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| In [[mathematics]], an '''exponential sum''' may be a finite [[Fourier series]] (i.e. a [[trigonometric polynomial]]), or other finite sum formed using the [[exponential function]], usually expressed by means of the function
| | Monsters Legends hack being a breeding game expects gamers to breed all the monsters, regardless of their kinds and get them expecting battleground with a view with a purpose to testing their fighting expertise.<br><br> |
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| :<math>e(x) = \exp(2\pi ix).\,</math>
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| Therefore a typical exponential sum may take the form
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| :<math>\sum e(x_n),</math>
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| summed over a finite sequence of [[real number]]s ''x''<sub>''n''</sub>.
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| ==Formulation==
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| If we allow some real coefficients ''a''<sub>''n''</sub>, to get the form
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| :<math>\sum a_n e(x_n)</math>
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| it is the same as allowing exponents that are [[complex number]]s. Both forms are certainly useful in applications. A large part of twentieth century [[analytic number theory]] was devoted to finding good estimates for these sums, a trend started by basic work of [[Hermann Weyl]] in [[diophantine approximation]].
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| ==Estimates==
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| The main thrust of the subject is that a sum
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| :<math>S=\sum e(x_n)</math>
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| is ''trivially'' estimated by the number ''N'' of terms. That is, the [[absolute value]] | |
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| :<math>|S| \le N\,</math>
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| by the [[triangle inequality]], since each summand has absolute value 1. In applications one would like to do better. That involves proving some cancellation takes place, or in other words that this sum of complex numbers on the [[unit circle]] is not of numbers all with the same [[Parameter|argument]]. The best that is reasonable to hope for is an estimate of the form
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| :<math>|S|= O(\sqrt{N})\,</math>
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| which signifies, up to the implied constant in the [[big O notation]], that the sum resembles a [[random walk]] in two dimensions.
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| Such an estimate can be considered ideal; it is unattainable in many of the major problems, and estimates
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| :<math>|S|= o(N)\,</math>
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| have to be used, where the o(''N'') function represents only a ''small saving'' on the trivial estimate. A typical 'small saving' may be a factor of log(''N''), for example. Even such a minor-seeming result in the right direction has to be referred all the way back to the structure of the initial sequence ''x''<sub>''n''</sub>, to show a degree of [[randomness]]. The techniques involved are ingenious and subtle.
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| A variant of 'Weyl differencing' investigated by Weyl involving a generating exponential sum
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| <math> G(\tau)= \sum e^{iaf(x)+ia\tau n} </math>
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| Was previously studied by Weyl himself, he developed a method to express the sum as the value <math> G(0)</math>, where 'G' can be defined via a linear differential equation similar to [[Dyson equation]] obtained via summation by parts.
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| ==History==
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| If the sum is of the form
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| :<math> S(x)= e^{ia f(x) } </math>
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| where ''ƒ'' is a smooth function, you could use the [[Euler–Maclaurin formula]] to convert the series into an integral, plus some corrections involving derivatives of ''S''(''x''), then for large values of ''a'' you could use "stationary phase" method to calculate the integral and give an approximate evaluation of the sum. Major advances in the subject were ''[[Van der Corput's method]]'' (c. 1920), related to the [[principle of stationary phase]], and the later ''[[Vinogradov method]]'' (c.1930).
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| The [[large sieve method]] (c.1960), the work of many researchers, is a relatively transparent general principle; but no one method has general application.
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| ==Types of exponential sum==
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| Many types of sums are used in formulating particular problems; applications require usually a reduction to some known type, often by ingenious manipulations. [[Partial summation]] can be used to remove coefficients ''a''<sub>''n''</sub>, in many cases.
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| A basic distinction is between a '''complete exponential sum''', which is typically a sum over all [[residue class]]es ''[[modular arithmetic|modulo]]'' some integer ''N'' (or more general [[finite ring]]), and an '''incomplete exponential sum''' where the range of summation is restricted by some [[inequality (mathematics)|inequality]]. Examples of complete exponential sums are [[Gauss sum]]s and [[Kloosterman sum]]s; these are in some sense [[finite field]] or finite ring analogues of the [[gamma function]] and some sort of [[Bessel function]], respectively, and have many 'structural' properties. An example of an incomplete sum is the partial sum of the quadratic Gauss sum (indeed, the case investigated by [[Carl Friedrich Gauss|Gauss]]). Here there are good estimates for sums over shorter ranges than the whole set of residue classes, because, in geometric terms, the partial sums approximate a [[Cornu spiral]]; this implies massive cancellation.
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| Auxiliary types of sums occur in the theory, for example [[character sum]]s; going back to [[Harold Davenport]]'s thesis. The [[Weil conjectures]] had major applications to complete sums with domain restricted by polynomial conditions (i.e., along an [[algebraic variety]] over a finite field).
