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| [[Image:Bump2D illustration.png|right|thumb|250px|A [[bump function]] is a smooth function with [[compact support]].]]
| | Each state and local areas properly check and provide certification to the proper person. Try applying a touch or bronzer in sandwiched between the cleavage to add depth to the area. Massage is a surefire way to get this breast enhancing hormone flowing. Have you ever wondered what it would be like to have bigger, fuller breasts. Make breast breasts appear firmer muscles strong, so helping to create in order to help in breast firming exercises such as push-ups too. <br><br>breast enlargement surgery and you may find you are suitable for cheap breast enlargement. The colloidal gel forms insert inside your brassiere and give you a natural feeling breast. "We try to eat a Mediterranean diet: lots of greens, lots of fish, lots of vegetables," she said. Additionally, it enhances and offers cell respiration and improved cell proliferation by 78% and breast tissue anti-fatique for firmer, tighter and healthier skin. Breast enhancement can be achieved by taking capsules, pills and the use of creams which increase both the size and firmness of the breast. <br><br>Today there are so many different methods like- pills, herbs, teas, suction brassieres etc. Everything from paraffin injections to ivory, glass balls and ox cartilage have been used to enhance the breast, with varying degrees of success. To ensure you’re psychologically, emotionally, and physically ready for breast implants, be 100 percent aware of your motivations for surgery, have reasonable expectations, and thoroughly understand the benefits and risks surrounding the procedure. Replace saline implants when you get breast enhancement surgery. sufficiently before undergoing your breast augmentation surgery. <br><br>The massage should always be started in the very center of the breast. Breast enlargement surgery can increase your breast size by one or more bra cup sizes. They stimulate the growth of the adipose tissue, i. Most women will be prescribed pain medication such as Vicodin or Percocet for about a week post-op surgery. more information about the clinic and it's facilities, as well as. <br><br>Using a breast enlargement cream is another way to stimulate breast growth, most breast creams contain an herb called Pueraria Mirifica which mimics natural estrogen production, when applied directly to the breast it not only stimulates breast growth making the breasts larger, it can also help women reduce sagging in their breasts while at the same time making them fuller and perkier. The patients own tissue is used to reconstruct the breast. It helps in throwing out toxins and impurities from the body. If a woman is unhappy with her legs, she can exercise and change her diet. You must have patience and wait for them to fade away naturally. <br><br>Oral breast enhancement work best when consumed with higher protein meals. And PMS may subside with intake of Breast Enlargement Pills. The venous drainage of the breast is mainly to the axillary vein, but there is some drainage to the internal thoracic vein and the intercostal veins. In the United States, cancer kills more people than nearly all other causes except heart disease. It is true that a better and more natural look can be obtained from breast uplift surgery.<br><br>If you are you looking for more information on how to increase your breast size naturally - [http://www.lucky-house.info/ www.lucky-house.info] - take a look at our web-site. |
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| In [[mathematical analysis]], a [[function (mathematics)|function]] that has [[derivative]]s of all [[Order of derivation|orders]] is called '''smooth'''.
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| ==Differentiability classes==
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| '''Differentiability class''' is a classification of functions according to the properties of their [[derivative]]s. Higher order differentiability classes correspond to the existence of more derivatives.
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| Consider an [[open set]] on the [[real line]] and a function ''f'' defined on that set with real values. Let ''k'' be a non-negative [[integer]]. The function ''f'' is said to be of (differentiability) '''class ''C<sup>k</sup>''''' if the derivatives ''f''′, ''f''′′, ..., ''f<sup>(k)</sup>'' exist and are [[continuous function|continuous]] (the continuity is implied by differentiability for all the derivatives except for ''f<sup>(k)</sup>''). The function ''f'' is said to be of '''class ''C<sup>∞</sup>''''', or '''smooth''', if it has derivatives of all orders.<ref name="def diff">Warner (1983), {{Google books quote|id=t6PNrjnfhuIC|page=5|text=f is differentiable of class Ck|p. 5, Definition 1.2}}.</ref> The function ''f'' is said to be of '''class ''C''<sup>ω</sup>''', or '''[[analytic function|analytic]]''', if ''f'' is smooth and if it equals its [[Taylor series]] expansion around any point in its domain.
