by any one of these

concept is the complex-variable version of the antiderivative of a real-valued function. The derivative of a constant function is the zero function. Therefore

named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also

In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging

Corput. The following result is stated by E. Stein: Suppose that a real-valued function ϕ ( x ) {\displaystyle \phi (x)} is smooth in an open interval (

In mathematics, the tent map with parameter μ is the real-valued function fμ defined by f μ ( x ) := μ min { x , 1 − x } , {\displaystyle f_{\mu }(x):=\mu

spaces, the Palais–Smale condition for a continuously differentiable real-valued function is satisfied automatically for proper maps: functions which do not

to be the set of points of non-differentiability of a continuous real-valued function, is that it be the union of a Gδ set and a G δ σ {\displaystyle {G_{\delta

symbolically x'′ = Rx'. Symbolically, the rotation invariance of a real-valued function of two real variables is f ( x ′ ) = f ( R x ) = f ( x ) {\displaystyle

a smooth, real-valued function exactly in a convenient manner. Hadamard's lemma—Let f {\displaystyle f} be a smooth, real-valued function defined on

volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may

measuring function φ {\displaystyle \varphi } is a non-negative extended real-valued function defined for all subsets of R {\displaystyle \mathbb {R} } . Translation

measuring function φ {\displaystyle \varphi } is a non-negative extended real-valued function defined for all subsets of R {\displaystyle \mathbb {R} } . Translation

particular in measure theory, a content μ {\displaystyle \mu } is a real-valued function defined on a collection of subsets A {\displaystyle {\mathcal {A}}}

functions. This article first explores the notion of a jet of a real valued function in one real variable, followed by a discussion of generalizations

functions of several complex variables, a pluriharmonic function is a real valued function which is locally the real part of a holomorphic function of several

σp under the exponential map at p). The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold. The sectional curvature

Luus–Jaakola (LJ) denotes a heuristic for global optimization of a real-valued function. In engineering use, LJ is not an algorithm that terminates with

In mathematical analysis, a Pompeiu derivative is a real-valued function of one real variable that is the derivative of an everywhere differentiable function

In calculus, the extreme value theorem states that if a real-valued function f {\displaystyle f} is continuous on the closed and bounded interval [ a

functions one would encounter. Stated precisely, suppose that f is a real-valued function defined on some open interval containing the point x and suppose

In mathematics, an unfolding of a smooth real-valued function ƒ on a smooth manifold, is a certain family of functions that includes ƒ. Let M {\displaystyle

{\displaystyle f:\mathbb {N} \rightarrow \mathbb {R} ^{+}} is a real-valued function taking on positive values, then for almost all α {\displaystyle \alpha

{\displaystyle P(y',y'',y''',...,y^{(n)})=0} is a UDE if for any continuous real-valued function f {\displaystyle f} and for any positive continuous function ε {\displaystyle

described as a minimization problem because the maximization of the real-valued function g ( x ) {\displaystyle g(x)} is equivalent to the minimization of

real-valued functions that is weaker than continuity. An extended real-valued function f {\displaystyle f} is upper (respectively, lower) semicontinuous

integral of the absolute value over the whole domain is finite. For a real-valued function, since ∫ | f ( x ) | d x = ∫ f + ( x ) d x + ∫ f − ( x ) d x {\displaystyle

Pierre-Simon Laplace and Eugenio Beltrami. For any twice-differentiable real-valued function f defined on Euclidean space Rn, the Laplace operator (also known

Hopf lemma, named after Eberhard Hopf, states that if a continuous real-valued function in a domain in Euclidean space with sufficiently smooth boundary

when we define metric space the distance function is taken to be a real-valued function. The real numbers form an ordered field which is Archimedean and

role played by graphs of real-valued function in real analysis. Specifically, the epigraph of an extended real-valued function provides geometric intuition

function f : A → R {\displaystyle f\colon A\to \mathbb {R} } is a real-valued function, then the unweighted sum of f {\displaystyle f} on A {\displaystyle

mathematician Oskar Schlömilch, who derived the series in 1857. The real-valued function f ( x ) {\displaystyle f(x)} has the following expansion: f ( x )

is the relation "is infinitely close to". In order to make f ' a real-valued function, the final term Δ x {\displaystyle \Delta x} is dispensed with. In

F(x,y,u,u_{x},u_{y})=0\qquad \qquad (1)} be a PDE for an unknown real-valued function u in two variables x and y. Assume that this PDE is non-degenerate

mathematical object reflecting the evolution of the level sets of a real-valued function on a manifold. According to a similar concept was introduced by G

Khinchin integrals. Let I = [a,b] be an interval and f : I → R be a real-valued function on I. Recall that f is absolutely continuous on a subset E of I if

Thomae's function is a real-valued function of a real variable that can be defined as:: 531  f ( x ) = { 1 q if  x = p q ( x  is rational), with  p ∈ Z

violation of the Baire category theorem. The continuity set of any real valued function is a Gδ subset of its domain (see the "Properties" section for a

( x , y , z ) {\displaystyle f(x,y,z)} is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another

