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{{primary sources|date=February 2012}}
[[Image:Total variation.gif|right|frame|As the green ball travels on the graph of the given function, the length of the path travelled by that ball's projection on the ''y''-axis, shown as a red ball, is the total variation of the function.]]


In [[mathematics]], the '''total variation''' identifies several slightly different concepts, related to the ([[local property|local]] or global) structure of the [[codomain]] of a [[Function (mathematics)|function]] or a [[measure (mathematics)|measure]]. For a [[real number|real-valued]] [[continuous function]] ''f'', defined on an [[interval (mathematics)|interval]] [''a'', ''b''] ⊂ ℝ, its total variation on the interval of definition is a measure of the one-dimensional [[arclength]] of the curve with parametric equation ''x'' ↦ ''f''(''x''), for ''x'' ∈ [''a'', ''b''].


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==Historical note==
The concept of total variation for functions of one real variable was first introduced by [[Camille Jordan]] in the paper {{Harv|Jordan|1881}}.<ref>According to {{Harvtxt|Golubov|Vitushkin|2001}}.</ref> He used the new concept in order to prove a convergence theorem for [[Fourier series]] of [[discontinuous function|discontinuous]] [[periodic function]]s whose variation is [[Bounded variation|bounded]]. The extension of the concept to functions of more than one variable however is not simple for various reasons.
 
==Definitions==
 
===Total variation for functions of one real variable===
{{EquationRef|1|Definition 1.1.}} The '''total variation''' of a [[real number|real]]-valued (or more generally [[complex number|complex]]-valued) [[function (mathematics)|function]] ''<math>f</math>'', defined on an [[interval (mathematics)|interval]] <math> [a , b] \subset \mathbb{R}</math> is the quantity
 
:<math> V^a_b(f)=\sup_P \sum_{i=0}^{n_P-1} | f(x_{i+1})-f(x_i) |, \,</math>
 
where the [[supremum]] runs over the [[Set (mathematics)|set]] of all [[partition of an interval|partitions]] <math> \scriptstyle \mathcal{P} =\left\{P=\{ x_0, \dots , x_{n_P}\}|P\text{ is a partition of } [a,b] \right\} </math> of the given [[interval (mathematics)|interval]].
 
===Total variation for functions of ''n'' > 1 real variables===
{{EquationRef|2|Definition 1.2.}} Let '''Ω''' be an [[open subset]] of ℝ<sup>n</sup>. Given a function ''f'' belonging to L<sup>1</sup>('''Ω'''), the '''total variation''' of ''f'' in '''Ω''' is defined as
 
:<math> V(f,\Omega):=\sup\left\{\int_\Omega f(x)\mathrm{div}\phi(x)\,\mathrm{d}x\colon \phi\in  C_c^1(\Omega,\mathbb{R}^n),\ \Vert \phi\Vert_{L^\infty(\Omega)}\le 1\right\}, </math>
 
where  <math> \scriptstyle C_c^1(\Omega,\mathbb{R}^n)</math> is the [[Set (mathematics)|set]] of [[Smooth function|continuously differentiable]] [[Vector valued function|vector functions]] of [[support (mathematics)#Compact support|compact support]] contained  in <math>\Omega</math>, and <math> \scriptstyle \Vert\;\Vert_{L^\infty(\Omega)}</math> is the [[essential supremum]] [[Norm (mathematics)|norm]]. Note that this definition ''does not require'' that the [[Domain of a function|domain]] <math>\Omega \subseteq \mathbb{R}^n</math> of the given function is a [[bounded set]].
 
===Total variation in measure theory===
 
====Classical total variation definition====
Following {{Harvtxt|Saks|1937|p=10}}, consider a [[signed measure]] ''<math>\mu</math>'' on a [[sigma-algebra|measurable space]] <math>(X,\Sigma)</math>: then it is possible to define two [[set function]]s <math>\scriptstyle\overline{\mathrm{W}}(\mu,\cdot)</math> and <math>\scriptstyle\underline{\mathrm{W}}(\mu,\cdot)</math>, respectively called '''upper variation''' and '''lower variation''', as follows
 