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| To put it differently, the class ''C''<sup>0</sup> consists of all continuous functions. The class ''C''<sup>1</sup> consists of all [[differentiable function]]s whose derivative is continuous; such functions are called '''continuously differentiable'''. Thus, a ''C''<sup>1</sup> function is exactly a function whose derivative exists and is of class ''C''<sup>0</sup>. In general, the classes ''C<sup>k</sup>'' can be defined [[recursion|recursively]] by declaring ''C''<sup>0</sup> to be the set of all continuous functions and declaring ''C<sup>k</sup>'' for any positive integer ''k'' to be the set of all differentiable functions whose derivative is in ''C''<sup>''k''−1</sup>. In particular, ''C<sup>k</sup>'' is contained in ''C''<sup>''k''−1</sup> for every ''k'', and there are examples to show that this containment is strict. ''C''<sup>∞</sup> is the intersection of the sets ''C<sup>k</sup>'' as ''k'' varies over the non-negative integers. ''C''<sup>ω</sup> is strictly contained in ''C''<sup>∞</sup>; for an example of this, see [[bump function]] or also below.
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| ===Examples===
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| [[Image:C0 function.svg|right|thumb|The ''C<sup>0</sup>'' function ''f''(''x'')=''x'' for ''x''≥0 and 0 otherwise.]]
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| [[Image:TV pic3.png|thumb|right|The function ''f''(''x'')=''x''<sup>2</sup> sin(1/''x'') for ''x''>0.]]
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| [[Image:Mollifier illustration.png|right|thumb|300px|A smooth function that is not analytic.]]
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| The function
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| : <math>f(x) = \begin{cases}x & \mbox{if }x \ge 0, \\ 0 &\mbox{if }x < 0\end{cases}</math>
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| is continuous, but not differentiable at ''x'' = 0, so it is of class ''C<sup>0</sup>'' but not of class ''C<sup>1</sup>''.
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| The function
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| :<math>f(x) = \begin{cases}x^2\sin{(\tfrac{1}{x})} & \mbox{if }x \neq 0, \\ 0 &\mbox{if }x = 0\end{cases}</math>
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| is differentiable, with derivative
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| :<math>f'(x) = \begin{cases}-\mathord{\cos(\tfrac{1}{x})} + 2x\sin(\tfrac{1}{x}) & \mbox{if }x \neq 0, \\ 0 &\mbox{if }x = 0.\end{cases}</math>
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| Because cos(1/''x'') oscillates as ''x'' → 0, ''f'' ’(''x'') is not continuous at zero. Therefore, this function is differentiable but not of class ''C''<sup>1</sup>. Moreover, if one takes {{nowrap begin}}''f''(''x'') = ''x''<sup>4/3</sup>sin(1/''x''){{nowrap end}} {{nowrap begin}}(''x'' ≠ 0){{nowrap end}} in this example, it can be used to show that the derivative function of a differentiable function can be unbounded on a [[compact set]] and, therefore, that a differentiable function on a compact set may not be locally [[Lipschitz continuous]].
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| The functions
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| : <math>f(x)=|x|^{k+1}</math>
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| where ''k'' is even, are continuous and ''k'' times differentiable at all ''x''. But at ''x'' = 0 they are not (''k''+1) times differentiable, so they are of class ''C<sup>k</sup>'' but not of class ''C<sup>j</sup>'' where ''j'' > ''k''.
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| The [[exponential function]] is analytic, so, of class ''C''<sup>ω</sup>. The [[trigonometric function]]s are also analytic wherever they are defined.
