Virial expansion: Difference between revisions

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In [[mathematics]], the '''Hasse–Weil zeta function''' attached to an [[algebraic variety]] ''V'' defined over an [[algebraic number field]] ''K'' is one of the two most important types of [[L-function]]. Such ''L''-functions are called 'global', in that they are defined as [[Euler product]]s in terms of [[local zeta-function|local zeta functions]]. They form one of the two major classes of global ''L''-functions, the other being the ''L''-functions associated to [[automorphic representations]]. Conjecturally there is just one essential type of global ''L''-function, with two descriptions (coming from an algebraic variety, coming from an automorphic representation); this would be a vast generalisation of the [[Taniyama–Shimura conjecture]], itself a very deep and recent result ({{As of|2009|lc=on}}) in [[number theory]].
Michael Jackson once known as one of the top most pop singer has had the same problem. If the initial signs or symptoms of the vitiligo patches are symmetrical (non localized vitiligo,) in various words, in the same the Region of both pieces of the bodily, the development is a lot slower than if the skin patches are in just you neighborhood of the person's body (segmental vitiligo). Basically, melanin is a derivative of tyrosine, an amino acid. Leucoderma can occur at any age, person and in any kind of skins. The initial course of treatment normally lasts one to two months by the end of which the patient's skin should be completely clear of all signs of Vitiligo. <br><br>In this connection, we will only discuss about the skin disorder when the immune system becomes dysfunctional. All issues in cure of vitiligo are due to the insufficient knowledge about vitiligo causes. By fighting the free radicals that damage and age your skin and the rest of your body, it helps your skin remain healthy and radiant, while also working to fight off infections. Vitiligo cure can be seen in many vitiligo skin patients who use the [http://discover-prague.info/ vitiligo natural treatment]. Use our local application and get a quick re-pigmentation of affected skin. <br><br>Anti vitiligo oil is famous in United States and Worldwide and easily available in world markets. This means that you need to control your activities. The immunofluorescense assay is done to detect antibodies to a specific virus. The element of this plastic surgery is that it only involves small puncture sites in the face which normally heal imperceptibly. “Although any part of your body may be affected by vitiligo, depigmentation usually develops first on sun-exposed areas of your skin, such as your hands, feet, arms, face and lips. <br><br>In early stage it can be appeared in some particular area of skin, small in size and after that it can be spread on whole body. The patches tend to distribute, sometimes extremely gradually over a wide range of years and sometimes wearing a thing of days. These natural ingredients were noted to be very helpful in the treatment of Leucoderma. The cause of hirsutism rarely remains unidentified because blood tests are performed and the levels of male hormones are easily established. Sometimes the best treatment for vitiligo is no treatment at all. <br><br>Benoquin helps in the reduction of these white patches. Dry the soaked seeds and then grind to form a paste. The skin of some people with this disease is seen to have some sort of patches. They cause graying when this happens and further investigations into the mechanisms may provide a way of tackling melanomas. It is very beneficial in the treatment of skin diseases like psoriasis, leucoderma and every leprosy. <br><br>It is a skin disorder mainly due to inappropriate diet combination intake regularly and formed due to disturbance of tridosas namely 'Vaat', 'Pitta' and 'Kapha'. Also, it is best to use physical sunblock only, or sunblock that sits atop the skin and only use a recommended moisturizer. It is suggested that the presence of oleoresin in psoralea esculenta can help in killing these intestinal parasites, thereby treating leucoderma. Jamaica's tropical climate, warm Caribbean waters, abundant golf courses, and luxurious resorts make the island a popular tourist destination. You need to conduct proper research on the herbal supplements for Vitiligo so that you can never fall prey to these false claims that most of the manufacturers make.
 
The description of the Hasse–Weil zeta function ''up to finitely many factors of its Euler product'' is relatively simple. This follows the initial suggestions of [[Helmut Hasse]] and [[André Weil]], motivated by the case in which ''V'' is a single point, and the [[Riemann zeta function]] results.
 
