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...[smoothing]]. The goal of surface fairing is to compute shapes that are as smooth as possible. ...in order to obtain, e.g., an as-smooth-as-possible surface patch or an as-smooth-as-possible shape deformation.<ref name="Botsch">Polygon Mesh Processing, B ...

The '''Griewank test function''' is a smooth multidimensional mathematical function used in unconstrained optimization. == Non-Smooth Variant of the Griewank Function == ...

...after [[James Serrin]] and [[Norman George Meyers]], states that [[smooth functions]] are dense in the [[Sobolev space]] <math>W^{k,p}(\Omega)</math> ...g Sobolev norm (obtained by summing over the <math>L^p</math> norms of the functions and all derivatives). The theorem establishes the equivalence <math>W^{k,p} ...

...hich contains ''X'' as an open subset.<ref>Griffiths, 1972, p. 286.</ref> Smooth completions exist and are unique over a [[perfect field]]. ...e curve can be embedded in a unique compact [[Riemann surface]] called its smooth completion. The projection of the Riemann surface to <math>\mathbb{C}\math ...

...eorem''', named after S.&nbsp;Ushiki, states that certain [[well-behaved]] functions cannot have certain kinds of well-behaved invariant manifolds. ...h> cannot have a 1-dimensional [[compact space|compact]] [[smooth function|smooth]] [[invariant manifold]]. In particular, such a map cannot have a [[homocl ...

...the Gevrey class <math>G^\sigma (\Omega)</math>, consists of those smooth functions <math>g \in C^\infty(\Omega)</math> such that for every [[compact subset]] ...</math>, but for <math>\sigma > 1</math> there are [[compactly supported]] functions in the class that are not identically zero (an impossibility in <math>C^\om ...

...rs ''S''(''x'',''y'') to be real-valued and smooth, and ''a''(''x'',''y'') smooth and [[compact support|compactly supported]]. Usually one is interested in t ...''C'' such that ''T''_λ, which is initially defined on [[smooth functions]], [[continuous linear extension|extends]] to a [[continuous operator]] fro ...

is flat at <math>x = 0</math>. Thus, this is an example of a [[non-analytic smooth function]]. The pathological nature of this example is partially illuminate [[Category:Smooth functions]] ...

...) of rational functions (which are natural generalizations for meromorphic functions in the non-complex realm) having a zero or a pole at ''P''. ...al ring at a point ''P'' is actually conformed by the germs of holomorphic functions vanishing at ''P''. ...

...i(x, y) = (\phi(x, y), y)</math> with <math>x \in V, y \in W</math>, and a smooth function ''h'' on ''W'' such that ...e extensions to infinite dimensions, to [[complex analytic function]]s, to functions invariant under the [[group action|action]] of a [[compact group]], ... ...

It is a smooth approximation (in fact, an [[analytic function]]) to the [[ramp function]], ...TS |quote=Rectifier and softplus activation functions. The second one is a smooth version of the first. }}</ref> and ''SmoothReLU''<ref>{{Cite web |url=https ...

{{Functions}} ...analysis]]. In particular, many [[function space]]s consist of real-valued functions. ...

...a ''fundamental period''. A definition is given for some of the following functions, though each function may have many equivalent definitions. == Smooth functions == ...

...t=Herbert|author2=John Harer|year=2002|title=Jacobi sets of multiple morse functions|journal=Foundations of Computational Mathematics|publisher=Cambridge Univer ...an also be defined as the set of points where the [[gradients]] of the two functions are parallel. ...

...o ''U'' is invertible is not a sheaf of rings (as adding two non-vanishing functions could provide one which vanishes), and we only get a sheaf of submonoids of ...h> \mathcal{O}_X </math> is considered as a monoid under multiplication of functions. ...

...atical]] study of the regularity of [[harmonic functions]] on sufficiently smooth domains by [[Oliver Dimon Kellogg]]. ...of the boundary are [[Dini continuity|Dini continuous]], then the harmonic functions are uniformly <math>C^k</math> as well. The second, more common version of ...

...rho</math> is said to be [[admissible representation|admissible]] if it is smooth and <math>V^K</math> is finite-dimensional for any open compact subgroup '' ...th> is then called the [[contragredient representation|contragredient]] or smooth dual of <math>(\rho, V)</math>. ...

...ial geometry]] and [[topology]], various types of [[Function (mathematics)|functions]] between [[manifold]]s are studied, both as objects in their own right and ...via [[PDIFF]], the category of [[piecewise]]-smooth maps between piecewise-smooth manifolds. ...

...ath|\alpha \to -\infty}}. The term can also be used loosely for a specific smooth function that behaves similarly to a maximum, without necessarily being par ...png|thumb|Smoothmax of (−x, x) versus x for various parameter values. Very smooth for <math>\alpha</math>=0.5, and more sharp for <math>\alpha</math>=8.]] ...

...ve algebras]]. However, graded manifolds are characterized by sheaves on [[smooth manifold]]s, while supermanifolds are constructed by gluing of sheaves of [ ...<math>(Z,A)</math>. Sections of the sheaf <math>A</math> are called graded functions on a graded manifold <math>(Z,A)</math>. They make up a graded commutative ...