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| The function
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| :<math>f(x) = \begin{cases}e^{-\frac{1}{1-x^2}} & \mbox{ if } |x| < 1, \\ 0 &\mbox{ otherwise }\end{cases}</math>
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| is smooth, so of class ''C<sup>∞</sup>'', but it is not analytic at ''x'' = ±1, so it is not of class ''C''<sup>ω</sup>. The function ''f'' is an example of a smooth function with [[compact support]].
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| ===Multivariate differentiability classes===
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| Let ''n'' and ''m'' be some positive integers. If ''f'' is a function from an open subset of '''R'''<sup>''n''</sup> with values in '''R'''<sup>''m''</sup>, then ''f'' has component functions ''f''<sub>1</sub>, ..., ''f<sub>m</sub>''. Each of these may or may not have [[partial derivative]]s. We say that ''f'' is of '''class ''C<sup>l</sup>''''' if all of the partial derivatives <math>\frac{\partial^\ell f_i}{\partial x_{i_1}^{\ell_1}\partial x_{i_2}^{\ell_2}\cdots\partial x_{i_n}^{\ell_n}}</math> exist and are continuous, where each of <math>i, i_1, i_2, \ldots, i_k</math> is an integer between 1 and ''n'', each of <math>\ell, \ell_1, \ell_2, \ldots, \ell_n </math> is an integer between 0 and ''l'', <math>\ell_1+\ell_2+\cdots + \ell_n = \ell </math>.<ref name="def diff" /> The classes ''C<sup>∞</sup>'' and ''C<sup>ω</sup>'' are defined as before.<ref name="def diff" />
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| These criteria of differentiability can be applied to the transition functions of a [[differential structure]]. The resulting space is called a ''C<sup>k</sup>'' [[manifold]].
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| If one wishes to start with a coordinate-independent definition of the '''class ''C<sup>k</sup>''''', one may start by considering maps between [[Banach spaces]]. A map from one Banach space to another is differentiable at a point if there is an affine map which approximates it at that point. The derivative of the map assigns to the point ''x'' the linear part of the affine approximation to the map at ''x''. Since the space of linear maps from one Banach space to another is again a Banach space, we may continue this procedure to define higher order derivatives. A map ''f'' is of '''class ''C<sup>k</sup>''''' if it has continuous derivatives up to order ''k'', as before.
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| Note that '''R'''<sup>''n''</sup> is a Banach space for any value of ''n'', so the coordinate-free approach is applicable in this instance. It can be shown that the definition in terms of partial derivatives and the coordinate-free approach are equivalent; that is, a function ''f'' is of '''class ''C<sup>k</sup>''''' by one definition iff it is so by the other definition.
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| ===The space of ''C''<sup>''k''</sup> functions===
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| Let ''D'' be an open subset of the real line. The set of all ''C<sup>k</sup>'' functions defined on ''D'' and taking real values is a [[Fréchet space|Fréchet vector space]] with the countable family of [[seminorm]]s
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| : <math>p_{K, m}=\sup_{x\in K}\left|f^{(m)}(x)\right|</math>
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| where ''K'' varies over an increasing sequence of [[compact set]]s whose [[union (set theory)|union]] is ''D'', and ''m'' = 0, 1, …, ''k''.
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| The set of ''C''<sup>∞</sup> functions over ''D'' also forms a Fréchet space. One uses the same seminorms as above, except that ''m'' is allowed to range over all non-negative integer values.
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| The above spaces occur naturally in applications where functions having derivatives of certain orders are necessary; however, particularly in the study of [[partial differential equation]]s, it can sometimes be more fruitful to work instead with the [[Sobolev space]]s.
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| ==Parametric continuity==
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| '''Parametric continuity''' is a concept applied to [[parametric curve]]s describing the smoothness of the parameter's value with distance along the curve.
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| ===Definition===
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| A curve can be said to have ''C''<sup>''n''</sup> continuity if <math>\displaystyle\frac{d^ns}{dt^n}</math> is continuous of value throughout the curve.
