Difference between revisions of "Regular Iteration"

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#redirect[[Regular Iteration]]
'''Regular Iteration''' or '''regular iterate''' refer to the [[fractional iterate]] a holomorphic function that is holomorphic in vicinity it its fixed point
 
<ref name="backer">
 
http://eretrandre.org/rb/files/Baker1962_53.pdf
 
I.N.Baker. Permutable power series and regular iterate. (2006)
 
</ref><ref name="hiperwiki">
 
http://math.eretrandre.org/hyperops_wiki/index.php?title=Regular_iteration
 
A [[fractional iterate]] $\phi$ of an analytic function $f$ at fixpoint $a$ is called regular, iff $\phi$ is analytic at $a$ or has an [[asymptotic powerseries development]] at $a$.
 
</ref>. In general, a holomorphic function may have several fixed points, and the fractional iterates, regular at different fixed points, have no need to coincide <ref name="sqrt2">
 
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
 
D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
 
</ref>.
 
 
==Transfer equation==
 
'''Regular Iteration''' may refer to the solution $F$ of the [[transfer equation]]
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ F(z\!+\!1)=T(F(z))$
 
where $~~T$ is known holomorphic function, that has physical sense of [[Transfer function]]. Then the [[regular iteration]] indicates an
 
algorithm that allows to construct the [[superfunction]] $~F~$, that asymptotically approaches $~L~$ in some directions (usually at $+\infty$ or at $-\infty$), in the form
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle F(z) = L+\sum_{n=1}^{N} a_n \varepsilon^n + o(\varepsilon^N)$ , where $\varepsilon=\exp(kz)$
 
for some constant $~k~$ and some constant coefficients $~a~$.
 
 
In [[TORI]], the representation of a [[fractional iterate]] of any function $~T~$ through its [[Superfunction]] and the [[Abel function]] is considered as principal, as it allows to deal with functions that have no real fixed points (and, perhaps, no fixed points at all).
 
However, for the transfer function $~T~$ with real fixed point, the fractional iterate, that is regular at this fixed point, can be defined, constructed and evaluated also through the [[Schroeder function]] (solution of the [[Schroeder equation]] <ref name="wSchroederEq">
 
http://en.wikipedia.org/wiki/Schröder's_equation
 
</ref>) and its inverse function.
 
The reason use of [[superfunciton]]s and not the [[Schroeder function]] in the consideration of the regular iterate is mainly [[history|historical]], "it happened so". The application in [[laser science]] of the [[Transfer equation]] is more transparent than that of the [[Schroeder equation]]. However, while a function $~T~$ has real fixed point $~L~$ and is holomorphic in vicinity of $~L~$, then the same can be done with the [[Schroeder function]] too, and the corresponding deduction is supposed to be loaded in future. While, the deduction through the [[Transfer equation]] is presented below.
 
 
==Deduction==
 
Let $T(L)=L$. Let $T\,'(L)\ne 1$. Then the solution of equation (1)
 
is searched in form of the expansion (2)
 
<!--: $(2) ~ ~ ~\displaystyle F(z) = L+\sum_{n=1}^{N} a_n \varepsilon^n + o(\varepsilon^N)$ , where $\varepsilon=\exp(kz) ~ $, $k=$constant!-->
 
for some constant coefficients $a$ ; usually, especially for [[real-holomorphic]] Transfer function $T$, it is assumed that $a_1= \pm 1$.
 
 
Such a representation allows to express
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~\displaystyle F(z\!+\!1) = L+\sum_{n=1}^{N} a_n K^n \varepsilon^n + o(\varepsilon^N)$ ,
 
where $K=\exp(k)$ and $N$ is [[natural number]].
 
 
The substitution of (2) and (3) into (1) give equation
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~\displaystyle T\!\left(L+\sum_{n=1}^N a_n \varepsilon^n + o(\varepsilon^N)\right)-
 
L-\sum_{n=1}^{N} a_n K^n \varepsilon^n + o(\varepsilon^N) = 0$
 
 
Expansion of the left hand side of equation (4) at $\varepsilon \rightarrow 0$
 
gives
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (5) ~ ~ ~ K=T\,'(L)$
 
Then, $k=\log(K)=\log\big(T\,'(L)\big)$ may be the convenient choise.
 
