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The '''Gauss–Newton algorithm''' is a method used to solve [[non-linear least squares]] problems. It can be seen as a modification of [[newton's method in optimization|Newton's method]] for finding a [[maxima and minima|minimum]] of a [[function (mathematics)|function]]. Unlike Newton's method, the Gauss–Newton algorithm can ''only'' be used to minimize a sum of squared function values, but it has the advantage that second derivatives, which can be challenging to compute, are not required.  
In [[abstract algebra]], an '''adelic algebraic group''' is a [[semitopological group]] defined by an [[algebraic group]] ''G'' over a [[number field]] ''K'', and the [[adele ring]] ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the definition of the appropriate [[topological space|topology]] is straightforward only in case ''G'' is a [[linear algebraic group]]. In the case of ''G'' an [[abelian variety]] it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers. Adelic algebraic groups are widely used in [[number theory]], particularly for the theory of [[automorphic representation]]s, and the [[arithmetic of quadratic form]]s.


Non-linear least squares problems arise for instance in [[non-linear regression]], where parameters in a model are sought such that the model is in good agreement with available observations.
In case ''G'' is a linear algebraic group, it is an [[affine algebraic variety]] in affine ''N''-space. The topology on the adelic algebraic group <math>G(A)</math> is taken to be the [[subspace topology]] in ''A''<sup>''N''</sup>, the [[Cartesian product]] of ''N'' copies of the adele ring.


The method is named after the mathematicians [[Carl Friedrich Gauss]] and [[Isaac Newton]].
==Ideles==
An important example, the '''idele group''' ''I''(''K''), is the case of <math>G = GL_1</math>. Here the set of '''ideles''' (also ''idèles'' {{IPAc-en|ɪ|ˈ|d|ɛ|l|z}}) consists of the invertible adeles; but the topology on the idele group is ''not'' their topology as a subset of the adeles. Instead, considering that <math>GL_1</math> lies in two-dimensional [[affine space]] as the '[[hyperbola]]' defined parametrically by


== Description ==
:{(''t'', ''t''<sup>−1</sup>)},
Given ''m'' functions '''r''' = (''r''<sub>1</sub>, …, ''r''<sub>''m''</sub>) of ''n'' variables '''''&beta;'''''&nbsp;=&nbsp;(''&beta;''<sub>1</sub>, …, ''&beta;''<sub>''n''</sub>), with ''m''&nbsp;≥&nbsp;''n'', the Gauss–Newton algorithm [[iterative method|iteratively]] finds the minimum of the sum of squares<ref name=ab>Björck (1996)</ref>


:<math> S(\boldsymbol \beta)= \sum_{i=1}^m r_i^2(\boldsymbol \beta).</math>
the topology correctly assigned to the idele group is that induced by inclusion in ''A''<sup>2</sup>; composing with a projection, it follows that the ideles carry a [[finer topology]] than the subspace topology from ''A''.


Starting with an initial guess <math>\boldsymbol \beta^{(0)}</math> for the minimum, the method proceeds by the iterations
Inside ''A''<sup>''N''</sup>, the product ''K''<sup>''N''</sup> lies as a [[discrete subgroup]]. This means that ''G''(''K'') is a discrete subgroup of ''G''(''A''), also. In the case of the idele group, the [[quotient group]]


:<math> \boldsymbol \beta^{(s+1)} = \boldsymbol \beta^{(s)} - \left(\mathbf{J_r}^\top \mathbf{J_r} \right)^{-1} \mathbf{ J_r} ^\top \mathbf{r}(\boldsymbol \beta^{(s)}) </math>
:''I''(''K'')/''K''<sup>×</sup>


where
is the '''idele class group'''. It is closely related to (though larger than) the [[ideal class group]]. The idele class group is not itself compact; the ideles must first be replaced by the ideles of norm 1, and then the image of those in the idele class group is a [[compact group]]; the proof of this is essentially equivalent to the finiteness of the class number.
:<math> \mathbf{J_r} = \frac{\partial r_i (\boldsymbol \beta^{(s)})}{\partial \beta_j}</math>


is the [[Jacobian matrix]] of '''r''' at <math>\boldsymbol \beta^{(s)}</math> and the symbol <math>^\top</math> denotes the [[matrix transpose]].
The study of the [[Galois cohomology]] of idele class groups is a central matter in [[class field theory]]. [[Character (group theory)|Characters]] of the idele class group, now usually called [[Hecke character]]s, give rise to the most basic class of [[L-function]]s.


