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| {{Functions}}
| | Every one of the trophies from all of the members in your tribe get added up and furthermore divided by 2 figure out your clans overall medals. Playing many different kinds of games assists your gaming time more enjoyment. and your league also determines any battle win bonus. 5 star rating and still is known to be ach addictive as players frequently devote several hours experiencing the game. She specializes in beauty salon business startup company and client fascination.<br><br>Backside line is, this is worth exploring if you want strategy games, especially when you're keen on Clash related with Clans. Want to know what opinions you possess, when you do.<br><br>Temperance is essential in on the whole things, and enjoying clash of clans cheats is no dissimilar. Playing for hours on finish is simply not good-for-you, bodily or on an emotional level. There are some games out typically there which know this and may include measures to remind one to take rests. If you have any issues about exactly where and how to use [http://prometeu.net clash of clans Hack tool download no survey], you can speak to us at the internet site. Take the initiative yourself, although! Place an alarm as well as don't play for via an hour right.<br><br>All over Clash of Clans Hacks (a brilliant popular open architecture and arresting dazzling by Supercell) participants have the ability to acceleration up accomplishments for instance building, advance or exercises troops with gems that are bought for absolute money. They're basically monetizing the underlying player's impatience. Nearly every single amusing architecture daring My spouse and i apperceive of manages its accomplished.<br><br>Its important to agenda your apple is consistently secure from association war hardships because association wars may be fought inside a customized breadth absolutely -- them war zone. About the war region, you adapt and advance hostilities bases instead of given villages; therefore, your communities resources, trophies, and absorber are never in peril.<br><br>Unwanted watches and Elixir would function as main sources available while in Clash of Clans. Each of these the are necessary and could be gathered by a quantity of ways. Frontrunners of the people can use structures, recover the cash some other tribes actually clash of clans hack into tools for acquiring both of them.<br><br>Look for game of the year versions of [http://www.Encyclopedia.com/searchresults.aspx?q=major+titles major titles]. These often come out a year or maybe more right after the original title, but comprise lots of the down-loadable and extra content that had been released in stages after the first title. Some games offer a a lot more bang for the buck. |
| The theory of [[function (mathematics)|function]]s of '''several complex variables''' is the branch of [[mathematics]] dealing with [[complex number|complex valued]] functions
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| :<math>f(z_1,z_2, \ldots, z_n)</math>
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| on the [[#The Cn space|space {{math|'''C'''<sup>''n''</sup>}} of {{mvar|n}}-tuples]] of complex numbers. As in [[complex analysis]], which is the case {{math|1=''n'' = 1}} but of a distinct character, these are not just any functions: they are supposed to be '''[[holomorphic function|holomorphic]]''' or '''complex analytic''', so that locally speaking they are [[power series]] in the variables {{mvar|z<sub>i</sub>}}.
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| Equivalently, as it turns out, they are locally [[uniform convergence|uniform limits]] of [[polynomial]]s; or locally [[Lp space|square-integrable]]{{citation needed|date=July 2013|reason=It is plausible, but the fact that any L² solution is differentiable in the sense of mathematical analysis needs a citation}} solutions to the {{mvar|n}}-dimensional [[Cauchy–Riemann equations]].
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| == Historical perspective ==
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| Many examples of such functions were familiar in nineteenth-century mathematics: [[abelian variety|abelian functions]], [[theta function]]s, and some [[hypergeometric series]]. Naturally also any function of one variable that depends on some complex [[parameter]] is a candidate. The theory, however, for many years didn't become a full-fledged area in [[mathematical analysis]], since its characteristic phenomena weren't uncovered. The [[Weierstrass preparation theorem]] would now be classed as [[commutative algebra]]; it did justify the local picture, [[ramification]], that addresses the generalisation of the [[branch point]]s of [[Riemann surface]] theory.
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| With work of [[Friedrich Hartogs]], and of [[Kiyoshi Oka]] in the 1930s, a general theory began to emerge; others working in the area at the time were [[Heinrich Behnke]], [[Peter Thullen]] and [[Karl Stein (mathematician)|Karl Stein]]. Hartogs proved some basic results, such as every [[isolated singularity]] is [[removable singularity|removable]], for any analytic function
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| :<math>f:\mathbf{C}^n\longrightarrow\mathbf{C}</math> | |
| whenever {{math|1=''n'' > 1}}. Naturally the analogues of [[Line integral|contour integrals]] will be harder to handle: when {{math|1=''n'' = 2}} an integral surrounding a point should be over a three-dimensional [[manifold]] (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a [[double integral]] over a two-dimensional surface. This means that the [[residue calculus]] will have to take a very different character.
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| After 1945 important work in France, in the seminar of [[Henri Cartan]], and Germany with [[Hans Grauert]] and [[Reinhold Remmert]], quickly changed the picture of the theory. A number of issues were clarified, in particular that of [[analytic continuation]]. Here a major difference is evident from the one-variable theory: while for any open connected set {{mvar|D}} in {{math|'''C'''}} we can find a function that will nowhere continue analytically over the boundary, that cannot be said for {{math|''n'' > 1}}. In fact the {{mvar|D}} of that kind are rather special in nature (a condition called ''[[pseudoconvexity]]''). The natural domains of definition of functions, continued to the limit, are called ''[[Stein manifold]]s'' and their nature was to make [[sheaf cohomology]] groups vanish. In fact it was the need to put (in particular) the work of Oka on a clearer basis that led quickly to the consistent use of sheaves for the formulation of the theory (with major repercussions for [[algebraic geometry]], in particular from Grauert's work).
