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Topic: Smooth function

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	Encyclopedia :: encyclopedia : Smooth function (Site not responding. Last check: 2007-10-09)
		For example, the exponential function is evidently smooth because the derivative of the exponential function is the exponential function itself.
		Smooth functions with given closed support are used in the construction of smooth partitions of unity (see topology glossary for partition of unity); these are essential in the study of smooth manifolds, for example to show that Riemannian metrics can be defined globally starting from their local existence.
		Smooth maps between smooth manifolds may be defined by means of charts, since the idea of smoothness of function is independent of the particular chart used.
	www.hallencyclopedia.com /topic/Smooth_function.html (610 words)

	Smooth Muscle
		Smooth muscle cells have a region around the nucleus that is filled with some smooth endoplasmic reticulum (SER) and lots of mitochondria.
		In the case of smooth muscle, alpha actinin is among the binding proteins.
		Smooth muscle cells are unique in that they produce the components of their extracellular matrix.
	www.cytochemistry.net /microanatomy/muscle/smooth_muscle_2001.htm (1729 words)

	SMOOTH MUSCLE
		Smooth muscle is responsible for the contractility of hollow organs, such as blood vessels, the gastrointestinal tract, the bladder, or the uterus.
		In the case of smooth muscle, this excitation-contraction coupling is termed electromechanical coupling; the link for the coupling is Ca that permeates from the extracellular space into the intracellular water of smooth muscle.
		Upon contraction of smooth muscle, the exchange of the bound-nucleotide and phosphate decreased and upon relaxation from the contracted state it increased, suggesting that polymerization-deplolymerization of actin is a part of the contraction-relaxation cycle of smooth muscle.
	www.uic.edu /classes/phyb/phyb516/smoothmuscleu3.htm (5492 words)

	PlanetMath: differentiable function
		Differentiable functions are often referred to as smooth.
		smooth function, differentiable mapping, differentiable map, smooth mapping, smooth map, continuously differentiable
		This is version 21 of differentiable function, born on 2002-05-19, modified 2006-06-08.
	planetmath.org /encyclopedia/Smooth.html (370 words)

	More on Smoothness
		A function is called C1 if it has a derivative that is a continuous function; such functions are also called continuously differentiable.
		The smooth functions are therefore those that lie in the class Cn for all n.
		Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals (-∞, c] and [d,+∞) to cover the whole line, such that the sum of the functions is always 1.
	www.artilifes.com /smoothness.htm (701 words)

	Smooth function - Wikipedia, the free encyclopedia
		In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i.e., has derivatives of all finite orders:
		The smooth functions are those that lie in the class C
		This shows that there is a large gap between smooth and analytic functions; and that in general smooth functions do not necessarily equal their Taylor's series.
	en.wikipedia.org /wiki/Smooth_function (661 words)

	PlanetMath: germ of smooth functions
		To be precise, a germ of smooth functions near
		"germ of smooth functions" is owned by draisma.
		This is version 1 of germ of smooth functions, born on 2002-10-01.
	planetmath.org /encyclopedia/GermOfSmoothFunctions.html (103 words)

	[No title]
		In distribution theory, the derivative of the delta function (at zero) is the distribution which assigns to each smooth function f with compact support the value -f'(0).
		The "physics" definition of the delta function as a functions such that integral(from -inf to inf) of delta(x)f(x) dx= f(0) is such a definition.
		A sequence {f_n} of functions is called "fundamental" if there exist some convergent sequence {g_n} and a positive integer k such that the kth derivative of each g_n equals f_n (Notice that a fundamental sequence is not necessarily convergent itself in the usual sense).
	www.math.niu.edu /~rusin/known-math/00_incoming/delta2 (1547 words)

	SIU SOM Histology GI
		Electrical activitation of smooth muscle is passed from cell to cell by gap junctions.
		Each smooth muscle cell (or "muscle fiber") is just a few microns in diameter but may be two hundred microns long.
		Smooth muscle may also be difficult to recognize when cut obliquely, so that nuclei appear neither small and round nor long and cigar shaped.
	www.siumed.edu /~dking2/ssb/smoothm.htm (622 words)

	SIU SOM Histology GI
		The muscle tissue of the GI tract is smooth muscle, except for that in the oral cavity (lips, tongue, palate) and upper esophagus.
		Smooth muscle also forms a delicate muscularis mucosae at the deep boundary of the mucosa (i.e., between lamina propria and submucosa).
		Each smooth muscle cell (or "muscle fiber") is just a few microns in diameter but a couple hundred microns long.
	www.siumed.edu /~dking2/erg/smoothm.htm (576 words)

