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searching for Differentiable function 162 found (299 total)

alternate case: differentiable function

Piecewise function (1,077 words) [view diff] no match in snippet view article find links to article

In mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose

derivative is equal to the right derivative. An example of a semi-differentiable function, which is not differentiable, is the absolute value function f

its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has

{x}})} with an equilibrium point at the origin, a continuously differentiable function V(x) such that the origin is a boundary point of the set G = {

implementable algorithm. They are called proximal because each non-differentiable function among f 1 , . . . , f n {\displaystyle f_{1},...,f_{n}} is involved

necessary condition for all values of k {\displaystyle k} . Consider a differentiable function f ( x ) {\displaystyle f(x)} on [ a , b ] {\displaystyle [a,b]}

stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R , {\displaystyle f\colon U\to \mathbb {R} ,} where U

and real-differentiability. On the real line, for example, the differentiable function f ( x ) = x 2 {\displaystyle f(x)=x^{2}} is not an open map, as

{\displaystyle u:\Omega \rightarrow \mathbb {R} } be a weakly differentiable function that vanishes on the boundary of Ω {\displaystyle \Omega } in the

for sparse or partially separable problems. A twice continuously differentiable function x ↦ f ( x ) {\displaystyle x\mapsto f(x)} has a gradient ( ∇ f

_{j}\left(U_{i}\cap U_{j}\right)} is an r {\displaystyle r} -times continuously differentiable function for every i , j ∈ I ; {\displaystyle i,j\in I;} that is, the r

mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966). This function

transposition, f(e) is scalar function, and f(0) = 0. Suppose, f(e) is a differentiable function and the following condition k 1 < f ′ ( e ) < k 2 . {\displaystyle

series is a Fourier series type expansion of twice continuously differentiable function in the interval ( 0 , π ) {\displaystyle (0,\pi )} in terms of

Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian

it can integrate the approximate derivative of an approximately differentiable function (see below for definitions). To do this, one first finds a condition

versions of the Wirtinger inequality: Let y be a continuous and differentiable function on the interval [0, L] with average value zero and with y(0) =

approximating solutions to equations. Given a twice continuously differentiable function f {\displaystyle f} of one real variable, Taylor's theorem for

functions with linear functions that are almost the same. Given a differentiable function f(x, y) with real values, one can approximate f(x, y) for (x, y)

case where the search space has underlying structure (e.g., is a differentiable function) that can be exploited more efficiently (e.g., Newton's method

differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains is a subset

is an intensive quantity, i.e., a single-valued, continuous and differentiable function of three-dimensional space (often called a scalar field), i.e.

distribution of compact support and f {\displaystyle f} is an infinitely differentiable function, the expression v ( f ) = v ( x ↦ f ( x ) ) {\displaystyle v(f)=v(x\mapsto

→ R {\displaystyle f\colon I\to \mathbb {R} } be a real-valued differentiable function. Then f ′ {\displaystyle f'} has the intermediate value property:

Peter Gustav Lejeune Dirichlet. Given an open set Ω ⊆ Rn and a differentiable function u : Ω → R, the Dirichlet energy of the function u is the real number

underlying topological space and to each p-times continuously differentiable function the underlying continuous function of topological spaces. Similarly

variational principle. The classical Fermat's theorem says that if a differentiable function attains its minimum at a point, and that point is an interior point

variable with expectation μ and variance σ2. Further suppose g is a differentiable function for which the two expectations E ⁡ ( g ( X ) ( X − μ ) ) {\displaystyle

non-differentiable functions (an example of an everywhere non-differentiable function is the Weierstrass function). Singh published about a dozen papers

\int _{a}^{b}e^{Mf(x)}\,dx,} where f {\displaystyle f} is a twice-differentiable function, M {\displaystyle M} is a large number, and the endpoints a {\displaystyle

10 17 {\displaystyle 10^{15}-10^{17}} eV) by the continuously differentiable function of energy E {\displaystyle E} taking into account the rate of change

whose derivative is nowhere vanishing, i.e. a strictly monotone differentiable function, either increasing or decreasing. The space of such functions consists

of global solutions. For example: There does not exist a twice-differentiable function u : ℝn → ℝ such that ∂ 2 u ∂ x 1 2 + ⋯ + ∂ 2 u ∂ x n 2 ≥ e u .

pairwise distinct points x0, ..., xn in the domain of an n-times differentiable function f there exists an interior point ξ ∈ ( min { x 0 , … , x n } ,

n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be a differentiable function and let v {\displaystyle \mathbf {v} } be a vector in R n {\displaystyle

the inverse function theorem which states that a continuously differentiable function between Euclidean spaces whose derivative matrix is invertible