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| As an example of a practical application of this concept, a curve describing the motion of an object with a parameter of time, must have ''C''<sup>1</sup> continuity for the object to have finite acceleration. For smoother motion, such as that of a camera's path while making a film, higher orders of parametric continuity are required.
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| ===Order of continuity===
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| [[Image:parametric continuity c0.gif|right|thumb|Two [[Bézier curve]] segments attached that is only C<sup>0</sup> continuous.]]
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| [[File:Parametric continuity vector.svg|right|thumb|Two Bézier curve segments attached in such a way that they are C<sup>1</sup> continuous.]]
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| The various order of parametric continuity can be described as follows:<ref>[http://www.cs.helsinki.fi/group/goa/mallinnus/curves/curves.html Parametric Curves<!-- Bot generated title -->]</ref>
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| * ''C''<sup>−1</sup>: curves include discontinuities
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| * ''C''<sup>0</sup>: curves are joined
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| * ''C''<sup>1</sup>: first derivatives are continuous
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| * ''C''<sup>2</sup>: first and second derivatives are continuous
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| * ''C<sup>n</sup>'': first through ''n''<sup>th</sup> derivatives are continuous
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| The term ''parametric continuity'' was introduced to distinguish it from ''geometric continuity'' (''G<sup>n</sup>'') which removes restrictions on the [[speed]] with which the parameter traces out the curve.<ref>{{Harv|Bartels|Beatty|Barsky|1987|loc=Ch. 13}}</ref>
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| ==Geometric continuity==
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| The concept of '''geometrical''' or '''geometric continuity''' was primarily applied to the [[conic sections]] and related shapes by mathematicians such as [[Gottfried Leibniz|Leibniz]], [[Johannes Kepler|Kepler]], and [[Jean-Victor Poncelet|Poncelet]]. The concept was an early attempt at describing, through geometry rather than algebra, the concept of [[continuous function|continuity]] as expressed through a parametric function.
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| The basic idea behind geometric continuity was that the five conic sections were really five different versions of the same shape. An [[ellipse]] tends to a [[circle]] as the [[eccentricity (mathematics)|eccentricity]] approaches zero, or to a [[parabola]] as it approaches one; and a [[hyperbola]] tends to a [[parabola]] as the eccentricity drops toward one; it can also tend to intersecting [[line (geometry)|line]]s. Thus, there was ''continuity'' between the conic sections. These ideas led to other concepts of continuity. For instance, if a circle and a straight line were two expressions of the same shape, perhaps a line could be thought of as a circle of infinite [[radius]]. For such to be the case, one would have to make the line closed by allowing the point ''x'' = ∞ to be a point on the circle, and for ''x'' = +∞ and ''x'' = −∞ to be identical. Such ideas were useful in crafting the modern, algebraically defined, idea of the [[continuous function|continuity]] of a function and of [[Real projective line|∞]].
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| === Smoothness of curves and surfaces ===
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| A [[curve]] or [[surface]] can be described as having ''G<sup>n</sup>'' continuity, ''n'' being the increasing measure of smoothness. Consider the segments either side of a point on a curve:
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| *''G''<sup>0</sup>: The curves touch at the join point.
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| *''G''<sup>1</sup>: The curves also share a common [[tangent]] direction at the join point.
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| *''G''<sup>2</sup>: The curves also share a common center of curvature at the join point.
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| In general, ''G<sup>n</sup>'' continuity exists if the curves can be reparameterized to have ''C''<sup>''n''</sup> (parametric) continuity.<ref>Brian A. Barsky and Tony D. DeRose, "Geometric Continuity of Parametric Curves: Three Equivalent Characterizations," IEEE Computer Graphics and Applications, 9(6), Nov. 1989, pp. 60–68.</ref> A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.
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| Equivalently, two vector functions ''f''(''t'') and ''g''(''t'') have ''G<sup>n</sup>'' continuity if ''f<sup>(n)</sup>''(''t'') ≠ 0 and ''f<sup>(n)</sup>''(''t'') ≡ ''kg<sup>(n)</sup>''(''t''), for a scalar ''k'' > 0 (i.e., if the direction, but not necessarily the magnitude, of the two vectors is equal).