 
Often, value $a_1=\pm 1$ gives the [[superfunction]] with appropriate asymptotical properties. This allows simple expression for other $a$.
 
In particular, the coefficients with $\varepsilon^2$ and $\varepsilon^3$ in the left hand side of (4) should be equal to zero; this gives
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (6) ~ ~ ~ \mathrm{e}^{2k} a_2=T\,' a_2+{T\,''}/{2}$
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (7) ~ ~ ~ \mathrm{e}^{3k} a_3=T\,' a_3 + T\,'' a_2+{T\,'''}/{6}$
 
and so on.
 
 
In the simple case, the equations for other $a$ have single solutions; this allow to extend the expansion of the superfunciotn to arbitrary large $N$. Usually, the series diverges; but at $\Re(k)\ne 0$, for fixed $N$, the representation
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (8) ~ ~ ~ F(z)=T\,^m(F(z\!-\!m))$
 
for some appropriate integer $m$ allows to get the required precision of the evaluation due the exponential decay of the resulting $\varepsilon$ at the replacement $z\mapsto z+m$. However, at $\Re(k)<0$, $m$ should be negative, and the inverse of the transfer function, id set, $T^{-1}$, should be implemented; for non-trivial $T$, the choice of the cut lines of $T^{-1}$ determines the behavior (and singularities) of the resulting superfunction.
 
 
For application of the [[regular iteration]] to tetration [[tetration|tet]]$_b$ at $1\!<\!b\!<\!\exp(1/\mathrm e)$, the natural choose of the cut of the logarithm leads to the set of parallel cut lines to the direction of imaginary axis, separated with distance
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (9) ~ ~ ~ \displaystyle \left| \frac{2 \pi} {k}\right| = \left| \frac{2\pi} {L ~ \ln(b)} \right| $.
 
 
The [[Regular Iteration]] allows to express the [[tetration]] and other superfunctions of $\exp_b$ at $1\!<\!b\!<\!\exp(1/\mathrm e)$
 
<ref name="sqrt2">
 
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
 
D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
 
</ref>, the superfunction of [[factorial]]
 
<ref name="factorial">
 
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1
 
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
 
</ref>, that of the [[logistic sequence]]<ref name="logistic">
 
http://www.springerlink.com/content/u712vtp4122544x4
 
D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.
 
</ref> and the [[holomorphic extension of the Collatz subsequence]]; as well as the superfunction of the [[Doya function]], that can be interpreted as transfer function of an idealized saturable optical amplifier
 
<ref name="simplyfied">
 
http://tori.ils.uec.ac.jp/2012OR/2012or.pdf
 
D. Kouznetsov. Superfunctions for optical amplifiers. Preprint ILS UEC, 2012
 
</ref>.
 
 
The scaling of $a_1$ woiuld caused the corresponding modification of other coefficients; this is equivalent of the displacement of the argument of the resulting superfunction with some constant. For superfunctions of $\exp_b$, id est, superexponentials, such a constant is choosen in such a way,
 
the superrunction $\mathrm {tet}_b(0)=1$ in the case of [[tetration]] and the superfunction is equal to the minimal integer values still bigger than $L$, for the growing along the real axis superexponential. (However, the last choice may be not good for the analysis of the dependence of the resulting growing superexponential on value of the base $b$ in the interval $1\!<\!b\!<\!\exp(1/\mathrm e)$ ).
 
 
The asymptotic expansion can be obtained also for the [[Abel function]] $~G=F^{-1}~$, either inverting series for $~F~$, or making the asymptotical expansion for the [[Abel equation]], considering $~G(L\!+\!\varepsilon)~$, where $~L~$ is [[fixed point]] of the [[Transfer funciton]], and $~\varepsilon~$ is small parameter.
 
Together, $~F~$ and $~G~$ determine the [[regular iteration]]s of the [[transfer function]] $~T~$ at fixed point $~L~$.
 
 
==Zooming and Schroeder==
 
The [[regular iterate]] of a transfer function $~T~$ at its fixed point $~L~$ can be constructed also with the [[zooming function]] $~f~$ and [[Schroeder function]] $~g=f^{-1}~$ for the new transfer function $~t~$ such that
 
 
$~ t(z)=T(L\!+\!z)-L~$
 
 
solving the corresponding [[Zooming equation]]
 
 
$~ T\big(f(z)\big)= f(K z)~$
 
 
and the [[Schroeder equation]]
 
 
$~ g\big(T(z))= K g(z)~$
 
 
Then, the [[regular iterate]] appears as
 
 
$~ T^r(z)=L+t^r(z\!-\!L)=L+f\Big(r g\big(\cdot(z\!-\!L)\big)\Big) ~$
 
 
which is almost equivalent of the representation through the [[Superfunction]] $~F~$ and the [[Abel function]] $~G~$.
 