If ''m''&nbsp;=&nbsp;''n'', the iteration simplifies to
==Tamagawa numbers==
{{see also|Weil conjecture on Tamagawa numbers}}


:<math> \boldsymbol \beta^{(s+1)} = \boldsymbol \beta^{(s)} - \left( \mathbf{J_r} \right)^{-1} \mathbf{r}(\boldsymbol \beta^{(s)}) </math>
For more general ''G'', the '''Tamagawa number''' is defined (or indirectly computed) as the measure of


which is a direct generalization of [[Newton's method]] in one dimension.
:''G''(''A'')/''G''(''K'').


In data fitting, where the goal is to find the parameters '''''&beta;''''' such that a given model function ''y''&nbsp;=&nbsp;''f''(''x'', '''''&beta;''''') best fits some data points (''x''<sub>''i''</sub>, ''y''<sub>''i''</sub>), the functions ''r''<sub>''i''</sub> are the [[residual (statistics)|residuals]]
[[Tsuneo Tamagawa]]'s observation was that, starting from an invariant [[differential form]] ω on ''G'', defined ''over K'', the measure involved was [[well-defined]]: while ω could be replaced by ''c''ω with ''c'' a non-zero element of ''K'', the [[product formula]] for [[valuation (algebra)|valuation]]s in ''K'' is reflected by the independence from ''c'' of the measure of the quotient, for the product measure constructed from ω on each effective factor. The computation of Tamagawa numbers for [[semisimple group]]s contains important parts of classical [[quadratic form]] theory.


: <math>r_i(\boldsymbol \beta)= y_i - f(x_i, \boldsymbol \beta).</math>
==History of the terminology==
 
Historically the ''idèles'' were introduced  by {{harvs|txt|last=Chevalley|authorlink=Claude Chevalley|year=1936}} under the name "élément idéal", which is "ideal element" in French, which {{harvtxt|Chevalley|1940}} then abbreviated to "idèle". (In these papers he also gave the ideles a rather non-[[Hausdorff topology]].) This was to formulate [[class field theory]] for infinite extensions in terms of topological groups. {{harvtxt|Weil|1938}} defined (but did not name) the ring of adeles in the function field case and pointed out that Chevalley's group of ''Idealelemente''  was the group of invertible elements of this ring. {{harvtxt|Tate|1950}} defined the ring of adeles as a restricted direct product, though he called its elements "valuation vectors" rather than adeles.  
Then, the Gauss-Newton method can be expressed in terms of the Jacobian of the function ''f'' as
 
:<math> \boldsymbol \beta^{(s+1)} = \boldsymbol \beta^{(s)} - \left(\mathbf{J_f}^\top \mathbf{J_f} \right)^{-1} \mathbf{ J_f} ^\top \mathbf{r}(\boldsymbol \beta^{(s)}). </math>


{{harvtxt|Chevalley|1951}} defined the ring of adeles in the function field case, under the name "repartitions". The term ''adèle'' (short for additive idèles, and also a French girls' name) was in use shortly afterwards {{harv|Jaffard|1953}} and may have been introduced by [[André Weil]].  The general construction of adelic algebraic groups by {{harvtxt|Ono|1957}} followed the algebraic group theory founded by [[Armand Borel]] and [[Harish-Chandra]].
<!--
==Notes==
==Notes==
 
{{reflist}}-->
The assumption ''m''&nbsp;≥&nbsp;''n'' in the algorithm statement is necessary, as otherwise the matrix '''J<sub>r</sub>'''<sup>T</sup>'''J<sub>r</sub>''' is not invertible and the normal equations cannot be solved (at least uniquely).
 
The Gauss–Newton algorithm can be derived by [[linear approximation|linearly approximating]] the vector of functions ''r''<sub>''i''</sub>. Using [[Taylor's theorem]], we can write at every iteration:
 
: <math>\mathbf{r}(\boldsymbol \beta)\approx \mathbf{r}(\boldsymbol \beta^s)+\mathbf{J_r}(\boldsymbol \beta^s)\Delta</math>
 
with <math>\Delta=\boldsymbol \beta-\boldsymbol \beta^s.</math> The task of finding &Delta; minimizing the sum of squares of the right-hand side, i.e.,
: <math>\mathbf{min}\|\mathbf{r}(\boldsymbol \beta^s)+\mathbf{J_r}(\boldsymbol \beta^s)\Delta\|_2^2</math>,
is a [[linear least squares (mathematics)|linear least squares]] problem, which can be solved explicitly, yielding the normal equations in the algorithm.
 