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| While it may be obvious that a curve would require ''G''<sup>1</sup> continuity to appear smooth, for good [[aesthetics]], such as those aspired to in [[architecture]] and [[sports car]] design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has ''G''<sup>2</sup> continuity.
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| A ''rounded rectangle'' (with ninety degree circular arcs at the four corners) has ''G''<sup>1</sup> continuity, but does not have ''G''<sup>2</sup> continuity. The same is true for a ''rounded cube'', with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with ''G''<sup>2</sup> continuity is required, then [[cubic splines]] are typically chosen; these curves are frequently used in [[industrial design]].
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| ==Smoothness==
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| ===Relation to analyticity===
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| While all [[analytic function]]s are smooth on the set on which they are analytic, the above example shows that the converse is not true for functions on the reals: there exist smooth real functions which are not analytic. Simple examples of functions which are [[Non-analytic_smooth_function#A_smooth_function_which_is_nowhere_real_analytic|smooth but not analytic at any point]] can be made by means of [[Fourier series]]; another example is the [[Fabius function]]. Although it might seem that such functions are the exception rather than the rule, it turns out that the analytic functions are scattered very thinly among the smooth ones; more rigorously, the analytic functions form a [[meagre set|meagre]] subset of the smooth functions. Furthermore, for every open subset A of the real line, there exist smooth functions which are analytic on A and nowhere else.
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| It is useful to compare the situation to that of the ubiquity of [[transcendental number]]s on the real line. Both on the real line and the set of smooth functions, the examples we come up with at first thought (algebraic/rational numbers and analytic functions) are far better behaved than the majority of cases: the transcendental numbers and nowhere analytic functions have full measure (their complements are meagre).
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| The situation thus described is in marked contrast to complex differentiable functions. If a complex function is differentiable just once on an open set it is both infinitely differentiable and analytic on that set.
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| ===Smooth partitions of unity===
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| Smooth functions with given closed [[support (mathematics)|support]] are used in the construction of '''smooth partitions of unity''' (see ''[[partition of unity]]'' and [[topology glossary]]); these are essential in the study of [[smooth manifold]]s, for example to show that [[Riemannian metric]]s can be defined globally starting from their local existence. A simple case is that of a '''[[bump function]]''' on the real line, that is, a smooth function ''f'' that takes the value 0 outside an interval [''a'',''b''] and such that
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| :<math>f(x) > 0 \quad \text{ for } \quad a < x < b.\,</math>
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| 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.
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| From what has just been said, partitions of unity don't apply to [[holomorphic function]]s; their different behavior relative to existence and [[analytic continuation]] is one of the roots of [[sheaf (mathematics)|sheaf]] theory. In contrast, sheaves of smooth functions tend not to carry much topological information.
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| ===Smooth functions between manifolds===
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| '''Smooth maps''' between [[smooth manifold]]s may be defined by means of [[chart (topology)|chart]]s, since the idea of smoothness of function is independent of the particular chart used. If ''F'' is a map from an ''m''-manifold ''M'' to an ''n''-manifold ''N'', then ''F'' is smooth if, for every ''p'' ∈ ''M'', there is a chart (''U'', φ) in ''M'' containing ''p'' and a chart (''V'', ψ) in ''N'' containing ''F''(''p'') with ''F''(''U'') ⊂ ''V'', such that <math>\scriptstyle\psi\circ F \circ \varphi^{-1}</math> is smooth from φ(''U'') to ψ(''V'') as a function from '''R'''<sup>''m''</sup> to '''R'''<sup>''n''</sup>.
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| Such a map has a first [[derivative]] defined on [[Tangent space|tangent vectors]]; it gives a fibre-wise linear mapping on the level of [[tangent bundle]]s.