	CALCULUS FOR STATICS
		As shown in the figure, the derivative of the function f(x) at point x gives the slope of the function at x in terms of the ratio of the rise divided by the run for the line AB that is tangent to the curve at point x.
		A point on a smooth function where the derivative is zero is a local maximum, a local minimum, or an inflection point of the function.
		The global maximum or minimum of a smooth function in a specific interval of its argument occurs either at the limits of the interval or at a point inside the interval where the function has a derivative of zero.
	em-ntserver.unl.edu /Math/mathweb/calculus/calcsb97.html (1954 words)

	Smoothing
		For example, "3" is the command for a median-of-three smooth and "H" is the command for Hanning.
		At the first appearance of a comma, the smoothed sequence will be saved, to be replaced (temporarily) by its "rough", which is the series of residuals (differences between the original and smoothed values).
		The smoothed value at location i in the series x[1], x[2],..., x[N] is the middle value of {x[i-K], x[i-K+1],..., x[i+K]}, provided i-K >= 1 and i+K <= N. Otherwise, the smoothed value is the original value (except for medians of three).
	www.quantdec.com /Excel/smoothing.htm (2062 words)

	: Class TableFunction
		Note that this class is meant to be used for functions that are defined by a fairly small number of points, since each function evaluation involves a linear search through the list of x-values of the defining points.
		If the style of the function is set to STEP, then the function is piecewise constant, and the value of the function at x is taken from the nearest point in the list of points that define the function.
		If the style of the function is set to STEP_LEFT, then the function is piecewise constant, and the value of the function at x is taken from the nearest point to the left in the list of points that define the function.
	math.hws.edu /javamath/javadoc/edu/hws/jcm/functions/TableFunction.html (1372 words)

	Implicit function Summary
		This technique is a consequence of a general theorem, called the implicit function theorem, which enables mathematicians to produce a limitless supply of smooth curves and surfaces.
		In general, the implicit function theorem states that an equation g(x, y) = C(where g is a continuously differentiable function and C is any constant) locally defines y as an implicit function of x in a neighborhood of any point where the partial derivative of g with respect to y is nonzero.
		It can be shown that an equation g(x, y) = C defines a smooth curve at all points except where both partial derivatives of g are 0, and similarly that an equation g(x, y, z) = C defines a smooth surface at all points except where all three partial derivatives are 0.
	www.bookrags.com /Implicit_function (931 words)

	R: Constructor functions for smooth terms in a GAM (Site not responding. Last check: 2007-10-09)
		Smooth terms in a GAM formula are turned into smooth specification objects of class
		is the number of covariates of which this smooth is a function.
		The dimension of the space of functions that have zero wiggliness according to the wiggliness penalty for this term.
	www.oulu.fi /atkk/tkpalv/unix/R/library/mgcv/html/smooth.construct.html (663 words)

	R: Define tensor product smooths in GAM formulae
		This parameterization represents any 1-d marginal smooths using a parameterization where the parameters are function values at `knots' spread evenly through the data.
		Smooths of several covariates can be constructed from tensor products of the bases used to represent smooths of one (or sometimes more) of the covariates.
		Tensor product smooths are especially useful for representing functions of covariates measured in different units, although they are typically not quite as nicely behaved as t.p.r.s.
	spider.stat.umn.edu /R/library/mgcv/html/te.html (734 words)

	Optimization Problem Types - Smooth Nonlinear Optimization
		A smooth nonlinear programming (NLP) or nonlinear optimization problem is one in which the objective or at least one of the constraints is a smooth nonlinear function of the decision variables.
		Nonlinear functions, unlike linear functions, may involve variables that are raised to a power or multiplied or divided by other variables.
		NLP problems and their solution methods require nonlinear functions that are continuous, and (usually) further require functions that are smooth -- which means that derivatives of these functions with respect to each decision variable, i.e.
	www.solver.com /probtype4.htm (626 words)

	475 Sectionally Smooth Note.nb
		We say a function f(x) is sectionally continuous on an interval [a,b] if and only if f is continuous on [a,b], except possibly for a finite number of jump or removable discontinuities.
		We say a function f is sectionally smooth on an interval [a,b] if and only if both f(x) and f '(x) are sectionally continuous on [a,b].
		The graph of a sectionally smooth function has a finite number of removable discontinuities, jumps, or corners in any finite interval in its domain.
	www.bsu.edu /web/mkarls/475SectionallySmoothNote (380 words)

	Non-analytic smooth function - Wikipedia, the free encyclopedia
		In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions.
		Indeed, all holomorphic functions are analytic, so that the failure of f to be analytic in spite of its being infinitely differentiable is an indication of one of the most dramatic differences between real-variable and complex-variable analysis.
		One of the most important applications of this function is the construction of so-called mollifiers, which are important in theories of generalized functions, like e.g.
	en.wikipedia.org /wiki/Non-analytic_smooth_function (322 words)