 
==Properties==
 
The superfunctions constructed with regular iteration, have the specific common feature, the exponential behavior is some direction. For real-holomorphic transfer function $T$, at real coefficient $a_1$, the superfunction decays to constant ([[fixed point]] of the transfer function) either in the right hand side of the complex plane, or in the left hand side. In the opposite direction, the superfunction may show complicated behavior, but may also asymptotically approach another fixed point, as the [[tetration|tet]]$_b$ at $1\!<\! b\!<\!\exp(1/ \mathrm e) ~ $ does.
 
 
The superfunction, constructed with the regular iteration, is periodic; the period
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (10) ~ ~ ~ \displaystyle \tau=\frac{2 \pi\mathrm i}{k} = \frac{2 \pi \mathrm i}{ \ln\left(f'(L)\right)}$
 
 
For [[real-holomorphic]] transfer function, the the [[complex map]] of such superfunction reproduces itself at the translations for $2\pi/k$ in the imaginary direction. In particular, a$1\!<b\!<\!\exp(1/ \mathrm e)$, the [[tetration]] $\mathrm{tet}_b$ is periodic with period
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (11) ~ ~ ~ \displaystyle \tau= \frac{2 \pi \mathrm i}{ \ln\left( L\, \ln(b) \right)}$
 
 
Such periodicity (and in particular, the case $b\!=\!\sqrt{2}$, whlle $L\!=\!2$ for the tetrational and $L\!=\!4$ for the growing superexponential) is considered in
 
<ref name="sqrt2">sqrt(2)
 
</ref>.
 
 
Similar periodicity may have the [[Abel function]] $~G=F^{-1}~$. However, the Abel functions, corresponding at different [fixed points]] have no need to have the
 
same periods. In general, the [[SuperFunction]]s constructed with regular iterations at different fixed points are not equivalent <ref name="karlin">
 
http://www.jstor.org/pss/1994886
 
Karlin, S., & Mcgregor, J. (1968). Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Am. Math. Soc., 132, 137–145.
 
</ref>, and, in general, different SuperFunctions have different periods. However, they may look similar at the plots versus real argument. In particular the two of [[SuperExponential]]s to base $\sqrt{2}$ considered in <ref name="sqrt2">sqrt(2)</ref>, the maximal deviation is of order of $10^{-24}$ along the whole real axis. Such a deep similarity bring to confusion those researchers who make conclusions only on the base of the numerical simulations along the real axis. However, the deviation is clearly seen for complex values of the argument.
 
Similar resemblance along the real axis may tale place also for the iterates of the transfer function $~T~$, constructed at the different fixed points $~L~$.
 
 
==Examples==
 
 
Regular iteration is used to construct the super-exponentials (including [[tetration]]) to base $b<\exp(1/\mathrm e)=\exp^2(-1)$, and in particilar, for $b=\sqrt{2}~$
 
<ref name="sqrt2">
 
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
 
D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
 
</ref>, the [[superFactorial]] (id est, [[superfunction]] of [[factorial]])
 
<ref name="factorial">
 
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1
 
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
 
</ref>
 
and the holomorphic extension of the [[logistic sequence]]
 
<ref name="logistic">
 
http://www.springerlink.com/content/u712vtp4122544x4
 
D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.
 
</ref>, which is [[superfunction]] of the quadratic function $z\mapsto rz(1\!-\!z)$ for positive values of $r$.
 
 
WIth regular iteration, the [[Holomorphic extension of the Collatz subsequence]] can be evaluated.
 
 
Below one more example is considered with the regular iteration of the transfer function of the idealized [[laser]] amplifier.
 
 
===Regular iteration of the Doya function===
 
[[File:DoyaplotTc.png|200px|thumb| $T=\mathrm{Doya}$ and its approximations]]
 
 
In this section, the [[Doya function]] is treated as the transfer function.
 