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| ===Smooth functions between subsets of manifolds===
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| There is a corresponding notion of '''smooth map''' for arbitrary subsets of manifolds. If ''f'' : ''X'' → ''Y'' is a [[function (mathematics)|function]] whose [[domain of a function|domain]] and [[range (mathematics)|range]] are subsets of manifolds ''X'' ⊂ ''M'' and ''Y'' ⊂ ''N'' respectively. ''f'' is said to be '''smooth''' if for all ''x'' ∈ ''X'' there is an open set ''U'' ⊂ ''M'' with ''x'' ∈ ''U'' and a smooth function ''F'' : ''U'' → ''N'' such that ''F''(''p'') = ''f''(''p'') for all ''p'' ∈ ''U'' ∩ ''X''.
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| ==See also==
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| * [[Non-analytic smooth function]]
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| * [[Quasi-analytic function]]
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| * [[Spline_(mathematics)|Spline]]
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| ==References==
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| {{More footnotes|date=May 2009}}
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| {{Reflist}}
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| * {{1911}}
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| * Guillemin, Pollack. Differential Topology. Prentice-Hall (1974).
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| * {{cite book|last=Warner| first=Frank Wilson| author-link=Frank Wilson Warner| year=1983| title=Foundations of differentiable manifolds and Lie groups| publisher=Springer| isbn = 978-0-387-90894-6}}
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| [[Category:Smooth functions]]
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Each state and local areas properly check and provide certification to the proper person. Try applying a touch or bronzer in sandwiched between the cleavage to add depth to the area. Massage is a surefire way to get this breast enhancing hormone flowing. Have you ever wondered what it would be like to have bigger, fuller breasts. Make breast breasts appear firmer muscles strong, so helping to create in order to help in breast firming exercises such as push-ups too.
breast enlargement surgery and you may find you are suitable for cheap breast enlargement. The colloidal gel forms insert inside your brassiere and give you a natural feeling breast. "We try to eat a Mediterranean diet: lots of greens, lots of fish, lots of vegetables," she said. Additionally, it enhances and offers cell respiration and improved cell proliferation by 78% and breast tissue anti-fatique for firmer, tighter and healthier skin. Breast enhancement can be achieved by taking capsules, pills and the use of creams which increase both the size and firmness of the breast.
Today there are so many different methods like- pills, herbs, teas, suction brassieres etc. Everything from paraffin injections to ivory, glass balls and ox cartilage have been used to enhance the breast, with varying degrees of success. To ensure you’re psychologically, emotionally, and physically ready for breast implants, be 100 percent aware of your motivations for surgery, have reasonable expectations, and thoroughly understand the benefits and risks surrounding the procedure. Replace saline implants when you get breast enhancement surgery. sufficiently before undergoing your breast augmentation surgery.
The massage should always be started in the very center of the breast. Breast enlargement surgery can increase your breast size by one or more bra cup sizes. They stimulate the growth of the adipose tissue, i. Most women will be prescribed pain medication such as Vicodin or Percocet for about a week post-op surgery. more information about the clinic and it's facilities, as well as.
Using a breast enlargement cream is another way to stimulate breast growth, most breast creams contain an herb called Pueraria Mirifica which mimics natural estrogen production, when applied directly to the breast it not only stimulates breast growth making the breasts larger, it can also help women reduce sagging in their breasts while at the same time making them fuller and perkier. The patients own tissue is used to reconstruct the breast. It helps in throwing out toxins and impurities from the body. If a woman is unhappy with her legs, she can exercise and change her diet. You must have patience and wait for them to fade away naturally.
Oral breast enhancement work best when consumed with higher protein meals. And PMS may subside with intake of Breast Enlargement Pills. The venous drainage of the breast is mainly to the axillary vein, but there is some drainage to the internal thoracic vein and the intercostal veins. In the United States, cancer kills more people than nearly all other causes except heart disease. It is true that a better and more natural look can be obtained from breast uplift surgery.
If you are you looking for more information on how to increase your breast size naturally - www.lucky-house.info - take a look at our web-site.