Taking the case of ''K'' the [[rational number]] field '''Q''', and ''V'' a [[non-singular]] [[projective variety]], we can for [[almost all]] [[prime number]]s ''p'' consider the reduction of ''V'' modulo ''p'', an algebraic variety ''V''<sub>''p''</sub> over the [[finite field]] '''F'''<sub>''p''</sub> with ''p'' elements, just by reducing equations for ''V''. Again for almost all ''p'' it will be non-singular. We define
 
:<math>Z_{V,Q}(s)</math>
 
to be the [[Dirichlet series]] of the [[complex variable]] ''s'', which is the [[infinite product]] of the [[local zeta-function|local zeta functions]]
 
:<math>\zeta_{V,p}\left(p^{-s}\right).</math>
 
Then ''Z''(''s''), according to our definition, is [[well-defined]] only up to multiplication by [[rational function]]s in a finite number of <math>p^{-s}</math>.
 
Since the indeterminacy is relatively harmless, and has [[meromorphic continuation]] everywhere, there is a sense in which the properties of ''Z(s)'' do not essentially depend on it. In particular, while the exact form of the [[functional equation (L-function)|functional equation]] for ''Z''(''s''), reflecting in a vertical line in the complex plane, will definitely depend on the 'missing' factors, the existence of some such functional equation does not.
 
A more refined definition became possible with the development of [[étale cohomology]]; this neatly explains what to do about the missing, 'bad reduction' factors. According to general principles visible in [[ramification|ramification theory]], 'bad' primes carry good information (theory of the ''conductor''). This manifests itself in the étale theory in the [[Ogg–Néron–Shafarevich criterion]] for [[good reduction]]; namely that there is good reduction, in a definite sense, at all primes ''p'' for which the [[Galois representation]] ρ on the étale cohomology groups of ''V'' is ''unramified''. For those, the definition of local zeta function can be recovered in terms of the [[characteristic polynomial]] of
 
:<math>\rho(\operatorname{Frob}(p)),</math>
 
Frob(''p'') being a [[Frobenius element]] for ''p''. What happens at the ramified ''p'' is that ρ is non-trivial on the [[inertia group]] ''I''(''p'') for ''p''. At those primes the definition must be 'corrected', taking the largest quotient of the representation ρ on which the inertia group acts by the [[trivial representation]]. With this refinement, the definition of ''Z''(''s'') can be upgraded successfully from 'almost all' ''p'' to ''all'' ''p'' participating in the Euler product. The consequences for the functional equation were worked out by [[Jean-Pierre Serre|Serre]] and [[Deligne]] in the later 1960s; the functional equation itself has not been proved in general.
 
==Example: elliptic curve over Q==
Let ''E'' be an [[Elliptic curve#Elliptic curves over a general field|elliptic curve over '''Q''']] of [[Conductor of an abelian variety|conductor]] ''N''. Then, ''E'' has good reduction at all primes ''p'' not dividing ''N'', it has [[Semistable elliptic curve|multiplicative reduction]] at the primes ''p'' that ''exactly'' divide ''N'' (i.e. such that ''p'' divides ''N'', but ''p''<sup>2</sup> does not; this is written ''p'' || ''N''), and it has [[additive reduction]] elsewhere (i.e. at the primes where ''p''<sup>2</sup> divides ''N''). The Hasse–Weil zeta function of ''E'' then takes the form
 
:<math>Z_{E,Q}(s)= \frac{\zeta(s)\zeta(s-1)}{L(s,E)}. \,</math>
 
Here, ζ(''s'') is the usual [[Riemann zeta function]] and ''L''(''s'',&nbsp;''E'') is called the ''L''-function of ''E''/'''Q''', which takes the form<ref>Section C.16 of {{Citation
| last=Silverman
| first=Joseph H.
| author-link=Joseph H. Silverman
| title=The arithmetic of elliptic curves
| publisher=[[Springer-Verlag]]
| location=New York
| series=[[Graduate Texts in Mathematics]]
| isbn=978-0-387-96203-0
| id={{MathSciNet | id = 1329092}}, ISBN 978-3-540-96203-8
| year=1992
| volume=106
}}</ref>
 
:<math>L(s,E)=\prod_pL_p(s,E)^{-1}\,</math>
 
where, for a given prime ''p'',
 
:<math>L_p(s,E)=\begin{cases}
            (1-a_pp^{-s}+p^{1-2s}), & \text{if }p\nmid N \\
            (1-a_pp^{-s}), & \text{if }p\|N \\
            1, & \text{if }p^2|N
      \end{cases}</math>
 
where, in the case of good reduction ''a''<sub>''p''</sub> is ''p''&nbsp;+&nbsp;1&nbsp;&minus;&nbsp;(number of points of ''E''&nbsp;mod&nbsp;''p''), and in the case of multiplicative reduction ''a''<sub>''p''</sub> is ±1 depending on whether ''E'' has split or non-split multiplicative reduction at&nbsp;''p''.
 