The normal equations are ''m'' linear simultaneous equations in the unknown increments, &Delta;. They may be solved in one step, using [[Cholesky decomposition]], or, better, the [[QR factorization]] of '''J'''<sub>'''r'''</sub>. For large systems, an [[iterative method]], such as the [[conjugate gradient]] method, may be more efficient. If there is a linear dependence between columns of '''J'''<sub>'''r'''</sub>, the iterations will fail as '''J<sub>r</sub>'''<sup>T</sup>'''J<sub>r</sub>''' becomes singular.
 
==Example==
[[File:Gauss Newton illustration.png|thumb|right|280px|Calculated curve obtained with <math>\hat\beta_1=0.362</math> and <math>\hat\beta_2=0.556</math> (in blue) versus the observed data (in red).]]
In this example, the Gauss–Newton algorithm will be used to fit a model to some data by minimizing the sum of squares of errors between the data and model's predictions.
 
In a biology experiment studying the relation between substrate concentration [''S''] and reaction rate in an enzyme-mediated reaction, the data in the following table were obtained.
:{|class="wikitable" style="text-align: center;"
|''i'' || 1 || 2 || 3 || 4 || 5 || 6 || 7
|-
| [S] || 0.038 || 0.194 || 0.425 || 0.626 || 1.253 ||  2.500 || 3.740
|-
| rate || 0.050 || 0.127 || 0.094  ||  0.2122 ||  0.2729 ||  0.2665 || 0.3317
|}
 
It is desired to find a curve (model function) of the form
 
:<math>\text{rate}=\frac{V_\text{max}[S]}{K_M+[S]}</math>
 
that fits best the data in the least squares sense, with the parameters <math>V_\text{max}</math> and <math>K_M</math> to be determined.
 
Denote by <math>x_i</math> and <math>y_i</math> the value of <math>[S]</math> and the rate from the table, <math>i=1, \dots, 7.</math> Let <math>\beta_1=V_\text{max}</math> and <math>\beta_2=K_M.</math> We will find <math>\beta_1</math> and <math>\beta_2</math> such that the sum of squares of the residuals
: <math>r_i = y_i - \frac{\beta_1x_i}{\beta_2+x_i}</math>  &nbsp; (<math>i=1,\dots, 7</math>)
is minimized.
 
The Jacobian <math>\mathbf{J_r}</math> of the vector of residuals <math>r_i</math> in respect to the unknowns <math>\beta_j</math> is an <math>7\times 2</math> matrix with the <math>i</math>-th row having the entries
 
:<math>\frac{\partial r_i}{\partial \beta_1}= -\frac{x_i}{\beta_2+x_i},\  \frac{\partial r_i}{\partial \beta_2}= \frac{\beta_1x_i}{\left(\beta_2+x_i\right)^2}.</math>
 
Starting with the initial estimates of <math>\beta_1</math>=0.9 and <math>\beta_2</math>=0.2, after five iterations of the Gauss–Newton algorithm the optimal values <math>\hat\beta_1=0.362</math> and <math>\hat\beta_2=0.556</math> are obtained. The sum of squares of residuals decreased from the initial value of 1.445 to 0.00784 after the fifth iteration. The plot in the figure on the right shows the curve determined by the model for the optimal parameters versus the observed data.
 
==Convergence properties==
 
It can be shown<ref>Björck (1996)  p260</ref> that the increment &Delta; is a [[descent direction]] for ''S'', and, if the algorithm converges, then the limit is a [[stationary point]] of ''S''. However, convergence is not guaranteed, not even [[local convergence]] as in [[Newton's method in optimization|Newton's method]].
 