:<math>\overline{\mathrm{W}}(\mu,E)=\sup\left\{\mu(A)|A\in\Sigma\text{ and }A\subset E \right\}\qquad\forall E\in\Sigma</math>
:<math>\underline{\mathrm{W}}(\mu,E)=\inf\left\{\mu(A)|A\in\Sigma\text{ and }A\subset E \right\}\qquad\forall E\in\Sigma</math>
 
clearly
 
:<math>\overline{\mathrm{W}}(\mu,E)\geq 0\geq \underline{\mathrm{W}}(\mu,E)\qquad\forall E\in\Sigma</math>
 
{{EquationRef|3|Definition 1.3.}} The '''variation''' (also called '''absolute variation''') of the signed measure ''<math>\mu</math>'' is the set function
 
:<math>|\mu|(E)=\overline{\mathrm{W}}(\mu,E)+\left|\underline{\mathrm{W}}(\mu,E)\right|\qquad\forall E\in\Sigma</math>
 
and its '''total variation''' is defined as the value of this measure on the whole space of definition, i.e.
 
:<math>\|\mu\|=|\mu|(X)</math>
 
====Modern definition of total variation norm====
{{Harvtxt|Saks|1937|p=11}} uses upper and lower variations to prove the [[Hahn decomposition theorem|Hahn&ndash;Jordan decomposition]]: according to his version of this theorem, the upper and lower variation are respectively a [[non-negative]] and a [[non-positive]] [[Measure (mathematics)|measure]]. Using a more modern notation, define
 
:<math>\mu^+(\cdot)=\overline{\mathrm{W}}(\mu,\cdot)\,,</math>
:<math>\mu^-(\cdot)=-\underline{\mathrm{W}}(\mu,\cdot)\,,</math>
 
Then ''<math>\mu^+</math>'' and ''<math>\mu^-</math>'' are two non-negative [[measure (mathematics)|measure]]s such that
 
:<math>\mu=\mu^+-\mu^-\,</math>
:<math>|\mu|=\mu^++\mu^-\,</math>
 
The last measure is sometimes called, by [[abuse of notation]], '''total variation measure'''.
 
====Total variation norm of complex measures====
If the measure ''<math>\mu</math>'' is [[Complex number|complex-valued]] i.e. is a [[complex measure]], its upper and lower variation cannot be defined and the Hahn&ndash;Jordan decomposition theorem can only be applied to its real and imaginary parts. However, it is possible to follow {{Harvtxt|Rudin|1966|pp=137&ndash;139}} and define the total variation of the complex-valued measure ''<math>\mu</math>'' as follows
 
{{EquationRef|4|Definition 1.4.}} The '''variation''' of the complex-valued measure ''<math>\mu</math>'' is the [[set function]]
 
:<math>|\mu|(E)=\sup_\pi \sum_{A\isin\pi} |\mu(A)|\qquad\forall E\in\Sigma</math>
 
where the [[supremum]] is taken over all partitions ''<math>\pi</math>'' of a [[measurable set]] ''<math>E</math>'' into a finite number of disjoint measurable subsets.
 
This definition coincides with the above definition ''<math>|\mu|=\mu^++\mu^-\,</math>'' for the case of real-valued signed measures.
 
====Total variation norm of vector-valued measures====
 
The variation so defined is a [[positive measure]] (see {{Harvtxt|Rudin|1966|p=139}}) and coincides with the one defined by {{EquationNote|3|1.3}} when ''<math>\mu</math>'' is a [[signed measure]]: its total variation is defined as above. This definition works also if ''<math>\mu</math>'' is a [[vector measure]]: the variation is then defined by the following formula
 
:<math>|\mu|(E) = \sup_\pi \sum_{A\isin\pi} \|\mu(A)\|\qquad\forall E\in\Sigma</math>
 
where the supremum is as above. Note also that this definition is slightly more general than the one given by {{Harvtxt|Rudin|1966|p=138}} since it requires only to consider ''finite partitions'' of the space ''<math>X</math>'': this implies that it can be used also to define the total variation on [[Sigma additivity|finite-additive measures]].
 