{\displaystyle V_{i}} is the value measure of partner i. It is a real-valued function that, for every piece of cake, returns a number that represents the

Morse Lemma. For a generalization, see Morse–Palais lemma). A smooth real-valued function on a manifold M {\displaystyle M} is a Morse function if it has no

motivate the more general notion of a distance function as follows: A real valued function d on a set X × X is called a metric (or a distance function) on X

{\displaystyle H(\omega )=A(\omega )\ e^{-j\omega \tau },} where: A(ω) is a real-valued function. τ {\displaystyle \tau } is the group delay. For a discrete-time

measure algebra is a Boolean algebra B with a measure m, which is a real-valued function on B such that: m(0)=0, m(1)=1 m(x) >0 if x≠0 m is countably additive:

zero vector can be inflated to include the set Bounded variation, a real-valued function whose total variation is bounded Bounded pointer, a pointer that

improper, the same answer will result. In the simplest case of a real-valued function of a single variable integrated in the sense of Riemann (or Darboux)

proved that an agent is (VNM-)rational if and only if there exists a real-valued function u defined by possible outcomes such that every preference of the

discontinuities of a real-valued function of a real variable. In this theorem Froda proves that the set of simple discontinuities of a real-valued function of a real

Tychonoff corkscrew, which contains two points such that any continuous real-valued function on the space has the same value at these two points. An even more

function is zero. The theorem is named after Michel Rolle. If a real-valued function f is continuous on a proper closed interval [a, b], differentiable

Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞;

the corresponding concept is the sample maximum and minimum. A real-valued function f defined on a domain X has a global (or absolute) maximum point

}{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2t}}\right)} ; does not exist as real valued function CF 2 Γ ( ν 2 ) ( − i t 2 ) ν 4 K ν 2 ( − 2 i t ) {\displaystyle {\frac

Equivalently, we can define the C*-enveloping algebra as follows: Define a real valued function on A by ‖ x ‖ C ∗ = sup f f ( x ∗ x ) {\displaystyle \|x\|_{\operatorname

(X,\tau )} be a Hausdorff locally convex space, for any extended real valued function f : X → R ∪ { ± ∞ } {\displaystyle f:X\to \mathbb {R} \cup \{\pm

to the Loewner differential equation, which depends on a certain real-valued function, the driving function, defined on the boundary of the domain. The

subset of ℝm. For each i and j between 1 and n, let gij be a smooth real-valued function on U, such that for each p in U, one has that the n × n matrix [gij

to the object of interest an infinite-dimensional manifold and a real valued function on it. In the symplectic version, this is the free loop space of

[a, b] be a fixed closed, bounded interval in the real line R. A real-valued function φ : [a, b] → R is called a step function if there exists a finite

Bethesda, Maryland Bounded mean oscillation, in harmonic analysis, a real-valued function whose mean oscillation is bounded BMO Field, a stadium located in

states that if f {\displaystyle f} is a continuously differentiable real-valued function on the real line, and f ( a ) = {\displaystyle f(a)=} f ( b ) = 0

characterize support functions: Any positive homogeneous, convex, real valued function on R n {\displaystyle \mathbb {R} ^{n}} is the support function of

Measure of local oscillation behavior Bounded mean oscillation – real-valued function whose mean oscillation is boundedPages displaying wikidata descriptions

true in general. Let X {\displaystyle X} be a topological space. A real-valued function f : X → R {\displaystyle f:X\rightarrow \mathbb {R} } is quasi-continuous

is a measure space, f {\displaystyle f} is a measurable extended real-valued function, and ε > 0, then μ ( { x ∈ X : | f ( x ) | ≥ ε } ) ≤ 1 ε ∫ X | f

used by Leibniz to define the derivative and the integral. For any real-valued function f , {\displaystyle f,} the differential d f {\displaystyle df} is

a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function

continuous functions follows almost immediately from this theorem. If a real-valued function f {\displaystyle f} is continuous on [ 0 , ∞ ) {\displaystyle [0

manifold forms a Poisson algebra. On a symplectic manifold, every real-valued function H on the manifold induces a vector field XH, the Hamiltonian vector

subset of a Banach space V and let f : K → R ∪ {+∞} be an extended real-valued function that is Fréchet differentiable with derivative df(x) : V → R at each

another if the former is a refinement of the latter. Let f be a real-valued function defined on the interval [a, b]. The Riemann sum of f with respect

series named after Joseph Fourier. In this article, f denotes a real-valued function on R {\displaystyle \mathbb {R} } which is periodic with period 2L