The rate of convergence of the Gauss–Newton algorithm can approach [[rate of convergence|quadratic]].<ref>Björck (1996)  p341, 342</ref> The algorithm may converge slowly or not at all  if the initial guess is far from the minimum or  the matrix <math>\mathbf{J_r^T  J_r}</math> is [[ill-conditioned]].  For example, consider the problem with <math>m=2</math> equations and <math>n=1</math> variable, given by
:<math> \begin{align}
r_1(\beta) &= \beta + 1 \\
r_2(\beta) &= \lambda \beta^2 + \beta - 1.
\end{align} </math>
The optimum is at <math>\beta = 0</math>. If <math>\lambda = 0</math> then the problem is in fact linear and the method finds the optimum in one iteration. If |λ| < 1 then the method converges linearly and the error decreases asymptotically with a factor |λ| at every iteration. However, if |λ| > 1, then the method does not even converge locally.<ref>Fletcher (1987) p.113</ref>
 
== Derivation from Newton's method ==
 
In what follows, the Gauss–Newton algorithm will be derived from [[Newton's method in optimization|Newton's method]] for function optimization via an approximation. As a consequence, the rate of convergence of the Gauss–Newton algorithm is at most quadratic. 
 
The recurrence relation for Newton's method for minimizing a function ''S'' of parameters, '''''&beta;''''', is
:<math> \boldsymbol\beta^{(s+1)} = \boldsymbol\beta^{(s)} - \mathbf H^{-1} \mathbf g \, </math>
where '''g''' denotes the [[gradient|gradient vector]] of ''S'' and '''H''' denotes the [[Hessian matrix]] of ''S''.
Since <math> S = \sum_{i=1}^m r_i^2</math>, the gradient is given by
:<math>g_j=2\sum_{i=1}^m r_i\frac{\partial r_i}{\partial \beta_j}.</math>
Elements of the Hessian are calculated by differentiating the gradient elements, <math>g_j</math>, with respect to <math>\beta_k</math>
:<math>H_{jk}=2\sum_{i=1}^m \left(\frac{\partial r_i}{\partial \beta_j}\frac{\partial r_i}{\partial \beta_k}+r_i\frac{\partial^2 r_i}{\partial \beta_j \partial \beta_k} \right).</math>
 
The Gauss–Newton method is obtained by ignoring the second-order derivative terms (the second term in this expression). That is, the Hessian is approximated by
 
:<math>H_{jk}\approx 2\sum_{i=1}^m J_{ij}J_{ik}</math>
 
where <math>J_{ij}=\frac{\partial r_i}{\partial \beta_j}</math> are entries of the Jacobian '''J<sub>r</sub>'''. The gradient and the approximate Hessian can be written in matrix notation as
 
:<math>\mathbf g=2\mathbf{J_r}^\top \mathbf{r}, \quad \mathbf{H} \approx 2 \mathbf{J_r}^\top \mathbf{J_r}.\,</math>
 
These expressions are substituted into the recurrence relation above to obtain the operational equations
 
:<math> \boldsymbol{\beta}^{(s+1)} = \boldsymbol\beta^{(s)}+\Delta;\quad \Delta = -\left( \mathbf{J_r}^\top \mathbf{J_r} \right)^{-1} \mathbf{J_r}^\top \mathbf{r}. </math>
 
Convergence of the Gauss–Newton method is not guaranteed in all instances. The approximation
:<math>\left|r_i\frac{\partial^2 r_i}{\partial \beta_j \partial \beta_k}\right| \ll \left|\frac{\partial r_i}{\partial \beta_j}\frac{\partial r_i}{\partial \beta_k}\right|</math>
that needs to hold to be able to ignore the second-order derivative terms may be valid in two cases, for which convergence is to be expected.<ref>Nocedal (1997) {{Page needed|date=December 2010}}</ref>
#The function values <math>r_i</math> are small in magnitude, at least around the minimum.
#The functions are only "mildly" non linear, so that <math>\frac{\partial^2 r_i}{\partial \beta_j \partial \beta_k}</math> is relatively small in magnitude.
 
== Improved versions ==
 
With the Gauss–Newton method the sum of squares ''S'' may not decrease at every iteration. However, since &Delta; is a descent direction, unless <math>S(\boldsymbol \beta^s)</math> is a stationary point, it holds that <math>S(\boldsymbol \beta^s+\alpha\Delta) < S(\boldsymbol \beta^s)</math> for all sufficiently small <math>\alpha>0</math>. Thus, if divergence occurs, one solution is to employ a fraction, <math>\alpha</math>, of the increment vector, &Delta; in the updating formula 
:<math> \boldsymbol \beta^{s+1} = \boldsymbol \beta^s+\alpha\  \Delta</math>.
In other words, the increment vector is too long, but it points in "downhill", so going just a part of the way will decrease the objective function ''S''. An optimal value for <math>\alpha</math> can be found by using a [[line search]] algorithm, that is, the magnitude of <math>\alpha</math> is determined by finding the value that minimizes ''S'', usually using a [[line search|direct search method]] in the interval <math>0<\alpha<1</math>.
 