====Total variation of probability measures====
{{unreferenced section|date=May 2012}}
{{main|Total variation distance of probability measures}}
The total variation of any [[probability measure]] is exactly one, therefore it is not interesting as a means of investigating the properties of such measures. However, when μ and ν are [[probability measure]]s, the '''[[total variation distance of probability measures]]''' can be defined as ''<math>\| \mu - \nu \|</math>'' where the norm is the total variation norm of signed measures.  Using the property that ''<math>(\mu-\nu)(X)=0</math>'', we eventually arrive at the equivalent definition
 
:<math>\|\mu-\nu\| = |\mu-\nu|(X)=\sup\left\{\,\left|\mu(A)-\nu(A)\right| : A\in \Sigma\,\right\}</math>
 
and its values are non-trivial. Informally, this is the largest possible difference between the probabilities that the two [[probability distribution]]s can assign to the same event. For a [[categorical distribution]] it is possible to write the total variation distance as follows
 
:<math>\delta(\mu,\nu) = \frac 1 2 \sum_x \left| \mu(x) - \nu(x) \right|\;.</math>
 
The total variational distance for categorical probability distributions is called [[statistical distance]]: sometimes, in the definition of this distance, the factor <math>\scriptstyle\frac 1 2</math> is omitted.
 
==Basic properties==
 
===Total variation of differentiable functions===
The total variation of a [[differentiable function]] <math>f</math> can be expressed as an [[integral]] involving the given function instead of as the [[supremum]] of the [[functional (mathematics)|functional]]s of definitions {{EquationNote|1|1.1}} and {{EquationNote|2|1.2}}.
 
====The form of the total variation of a differentiable function of one variable====
{{EquationRef|5|Theorem 1.}} The '''total variation''' of a [[differentiable function]] ''<math>f</math>'', defined on an [[interval (mathematics)|interval]] <math> [a , b] \subset \mathbb{R}</math>, has the following expression if f' is Riemann integrable
 
:<math> V^a_b(f) = \int _a^b |f'(x)|\mathrm{d}x</math>
 
====The form of the total variation of a differentiable function of several variables====
{{EquationRef|6|Theorem 2.}} Given a [[differentiable function]] '''<math>f</math>''' defined on a [[bounded set|bounded]] [[open set]] <math>\Omega \subseteq \mathbb{R}^n</math>, the '''total variation of <math>f</math>''' has the following expression
 
:<math>V(f,\Omega) = \int\limits_\Omega\left|\nabla f(x)\right|\mathrm{d}x</math>
 
Here <math>|.|</math> denotes the <math>l_2</math>-norm.
 
=====Proof=====
The first step in the proof is to first prove an equality which follows from the [[Gauss-Ostrogradsky theorem]].
 
=====Lemma=====
Under the conditions of the theorem, the following equality holds:
: <math> \int\limits_\Omega f\,\mathrm{div}\varphi = -\int_\Omega\nabla f\cdot\varphi </math>
 
======Proof of the lemma======
From the [[Gauss-Ostrogradsky theorem]]:
: <math> \int\limits_\Omega \text{div}\mathbf R = \int\limits_{\partial\Omega}\mathbf R\cdot \mathbf n </math>
by substituting <math>\mathbf R:= f\mathbf\varphi</math>, we have:
 
:<math> \int\limits_\Omega\text{div}\left(f\mathbf\varphi\right) =
\int\limits_{\partial\Omega}\left(f\mathbf\varphi\right)\cdot\mathbf n </math>
where <math>\mathbf\varphi </math> is zero on the border of <math>\Omega</math> by definition:
:<math> \int\limits_\Omega\text{div}\left(f\mathbf\varphi\right)=0</math>
:<math> \int\limits_\Omega \partial_{x_i} \left(f\mathbf\varphi_i\right)=0</math>
:<math> \int\limits_\Omega \mathbf\varphi_i\partial_{x_i} f + f\partial_{x_i}\mathbf\varphi_i=0</math>
:<math> \int\limits_\Omega f\partial_{x_i}\mathbf\varphi_i = - \int\limits_\Omega \mathbf\varphi_i\partial_{x_i} f </math>
:<math> \int\limits_\Omega f\text{div} \mathbf\varphi = - \int\limits_\Omega \mathbf\varphi\cdot\nabla f </math>
 
=====Proof of the equality=====
Under the conditions of the theorem, from the lemma we have:
:<math> \int\limits_\Omega f\text{div} \mathbf\varphi = - \int\limits_\Omega \mathbf\varphi\cdot\nabla f \leq \left| \int\limits_\Omega \mathbf\varphi\cdot\nabla f \right|\leq \int\limits_\Omega \left|\mathbf\varphi\right|\cdot\left|\nabla f\right|\leq \int\limits_\Omega \left|\nabla f\right| </math>
in the last part <math>\mathbf\varphi</math> could be omitted, because by definition its essential supremum is at most one.
 