( x , y ) = σ ( z ) {\displaystyle \sigma (x,y)=\sigma (z)} be a real-valued function defined in a bounded domain D {\displaystyle D} . If σ > 0 {\displaystyle

semidefinite programming. Given a real vector space X, a convex, real-valued function f : C → R {\displaystyle f:C\to \mathbb {R} } defined on a convex

formulate their continuous differentiability. For example, consider the real-valued function F {\displaystyle F} of two real variables defined by F ( x , y )

quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase

downwards, concave down, convex upwards, convex cap, or upper convex. A real-valued function f {\displaystyle f} on an interval (or, more generally, a convex

more fully in the main article Symmetry (geometry). Let f(x) be a real-valued function of a real variable, then f is even if the following equation holds

intuitive characterization of continuity is as follows: Theorem. A real-valued function f on the interval [a, b] is continuous if and only if for every hyperreal

theorem 4.4) give historical accounts of the theorem. If f is a real valued function defined on an interval, then with the possible exception of a set

parameterized by λ ( x ) {\displaystyle \lambda (x)} . This is a real-valued function λ : R 1 , 3 → R . {\displaystyle \lambda :\mathbb {R} ^{1,3}\rightarrow

→ R {\displaystyle f\colon {\mathcal {D}}\to \mathbb {R} } be a real-valued function. The sensitivity of a function, denoted Δ f {\displaystyle \Delta

null space of f {\displaystyle f} . If f {\displaystyle f} is a real-valued function of several real variables, the fibers of the function are the level

{\displaystyle \mathbb {C} } with this property. The essential supremum of a real valued function equals the supremum of its essential image and the essential infimum

[Ax]i / xi taken over all those i such that xi ≠ 0. Then f is a real valued function whose maximum over all non-negative non-zero vectors x is the Perron–Frobenius

^{m}\to \mathbb {R} \cup \{\pm \infty \}} be a continuous extended real-valued function. Define a nonlinear functional F {\displaystyle F} on functions u

ball. The standard proof uses the fact that a bounded continuous real-valued function f is strong-operator continuous. In other words, for a net {aα} of

{\displaystyle \sup _{\lambda \in \Lambda }f_{\lambda }} , is a real-valued function with modulus of continuity ω, provided it is finite valued at every

the Swedish probabilist Harald Cramér. Rate function An extended real-valued function I : X → [ 0 , + ∞ ] {\displaystyle I:X\to [0,+\infty ]} defined on

equation. To define the simplest kind of parabolic PDE, consider a real-valued function u ( x , y ) {\displaystyle u(x,y)} of two independent real variables

used to perform inference efficiently on it. The contour tree of a real-valued function on a vector space is a polytree that describes the level sets of

A simple criterion for checking if a given stationary point of a real-valued function F(x,y) of two real variables is a saddle point is to compute the

some n. A semialgebraic function on U is defined to be a continuous real-valued function on U whose restriction to any semialgebraic set contained in U has

separable if and only if it is σ-compact. Every sequentially continuous real-valued function in a metric space is a continuous function. Every accumulation point

(f\times f)^{-1}(V),} where f {\displaystyle f} is a continuous real-valued function on X {\displaystyle X} and V {\displaystyle V} is an entourage of

}}t\to 0;\end{cases}}} to have a smooth fundamental solution, i.e. a real-valued function p (0, +∞) × R2d → R such that p(t, ·, ·) is smooth on R2d for each

f(1) − 1 < 0. Thus the function g(x) = f(x) − x is a continuous real valued function which is positive at x = 0 and negative at x = 1. By the intermediate

these six functions is uniquely determined and defines a continuous real-valued function on M. According to Chern-Weil theory, if M is oriented then the Euler

follows: Weierstrass approximation theorem—Suppose f is a continuous real-valued function defined on the real interval [a, b]. For every ε > 0, there exists

φ) on M, one could consider the composition u ∘ φ−1, which is a real-valued function whose domain is an open subset of a Euclidean space; as such, if

the Cantor set. The Baire Characterisation Theorem states that a real valued function f defined on a Banach space X is a Baire-1 function if and only if

J {\displaystyle J} is preserved by parallel transport. A smooth real-valued function ρ {\displaystyle \rho } on a complex manifold is called strictly

limit of a single variable function: Let f {\displaystyle f} be a real-valued function of a real variable and x {\displaystyle x} , a {\displaystyle a}

various relationships are apparent, for example: The transform of a real-valued function ( s R E + s R O ) {\displaystyle (s_{_{RE}}+s_{_{RO}})} is the conjugate

distinct prime factors. A more precise version states that for every real-valued function ψ ( n ) {\displaystyle \psi (n)} that tends to infinity as n {\displaystyle

sufficiently smooth that derivatives can be taken. Let f(v) be a real valued function of the vector v. Then the derivative of f(v) with respect to v (or

p=\operatorname {prox} _{f}(x)\Leftrightarrow x-p=\nabla f(p)} . An (extended) real-valued function f on a Hilbert space is said to be proper if it is not identically

from reformulating the problem of value function approximation as real-valued function approximation on a graph or manifold. This results in broader applicability

continuous mapping theorem: if Tn is consistent for θ and g(·) is a real-valued function continuous at the point θ, then g(Tn) will be consistent for g(θ):

real-valued functions. If X {\displaystyle X} is any set then a real-valued function f : X → R {\displaystyle f:X\to \mathbb {R} } on X {\displaystyle

non-Archimedean metric or super-metric. An ultrametric on a set M is a real-valued function d : M × M → R {\displaystyle d\colon M\times M\rightarrow \mathbb