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| ==Weyl sums==
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| One of the most general types of exponential sum is the '''Weyl sum''', with exponents 2π''if''(''n'') where ''f'' is a fairly general real-valued [[smooth function]]. These are the sums involved in the distribution of the values
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| :''ƒ''(''n'') modulo 1,
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| according to [[Weyl's equidistribution criterion]]. A basic advance was [[Weyl's inequality]] for such sums, for polynomial ''f''.
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| There is a general theory of [[exponent pair]]s, which formulates estimates. An important case is where ''f'' is logarithmic, in relation with the [[Riemann zeta function]]. See also [[equidistribution theorem]].<ref name=Mont39>Montgomery (1994) p.39</ref>
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| ==Example: the quadratic Gauss sum==
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| Let ''p'' be an odd prime and let <math>\xi = e^{2\pi i / p}</math>. Then
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| the quadratic [[Gauss sum]] is given by
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| :<math>\sum_{n=0}^{p-1}\xi^{n^2} =
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| \begin{cases}
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| \sqrt{p}, & p = 1 \mod 4 \\
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| i\sqrt{p}, & p = 3 \mod 4
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| \end{cases}
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| </math>
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| where the square roots are taken to be positive.
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| This is the ideal degree of cancellation one could hope for without any ''a priori'' knowledge of the structure of the sum, since it matches the scaling of a [[random walk]].
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| ==See also==
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| * [[Hua's lemma]]
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| ==References==
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| {{reflist}}
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| * {{cite book | last=Montgomery | first=Hugh L. | authorlink=Hugh Montgomery (mathematician) | title=Ten lectures on the interface between analytic number theory and harmonic analysis | series=Regional Conference Series in Mathematics | volume=84 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0737-4 | zbl=0814.11001 }}
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| * {{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 }}
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| ==Further reading==
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| * {{cite book | zbl=0754.11022 | last=Korobov | first=N.M. | title=Exponential sums and their applications | others=Translated from the Russian by Yu. N. Shakhov | series=Mathematics and Its Applications. Soviet Series. | volume=80 | location=Dordrecht | publisher=Kluwer Academic Publishers | year=1992 | isbn=0-7923-1647-9 }}
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| ==External links==
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| * [http://mathworld.wolfram.com/WeylSum.html A brief introduction to Weyl sums on Mathworld]
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| [[Category:Exponentials]]
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| [[Category:Analytic number theory]]
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Monsters Legends hack being a breeding game expects gamers to breed all the monsters, regardless of their kinds and get them expecting battleground with a view with a purpose to testing their fighting expertise.
Monster Legends has grown well-liked for bringing a one-off gaming pleasure in contrast to many different video games accessible out there. Our hack software is nice for Fb's Monster Legends hack gems , together with them fast and guarded.
In Dragon Town, the main focus was centred on breeding as the Monster Legends crack has fight since the prime precedence. You just have to install it to laptop and hack this sport, while your telephone is plugged to your LAPTOP. It's safe to use, a hundred% undetectable. We're again with monster legends hack for limitless gems, gold and meals! is working bestly on PCs, but in addition works high-quality on Androids and iOS.
No root, or Jailbreak is required. We just found a brand new glitch in a recreation. All you need to do is obtain Monster Legends hack tool, put your account URL (no password required! MobileGamesHacks is the only web site the place you will get actual and dealing Monster Legends Hack Instrument at no cost and with NO SURVEY. Monster Legends Hack Instrument is up to date monthly, or after each Monster Legends update.
With our Monster Legends Hack Device you can get many new features. One thing which gamers would love probably probably the most in regards to the game is definitely combine breeding associated with monsters with struggle. The extent-based enterprise chart permits people to problem and uncover AI-managed enemies.
This program is very easy to make use of, will take most 2 minutes to add the resources to your account. Struggles carry out in a comparable means like in monster space. With Monster Legends Hack you can modify your level in a short time. When you've got any further questions, please do a comment on the internet internet web page.
The final recreation also comes upwards with content materials to assist keep folks centered all by way of the ready. The program have fb help so you should utilize your account to attach the hack with it. ), and patch the sport. All it's a must to do is to enter the specified worth in the correct area.
Just ensure you set the browser in the hack interface. This is the most powerfull Monster Legends Hack you will see that on the web, download it proper now and become the best participant! The software additionally works with the Android and iOS variations of the game.
The program works with all browsers so that you shouldn`t worry about this. There isn't any jailbreak or rooting required and the software is suitable with all phones and tablets. You select the expertise your choose and choose the objective to unleash the a look at, go to free gems in monster legends Be taught the whole tutorial on the web web site and you'll capable of do Monster Legends Cheats on Gems, Gold and Meals totally free.