	Schlumberger Fellowship Awarded to Student Studying Morse Theory
		And when it does, it means that strange things are going on with the physical or biological or sociological phenomenon that the function represents, such as the buckling of a beam or the outbreak of ecological catastrophe or the outbreak of a political revolution.
		For smooth functions, the key property is how they behave near a critical point since near a regular point the behavior is regular - essentially just that of a straight line or a plane in higher dimensions.
		is a function of more than one variable its local geometry near a nondegenerate critical point looks like a saddle, since the graph of the function is a surface that may bend in different directions at a given point.
	hypatia.math.uri.edu /~kulenm/mth381pr/morseth/morseth.html (656 words)

	SMOOTH
		The SMOOTH function is commonly used to take time averages and represent expectations.
		This is because the SMOOTH function has a level implicitly built into it.
		For the SMOOTH function itself, since it is so simple, it is often clearer to use the alternative INTEG formula instead of SMOOTH, but this is a matter of taste.
	www.vensim.com /documentation/html/20480.htm (407 words)

	SMOOTH N (Site not responding. Last check: 2007-10-09)
		If order is 1 this function is almost the same as SMOOTHI and if order is 3 it is almost the same as SMOOTH3I.
		The SMOOTH N function is treated as a discrete delay function, so that its output is constant for each Time Step.
		When this happens the behavior of the SMOOTH N function is essentially the same as the behavior of the DELAY INFORMATION function.
	www.vensim.com /documentation/html/22715.htm (190 words)

	Gamasutra - Features - "A Non-Integer Power Function on the Pixel Shader" (08.01.02]
		We call such a function a smooth conditional because it smoothly and rapidly goes from one value to the other as some threshold is crossed.
		This is therefore a good place where the smooth conditional function could be used.
		This model relies on the use of a power function where the exponent depends on the shininess of the material.
	www.gamasutra.com /features/20020801/beaudoin_03.htm (1693 words)

	SMOOTH
		The SMOOTH function returns a copy of Array smoothed with a boxcar average of the specified width.
		Normally, two-dimensional floating-point arrays are smoothed in one pass.
		SMOOTH should never be called without the NAN keyword if the input array may possibly contain NaN values.
	idlastro.gsfc.nasa.gov /idl_html_help/SMOOTH.html (738 words)

	Soil Water Retention Curve Description Using a Flexible Smooth Function -- Prunty and Casey 1 (1): 179 -- Vadose Zone ...
		Functions by Brooks and Corey (1964) and van Genuchten (1980)
		The Brooks and Corey (BC) function of Eq.
		Prunty, L. Curve fitting with smooth functions that are piecewise-linear in the limit.
	vzj.scijournals.org /cgi/content/full/1/1/179 (2405 words)

	The implicit function theorem
		The implicit function theorem is an important example where the linearized system qualitatively determines the behavior of the nonlinear system.
		The implicit function theorem asserts precisely that this is the case.
		A proof of the implicit function theorem can be found in advanced calculus texts.
	www.math.vt.edu /people/renardym/class_home/nova/bifs/node14.html (172 words)

	Filter/Smooth (Site not responding. Last check: 2007-10-09)
		To Smooth a texture is one of the standard filter operations that you can find in all image editors.
		The valence curve is given by the gaussian bell function.
		Utilize the Mixing Mode within the Mixer to achieve a number of useful effects with the Smooth function.
	www.i-tex.de /help/functions/filter_smooth.htm (117 words)

	Amazon.com: Cellular Aspects of Smooth Muscle Function: Books: C. Y. Kao,Mary E. Carsten (Site not responding. Last check: 2007-10-09)
		The editors have succeeded in developing a set of general principles of smooth muscle function, while still emphasizing the diversity in various organs.
		In particular, they cover the seven most important areas of smooth muscle function including morphology, electrophysiology, mechanisms of electromechanical and pharmacomechanical coupling, calcium homeostasis, signal transduction, mechanics of contraction, and the contractile proteins.
		Smooth muscle has a wide distribution in the body, and its functional specializations are extraordinarily varied.
	www.amazon.com /Cellular-Aspects-Smooth-Muscle-Function/dp/0521482100 (1033 words)

	gw_smooth_generalize
		While the generalization and smoothing of polylines is comparatively simple process, when applied to polygons there are certain complications connected to the topological relationships between the adjacent polygons.
		Smoothing - the process of introducing new vertices in a polyline or polygon boundary in order to achieve shapes with no sharp corners.
		Smooth the polylines (there is no duplicate polylines, so no topological problems will be introduced).
	www.ian-ko.com /ET_GeoWizards/WhitePapers/gw_smooth_generalize.htm (1018 words)

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