Normally, the output X w ( a , b ) {\displaystyle X_{w}(a,b)} is a real valued function except when the mother wavelet is complex. A complex mother wavelet

(fg)(a)=f(a)\nabla g(a)+g(a)\nabla f(a).} Chain rule Suppose that f : A → R is a real-valued function defined on a subset A of Rn, and that f is differentiable at a point

_{-\infty }^{\infty }f(t)\cdot \varphi (t-k)\,dt.\end{aligned}}} For a real-valued function f(t), a ~ j , k = a j , k ∗ {\displaystyle {\tilde {a}}_{j,k}=a_{j

on a single measurement alone Kernel smoother used to estimate a real valued function as the weighted average of neighboring observed data. most appropriate

definition requires the group to have a norm, which is a positive real-valued function ‖ ⋅ ‖ : G → R + {\textstyle \|\cdot \|:G\to \mathbb {R} _{+}} on

{\displaystyle {\mathcal {P}}(G):\Omega \rightarrow R^{m}} be a real valued function which assigns to each graph in Ω {\displaystyle \Omega } a vector

in which strict differentiability can be considered, is that of a real-valued function defined on an interval I of the real line. The function f:I → R is

\quad t\in \mathbb {R} ,} where g {\displaystyle g} is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion

point coordinates. It can be viewed as a creative way to plot a real-valued function of two real variables f ( x , y ) {\displaystyle f(x,y)} as a planar

( R ) {\displaystyle g\in C^{\infty }(\mathbb {R} )} is a smooth real-valued function. The solution u ( x , t ) {\displaystyle u(x,t)} is assumed to be

{\displaystyle f} ). Let f : X → R {\displaystyle f:X\to \mathbb {R} } be a real valued function defined on a set X . {\displaystyle X.} The supremum of a function

E {\displaystyle E} are dense in X , {\displaystyle X,} then the real-valued function which takes the value 1 {\displaystyle 1} on E {\displaystyle E}

E {\displaystyle E} are dense in X , {\displaystyle X,} then the real-valued function which takes the value 1 {\displaystyle 1} on E {\displaystyle E}

partial derivatives of multivariate functions. Suppose that some real-valued function of the real numbers f = f ( t 1 , t 2 , … , t k ) {\displaystyle

constant scalar curvature. In particular, the Mabuchi energy is a real-valued function on a Kähler class whose Euler-Lagrange equation is the constant scalar

equivalent to the Lebesgue integral of  f . Where  f  is a continuous real-valued function of a real variable and v is a non-decreasing real function, the Lebesgue–Stieltjes

points or probability density functions), denoted by P. There is a real-valued function denoted by G ("goal") on the set of all partitions. The map segmentation

notions: oscillation of a sequence of real numbers, oscillation of a real-valued function at a point, and oscillation of a function on an interval (or open

line). Thus, for example, a mapping of Rn into R1 is an ordinary real-valued function of n variables. In the case where the space R itself consists of

symmetric, which benefits the image processing: If the phase of real-valued function is symmetry, than the function has generalized linear phase, and

equation is most commonly written as the following equation for a real-valued function v = v ( x 1 , x 2 , t ) {\displaystyle v=v(x_{1},x_{2},t)} : ∂ t

response does is enable the inverse transform to distinguish between a real-valued function and a complex one. Perhaps the best-known application of negative

d-dimensional semimartingale and f is a twice continuously differentiable real valued function on Rd then f(X) is a semimartingale, and f ( X t ) = f ( X 0 ) +

quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase

where: s out of K attributes are specified with 0 < f(s/K) < 1, a real valued function of s/K Nearest neighbor queries: O(log n) k-d tree implicit k-d tree

the right hand side is a positive combination of exponentials, the real-valued function g is strictly convex if X ≠ 0, so has a unique minimum. On the other

function. In other words, let e {\displaystyle e} be a continuous real-valued function and [ a , b ] {\displaystyle [a,b]} an interval such that e ( a )

magnitude-squared coherence) between two signals x(t) and y(t) is a real-valued function that is defined as: C x y ( f ) = | G x y ( f ) | 2 G x x ( f ) G

subset of a linear topological space. If f {\displaystyle f} is a real-valued function on X × Y {\displaystyle X\times Y} with f ( ⋅ , y ) {\displaystyle

{\displaystyle M} is a C ∞ {\displaystyle C^{\infty }} manifold. A real-valued function f : M → R {\displaystyle f:M\to \mathbb {R} } is said to belong to

y))} , where u : X → R {\displaystyle u:X\to \mathbb {R} } is a real valued function, is minimal if and only if u {\displaystyle u} satisfies ( 1 + u

learning. Let ( X , ⪯ ) {\displaystyle (X,\preceq )} be a lattice. A real-valued function f : X → R {\displaystyle f:X\rightarrow \mathbb {R} } is called supermodular

  noncommutative torus norm 1.  A norm on a vector space X is a real-valued function ‖ ⋅ ‖ : X → R {\displaystyle \|\cdot \|:X\to \mathbb {R} } such that

geometric objects (see Hausdorff dimension). Weierstrass function, a real-valued function on the real line, that is continuous everywhere but differentiable

\pm \pi /2.} However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty argmax .