On the other hand we consider <math>\theta_n:=\mathbb I_{\left[-N,N\right]}\frac{\nabla f}{\left|\nabla f\right|}</math> and <math>\theta^*_n</math> which is the up to <math>\varepsilon</math> approximation of <math>\theta</math> in <math> C^1_c</math> with the same integral. We can do this since <math> C^1_c</math> is dense in <math> L^1 </math>. Now again substituting into the lemma:
 
:<math>\lim\limits_{N\rightarrow\infty}\int\limits_\Omega f\text{div}\theta^*_n =
\lim\limits_{N\rightarrow\infty}\int\limits_\Omega\mathbb I_{\left[-N,N\right]}\nabla f\cdot\frac{\nabla f}{\left|\nabla f\right|}=
\lim\limits_{N\rightarrow\infty}\int\limits_{\mathbb I_{\left[-N,N\right]}} \nabla f\cdot\frac{\nabla f}{\left|\nabla f\right|} = \int\limits_\Omega\left|\nabla f\right|
</math>
This means we have a convergent sequence of <math>\int\limits_\Omega f\text{div}\mathbf\varphi</math> that tends to <math>\int\limits_\Omega\left|\nabla f\right|</math> as well as we know that <math>\int\limits_\Omega f\text{div}\mathbf\varphi \leq \int\limits_\Omega\left|\nabla f\right| </math>. q.e.d.
 
It can be seen from the proof that the supremum is attained when
: <math>\varphi\to \frac{-\nabla f}{\left|\nabla f\right|}.</math>
 
The [[Function (mathematics)|function]] ''<math>f</math>'' is said to be of [[bounded variation]] precisely if its total variation is finite.
 
===Total variation of a measure===
The total variation is a [[norm (mathematics)|norm]] defined on the space of measures of bounded variation.  The space of measures on a σ-algebra of sets is a [[Banach space]], called the [[ca space]], relative to this norm.  It is contained in the larger Banach space, called the [[ba space]], consisting of ''[[Finitely additive measure|finitely additive]]'' (as opposed to countably additive) measures, also with the same norm. The [[distance function]] associated to the norm gives rise to the total variation distance between two measures ''μ'' and ''ν''.
 
For finite measures on ℝ, the link between the total variation of a measure ''μ'' and the total variation of a function, as described above, goes as follows. Given ''μ'', define a function <math>\scriptstyle\varphi\colon \mathbb{R}\to \mathbb{R}</math> by
:<math>\varphi(t) = \mu((-\infty,t])~.</math>
Then, the total variation of the signed measure ''μ'' is equal to the total variation, in the above sense, of the function ''φ''. In general, the total variation of a signed measure can be defined using [[Hahn decomposition theorem|Jordan's decomposition theorem]] by
:<math>\|\mu\|_{TV} = \mu_+(X) + \mu_-(X)~,</math>
for any signed measure ''μ'' on a measurable space <math>(X,\Sigma)</math>.
 
== Applications ==
Total variation can be seen as a [[non-negative]] [[real number|real]]-valued [[functional (mathematics)|functional]] defined on the space of [[real number|real-valued]] [[function (mathematics)|function]]s (for the case of functions of one variable) or on the space of [[integrable function]]s (for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, like [[optimal control]], [[numerical analysis]], and [[calculus of variations]], where the solution to a certain problem has to [[Maxima and minima|minimize]] its value. As an example, use of the total variation functional is common in the following two kind of problems
 
* '''Numerical analysis of differential equations''': it is the science of finding approximate solutions to [[differential equation]]s. Applications of  total variation to this problems are detailed in the article "''[[total variation diminishing]]''"
 
* '''Image denoising''': in [[image processing]], denoising is a collection of methods used to reduce the [[Electronic noise|noise]] in an [[image]] reconstructed from data obtained by electronic means, for example [[data transmission]] or [[Sensor|sensing]]. [[Total variation denoising]] is the name for the application of total variation to image noise reduction; further details can be found in the papers of {{Harv|Rudin|Osher|Fatemi|1992}} and {{Harv|Caselles|Chambolle|Novaga|2007}}. A sensible extension of this model to colour images, called Colour TV, can be found in {{Harv|Blomgren|Chan|1998}}.
 