{\displaystyle u(x)\in C_{c}^{\infty }(\mathbb {R} ^{n})} , infinitely differentiable function with compact support. Introduce u λ ( x ) := u ( λ x ) {\displaystyle

D_{q}f(x)={\frac {f(qx)-f(x)}{(q-1)x}}.} For x nonzero, if f is a differentiable function of x then in the limit as q → 1 we obtain the ordinary derivative

sequence Illustration of the central limit theorem An infinitely differentiable function that is not analytic Leech lattice Lewy's example on PDEs List

the curve does not vanish. Analytically, r(t) is a three times differentiable function of t with values in R3 and the vectors r ′ ( t ) , r ″ ( t ) {\displaystyle

statistical model, and θ {\textstyle \theta } is a continuously differentiable function of φ {\textstyle \varphi } , we say that the prior p θ ( θ ) {\textstyle

Differentiable vector-valued functions from Euclidean space – Differentiable function in functional analysis Infinite-dimensional vector function Hamilton

Critical point (aka stationary point), any value v in the domain of a differentiable function of any real or complex variable, such that the derivative of v

it is measurable and its Lebesgue measure is zero as well. Any differentiable function has the Luzin N property. This extends to functions that are differentiable

that is transformed by continuously differentiable function. If Y = g(X) is a continuous differentiable function, then the density of Y can be written

integral. It is also possible to show a converse – that every differentiable function is equal to the integral of its derivative, but this requires a

suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface

{1}{2}}f(X_{t})f'(X_{t})\mathrm {d} t+f(X_{t})\mathrm {d} W_{t}} for a given differentiable function f {\displaystyle f} is equivalent to the Stratonovich SDE d X t

S_{D}:{\vec {r}}={\vec {r}}(u,v),\quad (u,v)\in D} with a continuously differentiable function r → . {\displaystyle {\vec {r}}.} The area of an individual piece

positive, continuous, nondecreasing function. Then there exists a differentiable function G : [ 0 , ∞ ) → [ 0 , ∞ ) {\displaystyle G:[0,\infty )\rightarrow

X is an Rm-valued semimartingale and f is a twice continuously differentiable function from Rm to Rn, then f(X) is a semimartingale. This is a consequence

f ( x ) . {\displaystyle A=Df(x).} However, not every Gateaux differentiable function is Fréchet differentiable. This is analogous to the fact that the

\mathrm {d} W_{t},} where f {\displaystyle f} is any continuously differentiable function of two variables W {\displaystyle W} and t {\displaystyle t} and

x_{0}} . Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x 0 {\displaystyle x_{0}}

certain symbol class. For instance, if P(x,ξ) is an infinitely differentiable function on Rn × Rn with the property | ∂ ξ α ∂ x β P ( x , ξ ) | ≤ C α

exists in trace-class norm. If g ( t ) {\displaystyle g(t)} is a differentiable function with values in trace-class operators, then so too is exp ⁡ g (

real-valued function, Du is the gradient of u, and F is a continuously differentiable function that is regular in Du. Suppose that u(x;a) is an m-parameter family

{\displaystyle 0,} making the goal to prove that an everywhere differentiable function whose derivative is always zero must be constant: Choose a real

which states in the deterministic case, that for a continuously differentiable function f ∈ C 1 ( R k , R d ) {\displaystyle f\in C^{1}(\mathbb {R} ^{k}

with 2 input parameters consisting of (1) a real continuously differentiable function on a subinterval of the real numbers and (2) an n {\displaystyle

Itô's lemma. In its simplest form, for any twice continuously differentiable function f on the reals and Itô process X as described above, it states

expectation with respect to γn. The gradient operator ∇ acts on a (differentiable) function φ : Rn → R to give a vector field ∇φ : Rn → Rn. The divergence

{\displaystyle U} . By Cauchy's integral formula, which shows that a differentiable function is in fact infinitely differentiable, a function f {\displaystyle

leading up to an example of John Myhill of a computable continuously differentiable function whose derivative is not computable, the remaining two parts of

of coordinates from one coordinate map to another is always a differentiable function. In geometry and kinematics, coordinate systems are used to describe

Differentiable vector–valued functions from Euclidean space – Differentiable function in functional analysisPages displaying short descriptions of redirect

injective. For example, in calculus if f {\displaystyle f} is a differentiable function defined on some interval, then it is sufficient to show that the

bundle. An extremely special case of this is the following: if a differentiable function from reals to the reals has nonzero derivative at a zero of the