0)} . Indeed, this relation cannot be represented by a continuous real-valued function (in fact, it cannot be represented even by non-continuous functions)

that ∂ ∘ ∂ ≡ 0 {\displaystyle \partial \circ \partial \equiv 0} . A real-valued function μ : X → R {\displaystyle \mu \colon {\mathcal {X}}\to \mathbb {R}

is to the problem of prescribed Gauss curvature. Suppose that a real-valued function K is specified on a domain Ω in Rn, the problem of prescribed Gauss

{\displaystyle \operatorname {AC} (I)} . The following conditions on a real-valued function f on a compact interval [a,b] are equivalent: f is absolutely continuous;

written a i j {\displaystyle a_{ij}} . A graph signal is simply a real-valued function on the set of nodes of the graph. In graph neural networks, graph

naturally arise when working with semicontinuous functions, in that a real-valued function on a topological space is lower semicontinuous if and only if it

g {\displaystyle {\mathcal {L}}_{X}g=\phi g} where ϕ is a smooth real-valued function on M. A curvature collineation is a vector field which preserves

{\displaystyle A.} The rearrangement of a non-negative, measurable real-valued function f {\displaystyle f} whose level sets f − 1 ( y ) {\displaystyle f^{-1}(y)}

−1. We assume that the classification problem will be solved by a real-valued function f, by predicting a class label y = sign(f(x)). For many problems

definitions would tend to confound ready understanding. In the case of a real-valued function f ( ) {\displaystyle f()} (whose arguments might or might not be

subcover. Pseudocompact. A space is pseudocompact if every continuous real-valued function on the space is bounded. σ-compact. A space is σ-compact if it is

function; in particular, Ackermann function. Simple function: a real-valued function over a subset of the real line, similar to a step function. Measurable

(here o is little-o notation). Then φ(U) is measurable, and for any real-valued function f defined on φ(U): ∫ φ ( U ) f ( v ) d v = ∫ U f ( φ ( u ) ) | det

4  The extension is defined by a specific two-cocycle, which is a real-valued function C ( f , g ) {\displaystyle {\mathsf {C}}(f,g)} of pairs of diffeomorphisms

{\displaystyle \Omega } to R {\displaystyle \mathbb {R} } . As a real-valued function, X {\displaystyle X} often describes some numerical quantity of a

various relationships are apparent, for example: The transform of a real-valued function ( f R E + f R O ) {\displaystyle (f_{_{RE}}+f_{_{RO}})} is the conjugate

generated. A pseudocharacter on a discrete group G is defined to be a real-valued function f on a G such that f(gh) − f(g) − f(h) is uniformly bounded and f(gn)

also be seen from the following example.: 17–19  For example, for a real-valued function f : R 2 → R {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} } with

also be seen from the following example.: 17–19  For example, for a real-valued function f : R 2 → R {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} } with

viewed as a function that takes a vector field on U, and returns a real-valued function whose value at each point is the derivative along the vector field

function, and let g , {\displaystyle g,} the comparison function, be a real valued function. Let both functions be defined on some unbounded subset of the positive

converse to the four-vertex theorem states that any continuous, real-valued function of the circle that has at least two local maxima and two local minima

the expansion no longer converges uniformly. In mathematics, a real-valued function K(x,y) is said to fulfill Mercer's condition if for all square-integrable

depending on context. Suppose f ( x ) {\displaystyle f(x)} is a real-valued function on R {\displaystyle \mathbb {R} } . We can reinterpret the variable

) , g ( x ) ) {\displaystyle d(f(x),g(x))} is again a measurable real-valued function of x {\displaystyle x} . Fix ε > 0 {\displaystyle \varepsilon >0}

( x ) + i g ( x ) {\displaystyle F(x)=f(x)+i\,g(x)} where f is a real-valued function in L p ( R ) {\displaystyle L^{p}(\mathbb {R} )} and g is the Hilbert

following problem: Given a finite set X {\displaystyle X} and a real valued function g {\displaystyle g} on X {\displaystyle X} , find the largest subset

on a Riemannian manifold as the divergence of the gradient of a real-valued function f: Δ f := div ⁡ grad ⁡ f . {\displaystyle \Delta f:=\operatorname

to L∞. However, the two following properties are equivalent for a real valued function f the function f  is the real part of some function g ∈ H∞(T) the

following more general formulation is from Feller.: 445  Consider a real-valued function F : [ 0 , ∞ ) → R {\displaystyle F:[0,\infty )\to \mathbb {R} } of

numerically integrating the first 7 terms of their Taylor series. If a real-valued function f ( x ) {\textstyle f(x)} is differentiable at the point x = a {\textstyle

→ R {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } is a real-valued function which is intimately related to the diffusion coefficient. Then for

be a closed form. The Poincaré lemma then implies that there is a real-valued function u with ω = du. Due to the formula for the Ricci curvature under a