== See also ==
* [[Bounded variation]]
* [[Total variation diminishing]]
* [[Total variation denoising]]
* [[Quadratic variation]]
* [[Total variation distance of probability measures]]
* [[Kolmogorov–Smirnov test]]
 
==Notes==
{{more footnotes|date=February 2012}}
{{Reflist|2}}
 
==Historical references==
*{{Citation
| last = Arzelà
| first = Cesare
| author-link = Cesare Arzelà
| title = Sulle funzioni di due variabili a variazione limitata (On functions of two variables of bounded variation)
| journal = Rendiconto delle sessioni della [[Reale Accademia delle scienze dell'Istituto di Bologna]]
| series = Nuova serie
| volume = IX
| issue = 4
| pages = 100–107
| language = Italian
| date = 7 May 1905
| url = http://www.archive.org/details/rendicontodelle04bologoog
| archiveurl = http://www.archive.org/stream/rendicontodelle04bologoog#page/n121/mode/2up
| archivedate = 2007-08-07
| doi =
| jfm = 36.0491.02
}}.
*{{springer
| title= Arzelà variation
| id= a/a013470
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Fréchet variation
| id= f/f041400
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Hardy variation
| id= h/h046400
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Pierpont variation
| id= p/p072720
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Vitali variation
| id= h/h046400
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Tonelli plane variation
| id= t/t092990
| last= Golubov
| first= Boris I.
}}.
*{{springer
| title= Variation of a function
| id= V/v096110
| last= Golubov
| first= Boris I.
| last2= Vitushkin
| first2= Anatolii G.
| author2-link= Anatolii Georgievich Vitushkin
}}
*{{Citation
| last = Jordan
| first = Camille
| author-link = Camille Jordan
| title = Sur la série de Fourier
| journal = Comptes rendus hebdomadaires des séances de l'Académie des sciences
| language = French
| volume = 92
| pages = 228–230
| year = 1881
| url = http://gallica.bnf.fr/ark:/12148/bpt6k7351t/f227
| jfm = 13.0184.01
}} (available at [[Gallica]]). This is, according to Boris Golubov, the first paper on functions of bounded variation.
*{{Citation
| last = Hahn
| first = Hans
| author-link = Hans Hahn (mathematician)
| title = Theorie der reellen Funktionen
| place = Berlin
| publisher = [[Springer Verlag]]
| year = 1921
| language = German
| pages = VII+600
| url = http://www.archive.org/details/theoriederreelle01hahnuoft
| archiveurl = http://www.archive.org/details/theoriederreelle01hahnuoft
| archivedate = 2008-12-31
| doi =
| jfm = 48.0261.09
| isbn =
}}.
* {{Citation
| last = Vitali
| first = Giuseppe
| author-link= Giuseppe Vitali
| title = Sui gruppi di punti e sulle funzioni di variabili reali (On groups of points and functions of real variables)
| journal = [[Atti dell'Accademia delle Scienze di Torino]]
| origyear = 17 dicembre 1907
| year = 1908
| volume = 43
| language = Italian
| pages = 75–92
| url = http://www.archive.org/details/attidellarealeac43real
| archiveurl = http://www.archive.org/stream/attidellarealeac43real#page/228/mode/2up
| archivedate = 2009-03-31
| jfm= 39.0101.05
}}. The paper containing the first proof of [[Vitali covering theorem]].
 