In cases where the direction of the shift vector is such that the optimal fraction, <math> \alpha </math>, is close to zero, an alternative method for handling divergence is the use of the [[Levenberg–Marquardt algorithm]], also known as the "[[trust region]] method".<ref name="ab"/> The normal equations are modified in such a way that the increment vector is rotated towards the direction of [[steepest descent]],   
:<math>\left(\mathbf{J^TJ+\lambda D}\right)\Delta=\mathbf{J}^T \mathbf{r}</math>,
where '''D''' is a positive diagonal matrix. Note that when ''D'' is the identity matrix and <math>\lambda\to+\infty</math>, then  <math> \Delta/\lambda\to \mathbf{J}^T \mathbf{r}</math>, therefore the [[Direction (geometry, geography)|direction]] of &Delta; approaches the direction of the gradient <math> \mathbf{J}^T \mathbf{r}</math>.
 
The so-called Marquardt parameter, <math>\lambda</math>, may also be optimized by a line search, but this is inefficient as the shift vector must be re-calculated every time <math>\lambda</math> is changed. A more efficient strategy is this. When divergence occurs increase the Marquardt parameter until there is a decrease in S. Then, retain the value from one iteration to the next, but decrease it if possible until a cut-off value is reached when the Marquardt parameter can be set to zero; the minimization of ''S'' then becomes a standard Gauss–Newton minimization.
 
== Related algorithms ==
 
In a [[quasi-Newton method]], such as that due to [[Davidon-Fletcher-Powell (DFP) formula|Davidon, Fletcher and Powell]] or Broyden–Fletcher–Goldfarb–Shanno ([[BFGS method]]) an estimate of the full Hessian, <math>\frac{\partial^2 S}{\partial \beta_j \partial\beta_k}</math>, is built up numerically using first derivatives <math>\frac{\partial r_i}{\partial\beta_j}</math> only so that after ''n'' refinement cycles the method closely approximates to Newton's method in performance. Note that quasi-Newton methods can minimize general real-valued functions, whereas Gauss-Newton, Levenberg-Marquardt, etc. fits only to nonlinear least-squares problems.
 
Another method for solving minimization problems using only first derivatives is [[gradient descent]]. However, this method does not take into account the second derivatives even approximately. Consequently, it is highly inefficient for many functions, especially if the parameters have strong interactions.
 
==Notes==
<references />


==References==
==References==
*{{cite book
*{{Citation | last1=Chevalley | first1=Claude | title=Généralisation de la théorie du corps de classes pour les extensions infinies. | language=French | jfm=62.1153.02  | year=1936 | journal=Journal de Mathématiques Pures et Appliquées | volume=15 | pages=359–371}}
  | last      = Björck | first      = A.
*{{Citation | last1=Chevalley | first1=Claude | title=La théorie du corps de classes | jstor=1969013 | mr=0002357 | year=1940 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=41 | pages=394–418}}
| title     = Numerical methods for least squares problems
*{{Citation | last1=Chevalley | first1=Claude | title=Introduction to the Theory of Algebraic Functions of One Variable | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys, No. VI | mr=0042164  | year=1951}}
| publisher  = SIAM, Philadelphia | year       = 1996  | isbn      = 0-89871-360-9 }}
*{{Citation | last1=Jaffard | first1=Paul | title=Anneaux d'adèles (d'après Iwasawa) | url=http://www.numdam.org/item?id=SB_1954-1956__3__23_0 | publisher=Secrétariat mathématique, Paris | series=Séminaire Bourbaki, | mr=0157859 | year=1953}}
* {{Cite book | last1=Fletcher | first1=Roger | title=Practical methods of optimization | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | isbn=978-0-471-91547-8 | year=1987 }}.
*{{Citation | last1=Ono | first1=Takashi | title=Sur une propriété arithmétique des groupes algébriques commutatifs | url=http://www.numdam.org/item?id=BSMF_1957__85__307_0 | mr=0094362 | year=1957 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=85 | pages=307–323}}
*{{cite book
*{{Citation | last1=Tate | first1=John T. | title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) | publisher=Thompson, Washington, D.C. | isbn=978-0-9502734-2-6 | mr=0217026 | year=1950 | chapter=Fourier analysis in number fields, and Hecke's zeta-functions | pages=305–347}}
| last      = Nocedal  | first      = Jorge | coauthors  = Wright, Stephen
*{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Zur algebraischen Theorie der algebraischen Funktionen. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002174502 | language=German | doi=10.1515/crll.1938.179.129 | year=1938 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=179 | pages=129–133}}
| title     = Numerical optimization
| publisher  = New York: Springer | year       = 1999  | isbn       = 0-387-98793-2 }}
 