(Statement of the theorem ); we notice that for a continuously differentiable function f : R n + m → R m , f : ( x , y ) ↦ f ( x , y ) {\displaystyle

way that correctly accounts for a change-of-coordinates. Given a differentiable function f : R n → R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R}

an exact differential equation if there exists a continuously differentiable function F, called the potential function, so that ∂ F ∂ x = I {\displaystyle

parametrizations of an estimation problem, and θ is a continuously differentiable function of η, then I η ( η ) = I θ ( θ ( η ) ) ( d θ d η ) 2 {\displaystyle

function theorem gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Every fiber of a

\nu \}} . p-variation Hardy, G. H. (1916). "Weierstrass's Non-Differentiable Function". Transactions of the American Mathematical Society. 17 (3): 301–325

(where the term dζj is omitted). Suppose that f is a continuously differentiable function on the closure of a domain D in C {\displaystyle \mathbb {C} }

of Derrick's Theorem. (Above, f ( s ) {\displaystyle f(s)} is a differentiable function with f ′ ( 0 ) = 0 {\displaystyle f'(0)=0} .) The energy of the

vector field can be realized as the gradient of a continuously differentiable function outside a closed subset of a priori prescribed (small) measure

functions, and Markov chains. The Carleman matrix of an infinitely differentiable function f ( x ) {\displaystyle f(x)} is defined as: M [ f ] j k = 1 k

0)=(1,0)=f(-1,0)} ). Since the differential at a point (for a differentiable function) D f x : T x U → T f ( x ) V {\displaystyle Df_{x}:T_{x}U\to T_{f(x)}V}

Poincaré defines a manifold as the level set of a continuously differentiable function between Euclidean spaces that satisfies the nondegeneracy hypothesis

_{j\geq 1}(M_{j+1}/M_{j})^{1/j}<\infty } . For any infinitely differentiable function f {\displaystyle f} there are quasi-analytic rings C M {\displaystyle

of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue

) {\displaystyle \psi (a+iy)} is an n {\displaystyle n} times differentiable function of y {\displaystyle y} and such that the derivative | d n ψ ( a

{\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } is a differentiable function of variables x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}}

This process is illustrated below. In the case of a continuously differentiable function F, a coordinate descent algorithm can be sketched as: Choose an

\eta )=g(\xi )} , where g {\displaystyle g} is any real valued differentiable function defined on 1D, the image is invariant to rotations (around the

conformationally relevant dihedral angles, but can in principle be any differentiable function of the cartesian coordinates r {\displaystyle \mathbf {r} } . The

method is a root-finding algorithm for finding roots of a given differentiable function ⁠ f ( x ) {\displaystyle f(x)} ⁠. The iteration is x n + 1 = x

the curve as an example of a continuous everywhere yet nowhere differentiable function that was possible to represent geometrically at the time. From

beginning with an initial guess x0. If f is a three times continuously differentiable function and a is a zero of f but not of its derivative, then, in a neighborhood

literature the Van der Corput lemma. Let f {\displaystyle f} be a real differentiable function in the interval ] a , b ] , {\displaystyle ]a,b],} moreover, inside

⊆ R n {\displaystyle \Omega \subseteq \mathbb {R} ^{n}} and a differentiable function f : R n → R n {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb

solution of PDE, along with finite element methods. For a n-times differentiable function, by Taylor's theorem the Taylor series expansion is given as f

states: if c is a number, and f ( x ) {\displaystyle f(x)} is a differentiable function, then c ⋅ f ( x ) {\displaystyle c\cdot f(x)} is also differentiable

variables x and y. Consider a quasilinear PDE of the form For a differentiable function ( x , y ) ↦ u ( x , y ) {\displaystyle (x,y)\mapsto u(x,y)} , consider

be taken to find the utility maximising bundle as it is a non differentiable function. Therefore, intuition must be used. The consumer will maximise

These FFFs can provide a second-order approximation to any twice-differentiable function that meets the necessary regulatory conditions, including basic

{\displaystyle x^{*}(q)} may be viewed locally as a continuously differentiable function, and the local response of x ∗ ( q ) {\displaystyle x^{*}(q)} to

Differentiable vector-valued functions from Euclidean space – Differentiable function in functional analysis Differentiation in Fréchet spaces Fractal