==References==
*{{Citation
| last = Adams
| first = C. Raymond
| last2 = Clarkson
| first2 = James A.
| title = On definitions of bounded variation for functions of two variables
| journal = [[Transactions of the American Mathematical Society]]
| volume = 35
| pages = 824–854
| year = 1933
| url = http://www.ams.org/journals/tran/1933-035-04/S0002-9947-1933-1501718-2/home.html
| doi = 10.1090/S0002-9947-1933-1501718-2
| jfm = 59.0285.01
| mr = 1501718
| zbl = 0008.00602
}}.
*{{Citation
| last = Cesari
| first = Lamberto
| author-link = Lamberto Cesari
| title = Sulle funzioni a variazione limitata (On the functions of bounded variation)
| journal = [[Annali della Scuola Normale Superiore]]
| series = II
| volume = 5
| issue = 3-4
| language = Italian
| pages = 299&ndash;313
| date =
| year = 1936
| url = http://www.numdam.org/item?id=ASNSP_1936_2_5_3-4_299_0
| doi =
| jfm = 62.0247.03
| mr = 1556778
| zbl = 0014.29605
}}. Available at [http://www.numdam.org Numdam].
*{{Citation
| last = Saks
| first = Stanisław
| author-link = Stanislaw Saks
| title = Theory of the Integral
| place = [[Warszawa]]-[[Lwów]]
| publisher = G.E. Stechert & Co.
| year = 1937
| series = [http://matwbn.icm.edu.pl/ksspis.php?wyd=10&jez=pl Monografie Matematyczne]
| volume = 7
| edition = 2nd
| pages = VI+347
| url = http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=pl
| jfm = 63.0183.05
| mr = 1556778
| zbl = 0017.30004
}}. (available at the [http://matwbn.icm.edu.pl/ Polish Virtual Library of Science]). English translation from the original French by [[Laurence Chisholm Young]], with two additional notes by [[Stefan Banach]].
*{{Citation
| last = Rudin
| first = Walter
| author-link = Walter Rudin
| title = Real and Complex Analysis
| place = New York
| publisher = [[McGraw-Hill]]
| series = McGraw-Hill Series in Higher Mathematics
| year = 1966
| edition = 1st
| pages = xi+412
| mr = 210528
| zbl = 0142.01701
| isbn =
}}.
 
== External links ==
<!-- Most of these are not External links in the sense of [[WP:EL]] -->
 
'''One variable'''
* "[http://planetmath.org/encyclopedia/TotalVariation.html Total variation]" on [[Planetmath]].
 
'''One and more variables'''
*[http://www.encyclopediaofmath.org/index.php/Function_of_bounded_variation Function of bounded variation] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
 
'''Measure theory'''
*{{MathWorld
|author=Rowland, Todd
|title=Total Variation
|urlname=TotalVariation
}}.
*{{PlanetMath|urlname=JordanDecomposition|title=Jordan decomposition}}.
*[http://www.encyclopediaofmath.org/index.php/Jordan_decomposition_%28of_a_signed_measure%29 Jordan decomposition] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
 
=== Applications ===
*{{Citation
| last = Caselles | first = Vicent | last2 = Chambolle | first2 = Antonin
| last3 = Novaga | first3 = Matteo
| title = The discontinuity set of solutions of the TV denoising problem and some extensions
| url = http://cvgmt.sns.it/papers/caschanov07/
| publisher = [[Society for Industrial and Applied Mathematics|SIAM]], Multiscale Modeling and Simulation, vol. 6 n. 3,
| year = 2007}} (a work dealing with total variation application in denoising problems for [[image processing]]).
 
*{{Citation
| last = Rudin | first = Leonid I. | last2 = Osher | first2 = Stanley
| last3 = Fatemi | first3 = Emad
| title = Nonlinear total variation based noise removal algorithms
| url = http://www.sciencedirect.com/science/article/pii/016727899290242F
| publisher = Physica D: Nonlinear Phenomena 60.1: 259-268
| year = 1992}}.
 
*{{Citation
| last = Blomgren | first = Peter | last2 = Chan | first2 = Tony F.  
| title = Color TV: total variation methods for restoration of vector-valued images
| url = http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=661180&tag=1
| publisher = Image Processing, IEEE Transactions on, vol. 7, no. 3: 304-309
| year = 1998}}.
 
*[[Tony F. Chan]] and Jackie (Jianhong) Shen (2005), [http://jackieneoshen.googlepages.com/ImagingNewEra.html ''Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods''], [[Society for Industrial and Applied Mathematics|SIAM]], ISBN 0-89871-589-X (with in-depth coverage and extensive applications of Total Variations in modern image processing, as started by Rudin, Osher, and Fatemi).
 
{{DEFAULTSORT:Total Variation}}
[[Category:Mathematical analysis]]
[[Category:Probability theory]]

Latest revision as of 02:39, 11 September 2013

Template:Primary sources

As the green ball travels on the graph of the given function, the length of the path travelled by that ball's projection on the y-axis, shown as a red ball, is the total variation of the function.

In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ ℝ, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b].

Historical note

The concept of total variation for functions of one real variable was first introduced by Camille Jordan in the paper Template:Harv.[1] He used the new concept in order to prove a convergence theorem for Fourier series of discontinuous periodic functions whose variation is bounded. The extension of the concept to functions of more than one variable however is not simple for various reasons.