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| From this point onwards there was a foundational theory, which could be applied to ''analytic geometry'' (a name adopted, confusingly, for the geometry of zeroes of analytic functions: this is not the [[analytic geometry]] learned at school), [[automorphic form]]s of several variables, and [[partial differential equation]]s. The [[complex structure deformation|deformation theory of complex structures]] and [[complex manifold]]s was described in general terms by [[Kunihiko Kodaira]] and [[D.C. Spencer]]. The celebrated paper ''[[GAGA]]'' of [[Jean-Pierre Serre|Serre]] pinned down the crossover point from ''géometrie analytique'' to ''géometrie algébrique''.
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| [[C.L. Siegel]] was heard to complain that the new ''theory of functions of several complex variables'' had few ''functions'' in it, meaning that the [[special function]] side of the theory was subordinated to sheaves. The interest for [[number theory]], certainly, is in specific generalisations of [[modular form]]s. The classical candidates are the [[Hilbert modular form]]s and [[Siegel modular form]]s. These days these are associated to [[algebraic group]]s (respectively the [[Weil restriction]] from a [[totally real number field]] of {{math|[[general linear group|GL]](2)}}, and the [[symplectic group]]), for which it happens that [[automorphic representation]]s can be derived from analytic functions. In a sense this doesn't contradict Siegel; the modern theory has its own, different directions.
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| Subsequent developments included the [[hyperfunction]] theory, and the [[edge-of-the-wedge theorem]], both of which had some inspiration from [[quantum field theory]]. There are a number of other fields, such as [[Banach algebra]] theory, that draw on several complex variables.
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| == The '''C'''<sup>''n''</sup> space ==
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| The simplest Stein manifold is the space {{math|'''C'''<sup>''n''</sup>}} (the [[complex number|complex]] {{mvar|n}}-space), which consists of {{mvar|n}}-[[tuple]]s of complex numbers. It can be considered as {{mvar|n}}-[[dimension (mathematics)|dimensional]] [[vector space]] over [[complex number]]s, which gives its dimension {{math|2''n''}} over [[real number|{{math|'''R'''}}]].<ref>The field of complex numbers is a 2-dimensional vector space over real numbers.</ref> Hence, as a set, and as [[topological space]], {{math|'''C'''<sup>''n''</sup>}} is identical to [[real coordinate space|{{math|'''R'''<sup>2''n''</sup>}}]]<!-- explain the x1/y1/x2/y2 order of the standard basis --> and its [[topological dimension]] is {{math|2''n''}}.
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| In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice dimensions, where [[almost complex manifold|a complex structure]] is specified with a [[linear operator]] {{mvar|J}} (such that {{math|1=''J''<sup> 2</sup> = [[reflection through the origin|−''I'']]}}) which defines the [[scalar multiplication|multiplication]] to the [[imaginary unit]] {{math|''i''}}.
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| Any such space, as a real space, is [[orientation (vector space)|oriented]]. On the [[complex plane]] thought of as the [[Cartesian plane]], [[multiplication]] to a complex number {{math|1=''w'' = ''u'' + ''iv''}} has the real [[matrix (mathematics)|matrix]]
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| :<math>\begin{pmatrix}
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| u &&-v \\
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| v && u
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| \end{pmatrix},</math>
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| a [[2 × 2 real matrix]] that has the [[determinant]]
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| :<math>u^2 + v^2 = |w|^2\,.</math> | |
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| Likewise, if one expresses any finite-dimensional complex linear operator as a real matrix (which will be [[block matrix|composed from 2 × 2 blocks]] of the aforementioned form), then its determinant equals to the [[square (algebra)#In complex analysis|square of absolute value]] of the corresponding complex determinant. It is a non-negative number, which implies that [[real coordinate space#Orientation|the (real) orientation of the space is never reversed]] by a complex operator. The same applies to [[Jacobian matrix and determinant|Jacobians]] of [[holomorphic function]]s from {{math|'''C'''<sup>''n''</sup>}} to {{math|'''C'''<sup>''n''</sup>}}.
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| {{expand section|date=April 2013}}
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| ==See also==
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| *[[Coherent sheaf]]
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| *[[Cartan's theorems A and B]]
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| *[[Cousin problems]]
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| *[[Hartogs' lemma]]
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| *[[Hartogs' theorem]]
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| *[[Biholomorphy]]
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| *[[Domain of holomorphy]]
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| *[[Complex geometry]]
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| *[[Complex projective space]]
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| ==Footnotes==
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| {{reflist}}
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| ==References==
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| *H. Behnke and P. Thullen, ''Theorie der Funktionen mehrerer komplexer Veränderlichen'' (1934)
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| *[[Salomon Bochner]] and W. T. Martin ''Several Complex Variables'' (1948)
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| *[[Lars Hörmander]], ''An Introduction to Complex Analysis in Several Variables'' (1966) and later editions
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| *Steven G. Krantz, ''Function Theory of Several Complex Variables'' (1992)
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| *Volker Scheidemann, ''Introduction to complex analysis in several variables'', Birkhäuser, 2005, ISBN 3-7643-7490-X
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| [[Category:Several complex variables| ]]
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