 
Consider the transfer function $T(z)=\mathrm{Doya}(z)$.
 
The [[Doya function]] expresses the normalized output of the simple laser amplifier versus the normalized input, assuming, that the coefficient of amplification of a weak signal is
 
$\mathrm e\!=\!\exp(1)\!\approx\! 2.7$
 
 
In vicinity of the real axis (While $|\Im(z)| \!<\! \pi$), this function can be expressed through the [[LambertW]],
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (12) ~ ~ ~ \displaystyle T(z)=\mathrm{LambertW}\Big( z~ \mathrm{e}^{z+1} \Big)$
 
 
For real positive argument, this function is shown in figure at right with thick green line.
 
At large value of the argument, the function shows almost linear growth, $T(z)\approx z+1+O(1/z)$. However, for the Regular iteration, the expansion at zero is important;
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (13) ~ ~ ~ \displaystyle
 
T(z)=\mathrm{Doya}(z) = \,$ $\displaystyle
 
\mathrm e z-\mathrm e \left(\mathrm e\!-\!1\right) z^2
 
+\frac{1}{2} \mathrm e \left(-4 \mathrm e\!+\!3 \mathrm e^{2}\!+\!1\right) z^3
 
-\frac{1}{6} \mathrm e \left(12 \mathrm e\!-\!27 \mathrm e^{2}\!+\!16 \mathrm e^{3}\!-\!1\right) z^4 $ $
 
+O\left(z^5\right)$
 
 
Truncation of the series gives the linear and the quadratic approximations of the transfer function. These approximations are shown in the figure at right with thin lines.
 
The expansion determines the derivatives of the transfer function, determining parameters $k$, $K$ and coefficients $a$ in the expansion of superfunction:
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (14) ~ ~ ~ \displaystyle k\!=\!\ln(T'(0))\!=\!1 ~ ~$, $~ ~\varepsilon=e^z$
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (15) ~ ~ ~ \displaystyle T''(0)\!=\!-2 \mathrm e\,(\mathrm e\!-\!1)~$, $~a_2\!=\!-1$
 
 
[[File:SimudoyaTb.png|500px|thumb|primary approximations by (18) and (19), brown and red curves, and the iterations of $T$ by (17) for $n=1,2,3,4$. The exact superfunction $F$ by (20) is shown with black curve]]
 
Then, the expansion (3) gives the primary approximation of the superfunction
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (16) ~ ~ ~ \displaystyle \tilde F(x)\approx \exp(x) - \exp(2x) + ..$
 
for large negative values of $\Re(z)$. The primary approximation with single term in the expansion above is plotted with brown line in the figure at right; the primary approximation with two terms is plotted with red line.
 
 
Let $\tilde F$ be one of the primary approximation. Then, the precise approximation can be constructed with
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (17) ~ ~ ~ \displaystyle F(x)\approx T^n(\tilde F(x\!-\!n))$
 
For the two primary approximations,
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (18) ~ ~ ~ \displaystyle \tilde F(x)=\exp(x) ~$
 
and
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (19) ~ ~ ~ \displaystyle \tilde F(x)=\exp(x) - \exp(2x)$,
 
the four regular iterations are shown in figure at right.
 
The approximation becomes precise at $n>x\!+\!2$.
 
 
For this example, the [[superfunction]] $F$ can be espressed through the [[Tania function]] (and also through the [[LambertW]] function):
 
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (20) ~ ~ ~ \displaystyle F(x)= \mathrm{Tania}(x\!-\!1)= \mathrm{LambertW}(\exp(x))$
 
 
This "exact" superfunction is plotted with the black curve. It gives the exact representation for the iterates of the [[Doya function]],
 
 
$~ \mathrm{Doya}^r(z)=\mathrm{Tania}\Big(t+\mathrm{ArcTania}(z)\Big)$
 
 
In the similar way, the regular iteration can be used to approximate other superfunctions, in praticular, those, which are not yet expressed through the special function.
 
(However, after to reveal the properties and to provide the efficient algorithm for the expansion, any superfunction can be declared as known function, or "special function", and correspondently treated in future analysis and applications.)
 
 
==Generalization==
 
The '''Regular iteration''' is not universal way to express the [[superfunction]].
 