Definitions

Total variation for functions of one real variable

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The total variation of a real-valued (or more generally complex-valued) function

f

, defined on an interval

[a,b]⊂ℝ

is the quantity

Vba(f)=supP∑i=0nP−1|f(xi+1)−f(xi)|,

where the supremum runs over the set of all partitions 𝒫={P={x0,…,xnP}|P is a partition of [a,b]} of the given interval.

Total variation for functions of n > 1 real variables

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Let Ω be an open subset of ℝn. Given a function f belonging to L1(Ω), the total variation of f in Ω is defined as

V(f,Ω):=sup⁡{∫Ωf(x)divϕ(x)dx:ϕ∈Cc1(Ω,ℝn),‖ϕ‖L∞(Ω)≤1},

where Cc1(Ω,ℝn) is the set of continuously differentiable vector functions of compact support contained in Ω, and ‖‖L∞(Ω) is the essential supremum norm. Note that this definition does not require that the domain Ω⊆ℝn of the given function is a bounded set.

Total variation in measure theory

Classical total variation definition

Following Template:Harvtxt, consider a signed measure μ on a measurable space (X,Σ): then it is possible to define two set functions W‾(μ,⋅) and W_(μ,⋅), respectively called upper variation and lower variation, as follows

W‾(μ,E)=sup⁡{μ(A)|A∈Σ and A⊂E}∀E∈Σ
W_(μ,E)=inf⁡{μ(A)|A∈Σ and A⊂E}∀E∈Σ

clearly

W‾(μ,E)≥0≥W_(μ,E)∀E∈Σ

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The variation (also called absolute variation) of the signed measure

μ

is the set function

|μ|(E)=W‾(μ,E)+|W_(μ,E)|∀E∈Σ

and its total variation is defined as the value of this measure on the whole space of definition, i.e.

‖μ‖=|μ|(X)

Modern definition of total variation norm

Template:Harvtxt uses upper and lower variations to prove the Hahn–Jordan decomposition: according to his version of this theorem, the upper and lower variation are respectively a non-negative and a non-positive measure. Using a more modern notation, define

μ+(⋅)=W‾(μ,⋅),
μ−(⋅)=−W_(μ,⋅),

Then μ+ and μ− are two non-negative measures such that

μ=μ+−μ−
|μ|=μ++μ−

The last measure is sometimes called, by abuse of notation, total variation measure.

Total variation norm of complex measures

If the measure μ is complex-valued i.e. is a complex measure, its upper and lower variation cannot be defined and the Hahn–Jordan decomposition theorem can only be applied to its real and imaginary parts. However, it is possible to follow Template:Harvtxt and define the total variation of the complex-valued measure μ as follows

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The variation of the complex-valued measure

μ

is the set function

|μ|(E)=supπ∑A∈π|μ(A)|∀E∈Σ

where the supremum is taken over all partitions π of a measurable set E into a finite number of disjoint measurable subsets.

This definition coincides with the above definition |μ|=μ++μ− for the case of real-valued signed measures.

Total variation norm of vector-valued measures

The variation so defined is a positive measure (see Template:Harvtxt) and coincides with the one defined by Template:EquationNote when μ is a signed measure: its total variation is defined as above. This definition works also if μ is a vector measure: the variation is then defined by the following formula

|μ|(E)=supπ∑A∈π‖μ(A)‖∀E∈Σ

where the supremum is as above. Note also that this definition is slightly more general than the one given by Template:Harvtxt since it requires only to consider finite partitions of the space X: this implies that it can be used also to define the total variation on finite-additive measures.

Total variation of probability measures

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‖μ−ν‖=|μ−ν|(X)=sup⁡{|μ(A)−ν(A)|:A∈Σ}

and its values are non-trivial. Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event. For a categorical distribution it is possible to write the total variation distance as follows

δ(μ,ν)=12∑x|μ(x)−ν(x)|.

The total variational distance for categorical probability distributions is called statistical distance: sometimes, in the definition of this distance, the factor 12 is omitted.

Basic properties

Total variation of differentiable functions

The total variation of a differentiable function f can be expressed as an integral involving the given function instead of as the supremum of the functionals of definitions Template:EquationNote and Template:EquationNote.