The simplest form of a predicate is the condition on the sign of a real valued function resulting in the familiar representation of sets by equalities and

{\displaystyle \operatorname {div} _{g}{\frac {du}{|du|_{g}}}=|du|_{g}} for a real-valued function u on M. Weak solutions of this equation can be specified by a variational

various relationships are apparent, for example: The transform of a real-valued function ( s R E + s R O ) {\displaystyle (s_{\mathrm {RE} }+s_{\mathrm {RO}

\lim _{n\to \infty }f\left(x_{n}\right)} exists too. In fact, any real-valued function f {\textstyle f} is continuous if and only if it preserves the limits

Hilbert manifold (not necessarily finite dimensional) and f is a real-valued function then we say that p is a critical point of f if f is not a submersion

regular space W k,p(M). Assume that u is a continuously differentiable real-valued function on Rn with compact support. Then for 1 ≤ p < n there is a constant

{\displaystyle \mathbb {R} } . Let f be a continuously differentiable real-valued function over D. If the solid of revolution generated by revolving y = f(x)

information-theoretic lower bound and the true lower bound are much like a real-valued function lower-bounding an integer function. However, this is not exactly

called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that V = ∇ f = ( ∂ f ∂ x 1 , ∂ f ∂ x

the consumer has an ordinal utility function, called u. It is a real-valued function with domain being the set of all commodity bundles, or u : R + n

complexes known as the Vietoris–Rips filtration. Formally, consider a real-valued function on a simplicial complex f : K → R {\displaystyle f:K\rightarrow \mathbb

extreme value theorem of Karl Weierstrass states that a continuous real-valued function on a compact set attains its maximum and minimum value. More generally

provide a proof based on such definitions. A Darboux function is a real-valued function f that has the "intermediate value property," i.e., that satisfies

properties of the two functions. A triangle center function is a real valued function ⁠ F ( u , v , w ) {\displaystyle F(u,v,w)} ⁠ of three real variables

represented by a single cardinal utility function. I.e, there is no real-valued function U {\displaystyle U} such that x ≺ y {\displaystyle x\prec y} iff

gauge on [ a , b ] {\displaystyle [a,b]} is a strictly positive real-valued function δ : [ a , b ] → R + {\displaystyle \delta :[a,b]\to \mathbb {R} ^{+}}

a compact set X, a complete Lyapunov function V from X to R is a real-valued function on X satisfying: V is non-increasing along all solutions of the differential

compact, the extreme value theorem holds for such spaces: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum

\{f(x):g(x,u)\leq b,\forall u\in U\}} where g {\displaystyle g} is a real-valued function on X × U {\displaystyle X\times U} , and assume that there is no

of potential future observations. A scoring rule is any extended real-valued function S : F × Ω → R {\displaystyle \mathbf {S} :{\mathcal {F}}\times \Omega

{\displaystyle f:[a,b]\to (0,\infty )} is a positive continuous real-valued function, its geometric mean over this interval is GM [ f ] = exp ⁡ ( 1 b

optimization algorithms. For example, to find a local minimum of a real-valued function F ( x ) {\displaystyle F(\mathbf {x} )} using gradient descent, one

upper and lower (Darboux) integrals, which exist for any bounded real-valued function f {\displaystyle f} on the interval [ a , b ] . {\displaystyle [a

u:=-1} shows that every balanced function is a symmetric function. A real-valued function p : X → R {\displaystyle p:X\to \mathbb {R} } is a seminorm if and

three stages.: §2  In Gleason's terminology, a frame function is a real-valued function f {\displaystyle f} on the unit sphere of a Hilbert space such that

3-dimensional spaces. In 3 dimensions, a differential 0-form is a real-valued function f ( x , y , z ) {\displaystyle f(x,y,z)} ; a differential 1-form

standard bump function for the unit ball at the origin: a smooth real-valued function which is equal to 1 on B1/2(0), which vanishes to infinite order

changes may affect f. In the special case where k = 1, so that f is a real-valued function, then this formula simplifies even further: ∂ y ∂ x i = ∑ ℓ = 1 m

The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map

integrals for a class of functions called measurable functions. A real-valued function f on E is measurable if the pre-image of every interval of the form

that can take on the nominal values "red", "blue" or "green", the real-valued function [ X = green ] {\displaystyle [X={\text{green}}]} can be constructed;

_{n}|x_{n}|^{p}<\infty .} If p ≥ 1 , {\displaystyle p\geq 1,} then the real-valued function ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}} on ℓ p {\displaystyle \ell

y_{n})\in {\mathcal {X}}\times \mathbb {R} } , a strictly increasing real-valued function g : [ 0 , ∞ ) → R {\displaystyle g\colon [0,\infty )\to \mathbb {R}

relative to some background σ-finite measure ν on Ω′, there is a real-valued function fX on Ω' representing the Radon–Nikodym derivative of the two measures

the 1920s, the representability of a preference structure with a real-valued function is one of the major issues pervading the theory of preferences. This

by X ( 2 ) {\displaystyle X(2)} or X 2 {\displaystyle X_{2}} . A real-valued function f of three real variables a, b, c may have the following properties:

describable. In contrast, the possible singular behaviour of a continuous real-valued function is much more difficult to characterise. As a result of this, one