==Hasse–Weil conjecture==
 
The Hasse–Weil conjecture states that the Hasse–Weil zeta function should extend to a meromorphic function for all complex ''s'', and should satisfy a functional equation similar to that of the [[Riemann zeta function]]. For elliptic curves over the rational numbers, the Hasse–Weil conjecture follows from the [[modularity theorem]].
 
==See also==
 
*[[Arithmetic zeta function]]
 
==References==
<references/>
 
==Bibliography==
*[[Jean-Pierre Serre|J.-P. Serre]], ''Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures)'', 1969/1970, Sém. Delange–Pisot–Poitou, exposé 19
 
{{L-functions-footer}}
 
{{DEFAULTSORT:Hasse-Weil zeta function}}
[[Category:Zeta and L-functions]]
[[Category:Algebraic geometry]]

Latest revision as of 13:28, 10 November 2014

Michael Jackson once known as one of the top most pop singer has had the same problem. If the initial signs or symptoms of the vitiligo patches are symmetrical (non localized vitiligo,) in various words, in the same the Region of both pieces of the bodily, the development is a lot slower than if the skin patches are in just you neighborhood of the person's body (segmental vitiligo). Basically, melanin is a derivative of tyrosine, an amino acid. Leucoderma can occur at any age, person and in any kind of skins. The initial course of treatment normally lasts one to two months by the end of which the patient's skin should be completely clear of all signs of Vitiligo.

In this connection, we will only discuss about the skin disorder when the immune system becomes dysfunctional. All issues in cure of vitiligo are due to the insufficient knowledge about vitiligo causes. By fighting the free radicals that damage and age your skin and the rest of your body, it helps your skin remain healthy and radiant, while also working to fight off infections. Vitiligo cure can be seen in many vitiligo skin patients who use the vitiligo natural treatment. Use our local application and get a quick re-pigmentation of affected skin.

Anti vitiligo oil is famous in United States and Worldwide and easily available in world markets. This means that you need to control your activities. The immunofluorescense assay is done to detect antibodies to a specific virus. The element of this plastic surgery is that it only involves small puncture sites in the face which normally heal imperceptibly. “Although any part of your body may be affected by vitiligo, depigmentation usually develops first on sun-exposed areas of your skin, such as your hands, feet, arms, face and lips.

In early stage it can be appeared in some particular area of skin, small in size and after that it can be spread on whole body. The patches tend to distribute, sometimes extremely gradually over a wide range of years and sometimes wearing a thing of days. These natural ingredients were noted to be very helpful in the treatment of Leucoderma. The cause of hirsutism rarely remains unidentified because blood tests are performed and the levels of male hormones are easily established. Sometimes the best treatment for vitiligo is no treatment at all.

Benoquin helps in the reduction of these white patches. Dry the soaked seeds and then grind to form a paste. The skin of some people with this disease is seen to have some sort of patches. They cause graying when this happens and further investigations into the mechanisms may provide a way of tackling melanomas. It is very beneficial in the treatment of skin diseases like psoriasis, leucoderma and every leprosy.

It is a skin disorder mainly due to inappropriate diet combination intake regularly and formed due to disturbance of tridosas namely 'Vaat', 'Pitta' and 'Kapha'. Also, it is best to use physical sunblock only, or sunblock that sits atop the skin and only use a recommended moisturizer. It is suggested that the presence of oleoresin in psoralea esculenta can help in killing these intestinal parasites, thereby treating leucoderma. Jamaica's tropical climate, warm Caribbean waters, abundant golf courses, and luxurious resorts make the island a popular tourist destination. You need to conduct proper research on the herbal supplements for Vitiligo so that you can never fall prey to these false claims that most of the manufacturers make.

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