Examples of "hölder_condition"
for some formula_15, where formula_16 denotes the Hölder condition.
There is both an "interior" result, giving a Hölder condition for the solution in interior domains away from the boundary, and a "boundary" result, giving the Hölder condition for the solution in the entire domain. The former bound depends only on the spatial dimension, the equation, and the distance to the boundary; the latter depends on the smoothness of the boundary as well.
Let formula_160 be a formula_11-geometric rough path satisfying the Hölder condition that there exists formula_155, for all formula_163 and all formula_164,
This holds more generally for scalar functions on a compact metric space satisfying a Hölder condition with respect to the metric on .
for all "x" and "y" in "X". Sometimes a Hölder condition of order α is also called a uniform Lipschitz condition of order α > 0.
In mathematics, a real or complex-valued function "f" on "d"-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants "C", α, such that
More generally, a function "f" defined on "X" is said to be Hölder continuous or to satisfy a Hölder condition of order α > 0 on "X" if there exists a constant "M" > 0 such that
Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the "L"-setting. For formula_55 and formula_56 the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by
Riemannian manifolds are special cases of metric spaces, and thus one has a notion of Lipschitz continuity, Hölder condition, together with a coarse structure, which leads to notions such as coarse maps and connections with geometric group theory.
Due to the nonlinearity in the equation and the presence of space-time white-noise, the mathematical study of the KPZ equation has proven to be quite challenging: indeed, even without the nonlinear term, the equation reduces to the stochastic heat equation, whose solution is not differentiable in the space variable but verifies a Hölder condition with exponent < 1/2. Thus, the nonlinear term formula_21 is ill-defined in a classical sense.
for all "x" and "y" in the domain of "f". More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the "exponent" of the Hölder condition. If α = 1, then the function satisfies a Lipschitz condition. If α > 0, the condition implies the function is continuous. If α = 0, the function need not be continuous, but it is bounded. The condition is named after Otto Hölder.
KZ filter can be used to smooth the periodogram. For a class of stochastic processes, Zurbenko considered the worst-case scenario where the only information available about a process is its spectral density and smoothness quantified by Hölder condition. He derived the optimal bandwidth of the spectral window, which is dependent upon the underlying smoothness of the spectral density. Zurbenko compared the performance of Kolmogorov-Zurbenko (KZ) window to the other typically used spectral windows including Bartlett window, Parzen window, Tukey-Hamming window and uniform window and showed that the result from KZ window is closest to optimum.
He is noted for many theorems including: Hölder's inequality, the Jordan–Hölder theorem, the theorem stating that every linearly ordered group that satisfies an Archimedean property is isomorphic to a subgroup of the additive group of real numbers, the classification of simple groups of order up to 200, the anomalous outer automorphisms of the symmetric group "S" and Hölder's theorem which implies that the Gamma function satisfies no algebraic differential equation. Another idea related to his name is the Hölder condition (or Hölder continuity) which is used in many areas of analysis, including the theories of partial differential equations and function spaces.
Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations, and in dynamical systems. The Hölder space "C"(Ω), where Ω is an open subset of some Euclidean space and "k" ≥ 0 an integer, consists of those functions on Ω having continuous derivatives up to order "k" and such that the "k"th partial derivatives are Hölder continuous with exponent α, where 0 < α ≤ 1. This is a locally convex topological vector space. If the Hölder coefficient