{{Least Squares and Regression Analysis}}


{{Optimization algorithms}}
==External links==
*{{springer|first=A.S. |last=Rapinchuk|id=T/t092060|title=Tamagawa number}}


{{DEFAULTSORT:Gauss-Newton algorithm}}
[[Category:Topological groups]]
[[Category:Optimization algorithms and methods]]
[[Category:Algebraic number theory]]
[[Category:Least squares]]
[[Category:Algebraic groups]]
[[Category:Statistical algorithms]]

Revision as of 09:01, 13 August 2014

In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only in case G is a linear algebraic group. In the case of G an abelian variety it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphic representations, and the arithmetic of quadratic forms.

In case G is a linear algebraic group, it is an affine algebraic variety in affine N-space. The topology on the adelic algebraic group is taken to be the subspace topology in AN, the Cartesian product of N copies of the adele ring.

Ideles

An important example, the idele group I(K), is the case of . Here the set of ideles (also idèles Template:IPAc-en) consists of the invertible adeles; but the topology on the idele group is not their topology as a subset of the adeles. Instead, considering that lies in two-dimensional affine space as the 'hyperbola' defined parametrically by

{(t, t−1)},

the topology correctly assigned to the idele group is that induced by inclusion in A2; composing with a projection, it follows that the ideles carry a finer topology than the subspace topology from A.

Inside AN, the product KN lies as a discrete subgroup. This means that G(K) is a discrete subgroup of G(A), also. In the case of the idele group, the quotient group

I(K)/K×

is the idele class group. It is closely related to (though larger than) the ideal class group. The idele class group is not itself compact; the ideles must first be replaced by the ideles of norm 1, and then the image of those in the idele class group is a compact group; the proof of this is essentially equivalent to the finiteness of the class number.

The study of the Galois cohomology of idele class groups is a central matter in class field theory. Characters of the idele class group, now usually called Hecke characters, give rise to the most basic class of L-functions.

Tamagawa numbers

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For more general G, the Tamagawa number is defined (or indirectly computed) as the measure of

G(A)/G(K).

Tsuneo Tamagawa's observation was that, starting from an invariant differential form ω on G, defined over K, the measure involved was well-defined: while ω could be replaced by cω with c a non-zero element of K, the product formula for valuations in K is reflected by the independence from c of the measure of the quotient, for the product measure constructed from ω on each effective factor. The computation of Tamagawa numbers for semisimple groups contains important parts of classical quadratic form theory.

History of the terminology

Historically the idèles were introduced by Template:Harvs under the name "élément idéal", which is "ideal element" in French, which Template:Harvtxt then abbreviated to "idèle". (In these papers he also gave the ideles a rather non-Hausdorff topology.) This was to formulate class field theory for infinite extensions in terms of topological groups. Template:Harvtxt defined (but did not name) the ring of adeles in the function field case and pointed out that Chevalley's group of Idealelemente was the group of invertible elements of this ring. Template:Harvtxt defined the ring of adeles as a restricted direct product, though he called its elements "valuation vectors" rather than adeles.

Template:Harvtxt defined the ring of adeles in the function field case, under the name "repartitions". The term adèle (short for additive idèles, and also a French girls' name) was in use shortly afterwards Template:Harv and may have been introduced by André Weil. The general construction of adelic algebraic groups by Template:Harvtxt followed the algebraic group theory founded by Armand Borel and Harish-Chandra.

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
  • 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
  • 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
  • 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

  • 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.

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