In calculus, Rolle's theorem says that if a real-valued function f is continuous on a closed interval [a, b], differentiable on the open interval (a, b)

n {\displaystyle \mathbb {R} ^{n}} and F {\displaystyle F} is a real-valued function on Ω × R m × R m n {\displaystyle \Omega \times \mathbb {R} ^{m}\times

Theorem 1: Assume that f {\displaystyle f} is a continuous, real-valued function, defined on an arbitrary interval I {\displaystyle I} of the real

to y (new value) R ( x , y ) {\displaystyle R(x,y)} is a binary real-valued function defined for the domain of interest which satisfies the following

to the Markov numbers. Let ψ {\displaystyle \psi } be a positive real-valued function on positive integers (i.e., a positive sequence) such that q ψ (

\|x+sy\|} for all scalars s , {\displaystyle s,} and if and only if the real-valued function f ( s ) := ‖ x + s y ‖ 2 − ‖ x ‖ 2 {\displaystyle f(s):=\|x+sy\|^{2}-\|x\|^{2}}

is a variable with values in an open set Ω ⊂ Rn, u is an unknown real-valued function, Du is the gradient of u, and F is a continuously differentiable

H(x_{1},x_{2},\dots )\right)} The function H is understood to be a real-valued function on the space of states { X 1 , X 2 , … } {\displaystyle \{X_{1},X_{2}

1/f\right)^{(d)}(x_{n})}}} . Householder's method applied to the real-valued function f(x) is the same as applying Newton's method x n + 1 = x n − g (

mechanics is achieved with the tautological one-form. Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system

which is useful for proving Hölder's inequality. Proof Define a real-valued function f {\displaystyle f} on the positive real numbers by f ( t )   =  

characterizes an important vertex?" The answer is given in terms of a real-valued function on the vertices of a graph, where the values produced are expected

{\displaystyle q} denote the desired quantile (e.g., 0.025) of a real-valued function g ( X ) {\displaystyle g(X)} : in other words, the goal is to find

Campbell's theorem offers a way to calculate expectations of a real-valued function f {\displaystyle f} defined also on R d {\displaystyle {\textbf {R}}^{d}}

This indicates that, although χ k {\displaystyle \chi _{k}} is a real-valued function of r {\displaystyle \mathbf {r} } , its functional form depends on

where ⟨ η , ζ ⟩ {\displaystyle \langle \eta ,\zeta \rangle } is a real-valued function on M {\displaystyle M} , and the volume form ω {\displaystyle \omega

Distance from a point to the boundary of a set Similarity measure – Real-valued function that quantifies similarity between two objects Space (mathematics) –

the output function of A {\displaystyle {\mathcal {A}}\,\!} as a real valued function (called as the transcript output by A {\displaystyle {\mathcal {A}}\

{\displaystyle \;\mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})\;} gives a real-valued function, L n ( θ ) = L n ( θ ; y ) = f n ( y ; θ ) , {\displaystyle {\mathcal

with the classical phase space. A classical phase space contains a real-valued function in 6N dimensions (each particle contributes 3 spatial coordinates

under-developed. Weil function A Weil function on an algebraic variety is a real-valued function defined off some Cartier divisor which generalises the concept of

true in general. Let X {\displaystyle X} be a topological space. A real-valued function f : X → R {\displaystyle f:X\rightarrow \mathbb {R} } is cliquish

\langle {\hat {G}}\rangle =\int dx\,dp\,W(x,p)g(x,p).} 1. W(x, p) is a real-valued function. 2. The x and p probability distributions are given by the marginals:

{\displaystyle a\in \mathbb {R} } and a "well-behaved" arbitrary real-valued function y(x), ∫ y ( x ) δ ( x − a ) d x = y ( a ) H ( x − a ) + c , {\displaystyle

value property. That has a natural and important discrete version: a real-valued function f {\displaystyle f} on a finite or infinite graph G ( V , E ) {\displaystyle

displaying wikidata descriptions as a fallback Similarity measure – Real-valued function that quantifies similarity between two objects Type–token distinction –

eigenfunctions (the Titchmarsh–Kodaira formula). Let q(x) be a continuous real-valued function on (0, ∞) and let D be the second order differential operator D f

a b {\displaystyle \textstyle \int _{a}^{b}} f (x) dx of a given real-valued function f over a given interval [a,b] of the real line. The numerical method

the notion of ridge generalizes the idea of a local maximum of a real-valued function. A point x 0 {\displaystyle \mathbf {x} _{0}} in the domain of a

\cdot ,\cdot \rangle )} will form a dual system. The norm is the real-valued function ‖ x ‖ = ⟨ x , x ⟩ , {\displaystyle \|x\|={\sqrt {\langle x,x\rangle