SO(3). In a rectangular coordinate system, the gradient of a (differentiable) function f : R 3 → R {\displaystyle f:\mathbb {R} ^{3}\rightarrow \mathbb

as infinite or zero capital accumulation. Given a continuously differentiable function f : X → Y {\displaystyle f\colon X\to Y} , where X = { x : x ∈

temperature T is an intensive quantity, i.e., a single-valued, differentiable function of three-dimensional space (a scalar field), i.e., that T = T (

the following fact about averages. Given a twice continuously differentiable function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }

_{h\to 0}{\frac {f(x+h)-f(x)}{h}},\qquad \forall x\in [0,1].} Every differentiable function is continuous, so C1([0, 1]) ⊆ C([0, 1]). We claim that ⁠d/dx⁠ :

right it is expressed in terms of u. If y = f(x) where f is a differentiable function that is invertible, the derivative of the inverse function, if

functions on convex sets. Given a strictly convex, continuously differentiable function F on a convex set, known as the Bregman generator, the Bregman

known that the modulus (absolute value) of a holomorphic (complex differentiable) function in the interior of a bounded region is bounded by its modulus on

general formula for the surface area of the graph of a continuously differentiable function z = f ( x , y ) , {\displaystyle z=f(x,y),} where ( x , y ) ∈ D

differential geometry can be calculated from In(x) if it is a differentiable function. This is the original motivation behind its definition. Vector

{\displaystyle k.} If P : U → Y {\displaystyle P:U\to Y} is a continuously differentiable function, then the differential equation x ′ ( t ) = P ( x ( t ) ) , x (

the inverse function theorem one can show that a continuously differentiable function f : U → R n {\displaystyle f:U\to \mathbb {R} ^{n}} (where U {\displaystyle

satisfactory version of the second fundamental theorem of calculus: each differentiable function is, up to a constant, the integral of its derivative: F ( x ) −

far as possible from being associated with the level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane

can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma. As a

can be simplified by considering their effect on an arbitrary differentiable function f ( q ) , {\displaystyle f(q),} ( d d q q − q d d q ) f ( q ) =

There are two concepts of solution of this problem: a continuously differentiable function u: [0, ∞) → X is called a classical solution of the Cauchy problem

circles containing an S-like curve represent the application of a differentiable function (like the sigmoid function) to a weighted sum. Peephole convolutional

\cdot \mathbf {n} \,dS} Let Velocity U be a twice continuously differentiable function in a region of volume V in space. This function is the stream function

properties from marginalist theory which assumed utility to be a differentiable function of quantified goods and services. Later work attempted to generalize

f}{\partial t}}\end{aligned}}} where f is any twice continuously differentiable function that depends on position and time. The electromagnetic fields remain

_{0}} is a constant vector, f {\displaystyle f} is any second differentiable function, k ^ {\displaystyle {\hat {\mathbf {k} }}} is a unit vector in

{\displaystyle i} th group/subject, f {\displaystyle f} is a real-valued differentiable function of a group-specific parameter vector ϕ i j {\displaystyle \phi

^{2}\geq 0} . Let φ ( x ) {\displaystyle \varphi (x)} be a twice differentiable function, and define the function h ( x ) ≜ φ ( x ) − φ ( μ ) ( x − μ )

\mathbb {R} ^{mn}} . The argument of J {\displaystyle J} is a differentiable function u : Ω → R m {\displaystyle u:\Omega \to \mathbb {R} ^{m}} , and

beginning with an initial guess x0. If f is a d + 1 times continuously differentiable function and a is a zero of f but not of its derivative, then, in a neighborhood

properties from marginalist theory which assumed utility to be a differentiable function of quantified goods and services. But it came to be seen that indifference

{\delta G[\rho ](x)}{\delta \rho (y)}}\ .} If G is an ordinary differentiable function (local functional) g, then this reduces to δ F [ g ( ρ ) ] δ ρ

\mu )} should converge to a solution of (1). The gradient of a differentiable function h : R n → R {\displaystyle h:\mathbb {R} ^{n}\to \mathbb {R} }

Let u : M → R {\displaystyle u:M\to \mathbb {R} } be a twice-differentiable function which attains its maximum value C. Suppose that a i j ∂ 2 u ∂ x

information learned from each event. I(p) is a twice continuously differentiable function of p. Given two independent events, if the first event can yield

worth function, for example, assuming v can be represented as differentiable function of a non-atomic measure on I, μ, v ( c ) = f ( μ ( c ) ) {\displaystyle

be much more difficult. Notice that by adding a suitable known differentiable function to y, whose values at a and b satisfy the desired boundary conditions