The form of the total variation of a differentiable function of one variable

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The total variation of a differentiable function

f

, defined on an interval

[a,b]⊂ℝ

, has the following expression if f' is Riemann integrable

Vba(f)=∫ab|f′(x)|dx

The form of the total variation of a differentiable function of several variables

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Given a differentiable function

f

defined on a bounded open set

Ω⊆ℝn

, the

total variation of f

has the following expression

V(f,Ω)=∫Ω|∇f(x)|dx

Here |.| denotes the l2-norm.

Proof

The first step in the proof is to first prove an equality which follows from the Gauss-Ostrogradsky theorem.

Lemma

Under the conditions of the theorem, the following equality holds:

∫Ωfdivφ=−∫Ω∇f⋅φ
Proof of the lemma

From the Gauss-Ostrogradsky theorem:

∫ΩdivR=∫∂ΩR⋅n

by substituting R:=fφ, we have:

∫Ωdiv(fφ)=∫∂Ω(fφ)⋅n

where φ is zero on the border of Ω by definition:

∫Ωdiv(fφ)=0
∫Ω∂xi(fφi)=0
∫Ωφi∂xif+f∂xiφi=0
∫Ωf∂xiφi=−∫Ωφi∂xif
∫Ωfdivφ=−∫Ωφ⋅∇f
Proof of the equality

Under the conditions of the theorem, from the lemma we have:

∫Ωfdivφ=−∫Ωφ⋅∇f≤|∫Ωφ⋅∇f|≤∫Ω|φ|⋅|∇f|≤∫Ω|∇f|

in the last part φ could be omitted, because by definition its essential supremum is at most one.

On the other hand we consider θn:=𝕀[−N,N]∇f|∇f| and θn* which is the up to ε approximation of θ in Cc1 with the same integral. We can do this since Cc1 is dense in L1. Now again substituting into the lemma:

lim⁡\limits N→∞∫Ωfdivθn*=lim⁡\limits N→∞∫Ω𝕀[−N,N]∇f⋅∇f|∇f|=lim⁡\limits N→∞∫𝕀[−N,N]∇f⋅∇f|∇f|=∫Ω|∇f|

This means we have a convergent sequence of ∫Ωfdivφ that tends to ∫Ω|∇f| as well as we know that ∫Ωfdivφ≤∫Ω|∇f|. q.e.d.

It can be seen from the proof that the supremum is attained when

φ→−∇f|∇f|.

The function f is said to be of bounded variation precisely if its total variation is finite.

Total variation of a measure

The total variation is a norm defined on the space of measures of bounded variation. The space of measures on a σ-algebra of sets is a Banach space, called the ca space, relative to this norm. It is contained in the larger Banach space, called the ba space, consisting of finitely additive (as opposed to countably additive) measures, also with the same norm. The distance function associated to the norm gives rise to the total variation distance between two measures μ and ν.

For finite measures on ℝ, the link between the total variation of a measure μ and the total variation of a function, as described above, goes as follows. Given μ, define a function φ:ℝ→ℝ by

φ(t)=μ((−∞,t]).

Then, the total variation of the signed measure μ is equal to the total variation, in the above sense, of the function φ. In general, the total variation of a signed measure can be defined using Jordan's decomposition theorem by

‖μ‖TV=μ+(X)+μ−(X),

for any signed measure μ on a measurable space (X,Σ).

Applications

Total variation can be seen as a non-negative real-valued functional defined on the space of real-valued functions (for the case of functions of one variable) or on the space of integrable functions (for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, like optimal control, numerical analysis, and calculus of variations, where the solution to a certain problem has to minimize its value. As an example, use of the total variation functional is common in the following two kind of problems

See also

Notes

Template:More footnotes 43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Historical references

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  • Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

    my web-site http://himerka.com/.
  • Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

    my web-site http://himerka.com/.
  • Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

    my web-site http://himerka.com/.
  • Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

    my web-site http://himerka.com/.
  • Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

    my web-site http://himerka.com/.
  • Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

    my web-site http://himerka.com/
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. The paper containing the first proof of Vitali covering theorem.

References

  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. Available at Numdam.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. (available at the Polish Virtual Library of Science). English translation from the original French by Laurence Chisholm Young, with two additional notes by Stefan Banach.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

External links

One variable

One and more variables

Measure theory



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  • Template:PlanetMath.
  • Jordan decomposition at Encyclopedia of Mathematics

Applications

  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (a work dealing with total variation application in denoising problems for image processing).
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
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