Pseudometric space A pseudometric space (M, d) is a set M equipped with a real-valued function d : M × M → R {\displaystyle d:M\times M\to \mathbb {R} } satisfying

the real unit interval [0, 1]. They are uniquely determined by the real-valued function that interprets strong conjunction, which can be any left-continuous

sake of the comparison. This method for finding fixed points of a real-valued function has been generalized for functions F : X → X {\displaystyle F:X\to

( a , u ) {\displaystyle {\mathfrak {u}}:=(a,u)} consisting of a real-valued function a {\displaystyle a} on M {\displaystyle M} and a vector field u {\displaystyle

sufficiently smooth that derivatives can be taken. Let f(v) be a real valued function of the vector v. Then the derivative of f(v) with respect to v (or

f(\mathbf {x} )=\mathbf {x} ^{*}\mathbf {M} \mathbf {x} } ⁠ is a real-valued function of ⁠ 2 n {\displaystyle 2n} ⁠ real variables. Singular values are

  . {\textstyle f(x)=f^{\star \star }(x)~.} For a differentiable real-valued function on an open convex subset U of Rn the Legendre conjugate of the pair

{\displaystyle X} is reflexive then this theorem solves the vector problem. A real-valued function f : M → R {\displaystyle f:M\to \mathbb {R} } defined on a subset

argument of the optical transfer function can be depicted as a second real-valued function, commonly referred to as the phase transfer function (PhTF). The

definitions apply, but we discuss only matrices, for simplicity. For any real-valued function f {\displaystyle f} on an interval I ⊆ R , {\displaystyle I\subseteq

Let F {\displaystyle \mathbb {F} } be a fixed field. Consider a real-valued function on a chain complex f : K → R {\displaystyle f:K\rightarrow \mathbb

compact subsets of U . {\displaystyle U.} If f {\displaystyle f} is a real-valued function on U {\displaystyle U} , then f {\displaystyle f} is an element of

vector subspace is an F-seminorm (resp. F-norm). A non-negative real-valued function on X {\displaystyle X} is a seminorm if and only if it is a convex

{\displaystyle x''(t)+p(t)x(t)=0} where x {\displaystyle x} is a real-valued function of a real parameter t {\displaystyle t} . Let X {\displaystyle X}

1]K=\{tk:t\in [0,1],k\in K\}.} If q {\textstyle q} is a nonnegative real-valued function on X {\textstyle X} that is positive homogeneous, then the sets U

regular surface S, the Gaussian curvature of S can be regarded as a real-valued function on S; relative to a choice of unit normal vector field on all of

is a brief review of function theory in one-variable calculus. A real-valued function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is continuous

potentials. To be mathematically precise, let f be a decreasing real-valued function, and define the energy functional ∑ i < j f ( | x i − x j | ) . {\displaystyle

constant such that g(W) lies in the strip a < Im z < b. Define a real-valued function by h = φ(u)ψ(v) on W and 0 off W. Then h, so defined, cannot be a

2 , r 3 , r 4 ) {\displaystyle f(r_{1},r_{2},r_{3},r_{4})} is a real valued function. Now, we shall consider the MPF problem where the commutative semiring

manifold M {\displaystyle M} , and φ {\displaystyle \varphi } a smooth real-valued function on M {\displaystyle M} . Then g ~ = e 2 φ g {\displaystyle {\tilde

discussion on how the result connects with the triangle centers. A real-valued function f of three real variables a, b, c may have the following properties:

complex-valued function is often much easier than treating the corresponding real-valued function. ( ∇ 2 + k 2 ) ψ ( r ) = 0. {\displaystyle \left(\nabla ^{2}+k^{2}\right)\psi

(h)}\left|\psi \right\rangle \,,} where ω {\displaystyle \omega } is a real-valued function of h {\displaystyle h} . Let X = G / H {\displaystyle X=G/H} be the

0 , 1 } {\displaystyle \{0,1\}} -valued functions instead. Every real-valued function f : { − 1 , 1 } n → R {\displaystyle f\colon \{-1,1\}^{n}\to \mathbb

ds^{2}=(1+t\psi (t)h(\theta ))^{2}d\theta ^{2}+dt^{2},} where ψ is an analytic real-valued function of t: ψ ( t ) = 1 + a 1 t + a 2 t 2 + ⋯ {\displaystyle \displaystyle

such as probability theory, numerical analysis, and elasticity. A real valued function f ( x ) {\displaystyle f(x)} defined over an interval I {\displaystyle

representing the resource to be allocated. Let v {\displaystyle v} be any real-valued function on subsets of C {\displaystyle C} , representing their "value". The

|x_{0}|}{|x|\neq |x_{0}|}}f(x)\to y.} Divergence to infinity Divergence of a real-valued function to infinity can be defined/characterized by using the prefilters

Weierstrass approximation theorem, which states that any continuous real-valued function defined on a closed interval can be uniformly approximated by polynomials

over K . {\displaystyle \mathbb {K} .} If f {\displaystyle f} is a real-valued function on U, then f {\displaystyle f} is an element of C c k ( U ) {\displaystyle