{\displaystyle f} is a k {\displaystyle k} -times continuously differentiable function and all derivatives up to the k {\displaystyle k} th one decay

by Karl Weierstrass as an example of a continuous but nowhere differentiable function. The Weierstrass–Mandelbrot function builds on this foundation

that giving up this demand means that the wave function is not a differentiable function at the boundary of the box, and thus it can be said that the wave

\mathbb {R} ^{2}} , but is not complex differentiable. A real differentiable function is complex differentiable if and only if it satisfies the Cauchy–Riemann

application of differential calculus. The basic way to maximize a differentiable function is to find the stationary points (the points where the derivative

(56) follows that i.e. e {\displaystyle \mathbf {e} } is a smooth differentiable function of the state vector ( r , v ) {\displaystyle (\mathbf {r} ,\mathbf

dt+\sigma _{t}\,dB_{t}} says that the differential of a twice-differentiable function f(t, x) is given by d f = ( ∂ f ∂ t + μ t ∂ f ∂ x + σ t 2 2 ∂ 2

given as follows: The time evolution of the state is given by a differentiable function from the real numbers R, representing instants of time, to the

∂x⁠ then gives the incompressibility constraint ∇ · v = 0. Any differentiable function may be used for f. The examples that follow use a variety of elementary

} where p is a strictly positive continuously differentiable function and q and r are continuous real-valued functions. For x0 in (a

{f} ].} Let U be an open set in ℝn, and let φ be a continuously differentiable function from U into the Euclidean space ℝm, where m > n. The mapping φ

( x , σ ) {\displaystyle f(\mathbf {x} ,\sigma )} be a smooth differentiable function on U ⊂ R 2 × R + {\displaystyle U\subset \mathbb {R} ^{2}\times

}{\mathrm {d} x}}f(x)\,,} where i is the imaginary unit and f is a differentiable function of compact support. The multiplication-by-x operator: ( B f ) (

transformation, one starts by considering an arbitrary increasing, twice-differentiable function of r {\displaystyle r} , say G ( r ) {\displaystyle G(r)} . Finding

into an output volume (e.g. holding the class scores) through a differentiable function. A few distinct types of layers are commonly used. These are further

degree k {\displaystyle k} ) is a k + 1 {\displaystyle k+1} times differentiable function f {\displaystyle f} defined by [ f ] ( [ x ] ) := f ( y ) + ∑ i

continuation shows that the principal nth root is the unique complex differentiable function that extends the usual nth root to the complex plane without the

^{+}\rightarrow \mathbb {R} ^{+}} is a strictly increasing and differentiable function such that g ( 0 ) = 0 {\displaystyle g(0)=0} , then we can define

M\}\}_{M=0}=0,} is where O {\displaystyle O} is at least a twice differentiable function on phase space is equivalent to O {\displaystyle O} being a weak

Euler–Maclaurin formula. Assuming that f is a sufficiently often differentiable function the Euler–Maclaurin formula can be written as ∑ k = a b − 1 f (

set up to apply the implicit function theorem. Our continuously differentiable function is the rigidity map ρ : R 2 | V | − 3 → R | E | {\displaystyle

functions. Let 1 < p < ∞ {\displaystyle 1<p<\infty } and define a differentiable function F p : [ 0 , 1 ] → R {\displaystyle F_{p}:[0,1]\rightarrow {\mathbb

Ω is a bounded open region with smooth boundary ∂Ω and h is a differentiable function on Ω extending to a continuous function on the closure, then, by

under the above redefinition of the potentials for an arbitrary differentiable function F {\displaystyle F} . In order to construct a stochastic Lagrangian

the Bochner–Martinelli formula. Suppose that f is a continuously differentiable function on the closure of a domain D on C n {\displaystyle \mathbb {C}

10^{-3}} . The most widely used method for computing a root of any differentiable function f {\displaystyle f} is Newton's method, in which an initial guess

^{n}} be an ⌊ γ ⌋ {\displaystyle \lfloor \gamma \rfloor } -times differentiable function and the ⌊ γ ⌋ {\displaystyle \lfloor \gamma \rfloor } -th derivative

(t_{0})=\mathbf {x} _{0}} , where f {\displaystyle \mathbf {f} } is a differentiable function. Let ( t ) h = { t n : n = 0 , . . , N } {\displaystyle

N_{n})}{\partial \alpha _{i}}}=\{f,H_{i}\},\quad 1\leq i\leq n} for any differentiable function f ( N 1 , … , N n ) {\displaystyle f(N_{1},\dots ,N_{n})} . The

)} . subanalytic subanalytic. subharmonic A twice continuously differentiable function f {\displaystyle f} is said to be subharmonic if Δ f ≥ 0 {\displaystyle