 
Terms with different $k$ (various solutions of equation $~\exp(k)\!=\!K~$)
 
can be added to the right hand side of equation (2), giving the more complicated expansions <ref name="kneser">
 
http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002175851 H.Kneser. Reelle analytische Lösungen der Gleichung φ(φ(x))=ex. Equationes Mathematicae, Journal fur die reine und angewandte Mathematik '''187''', 56–67 (1950)
 
<!-- http://www.ils.uec.ac.jp/~dima/Relle.pdf !-->
 
</ref><ref name="vladi"> http://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf
 
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.</ref>
 
and more complicated superfunction. It is expected that namely the regular iteration corresponds to the physicaly–meaningful solution of the transfer equation of a ralistic amplifier; at least, it is so for the exampe above with Doya transfer functions.
 
 
===Case without real fixed points===
 
Some transfer functions have no fixed points; $~f(z)=\exp(z)\!+\!z~$ is an example of such a function.
 
 
Even the fixed point exist, it may be not real; also, even if it is real, the derivative at this fixed point may happen to be unity,
 
the example of such transfer function is the exponential to base $b\!=\!\exp(1/ \mathrm e)$; in this case the expansion (2) should be modified <ref name="e1e">http://www.ils.uec.ac.jp/~dima/PAPERS/2011e1e.pdf
 
H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). (Mathematics of computation, under consideration) (preprint, 2011)
 
</ref>. In these cases, the regular iteration requires some generalization, or other methods
 
(for example, the [[Iterated Cauchi]]) should be used for the construction of the [[superfunction]]
 
<ref name="moc1">
 
http://www.ams.org/journals/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
 
D.Kouznetsov. Solution of F(z+1)=exp(F(z)) in complex z-plane.
 
[[Math.Comp.]] '''78''' (2009), 1647-1670.
 
</ref>.
 
 
However, the regular iteration is expected to work for the most of amplifiers,
 
that convert zero to zero, having some range in vicinity of zero, where the output can be considered as proportional to the input.
 
 
==Uniqueness==
 
The solution of the [[Transfer equation]] (as well as the solution of the [[Abel equation]]) is not unique.
 
If $F$ is solution of the [[Transfer equation]] (0), then another solution, id est, [[superfunction]] $\tilde F$ can be expressed as
 
:$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (21) ~ ~ ~ \displaystyle F(z)=F(z+\eta(z))$
 
where $\eta$ is periodic holomorphic function with period unity. For the uniqueness, the additional requirements on the range of holomorphism of the [[superfunction]] should be postulated.
 
 
There is conjecture that for a physical unidimensional distributed homegeneous systems (which in general can be called [[amplifier]]), the namely [[Regular Iteration]], if it can be applied, gives the meaningful ("physical") solution
 
<ref name="conje1">
 
http://tori.ils.uec.ac.jp/PAPERS/2011recovery.pdf
 
D.Kouznetsov. Recovery of intensity inside a uniform amplifier from its transfer function. preprint ILS UEC, 2011.
 
</ref><ref name="conje2">
 
http://tori.ils.uec.ac.jp/PAPERS/2012transfer.pdf
 
D.Kouznetsov. Transfer function of an amplifier and characterization of Materials. 2012, under consideration.
 
</ref>.
 
 
==Conclusion==
 
Not every superfunction can be represented through the regular iteration above, but in many cases, the '''regular iteration''' provides the simple and efficient way of the evaluation.
 
<!--
 
With '''regular iteration''' , the [[complex maps]] of the superfunction can be plotted in real time.
 
!-->
 
 
The simplified version of this article exists
 
<ref name="simplyfied">
 
http://tori.ils.uec.ac.jp/2012OR/2012or.pdf
 
D. Kouznetsov. Superfunctions for optical amplifiers. Preprint ILS UEC, 2012
 
</ref>;
 
the slideshow is also available
 
<ref>
 
http://tori.ils.uec.ac.jp/2012BANGKOK/show.pdf
 
D.Kouznetsov. RECOVERY OF PROPERTIES OF A MATERIAL FROM THE TRANSFER FUNCTION OF A BULK SAMPLE (THEORY). 2012.
 
</ref>.
 
 
==References==
 
<references/>
 
 
[[Category:Iteration]]
 
[[Category:Articles in English]]
 
[[Category:Superfunction]]
 
[[Category:Algotirhm]]
 
[[Category:Regular iteration]]
 

Revision as of 07:04, 1 December 2018

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