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| In [[mathematics]], the equations governing the '''isomonodromic deformation''' of [[meromorphic]] linear systems of [[ordinary differential equations]] are, in a fairly precise sense, the most fundamental [[exact]] [[nonlinear]] differential equations. As a result, their solutions and properties lie at the heart of the field of exact nonlinearity and [[integrable systems]].
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| Isomonodromic deformations were first studied by [[Lazarus Fuchs]], with early pioneering contributions from [[Paul Painlevé]], [[René Garnier]], and [[Ludwig Schlesinger]]. Inspired by results in [[statistical mechanics]], a seminal contribution to the theory was made by [[Michio Jimbo]], [[Tetsuji Miwa]] and [[Kimio Ueno]], who studied cases with arbitrary singularity structure.
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| ==Fuchsian systems and Schlesinger's equations==
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| We consider the [[Fuchsian system]] of linear differential equations
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| :<math>\frac{dY}{dx}=AY=\sum_{i=1}^{n}\frac{A_i}{x-\lambda_i}Y</math>
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| where the dependent variable ''x'' takes values in the complex projective line '''P'''<sup>1</sup>('''C'''), the solution ''Y'' takes values in '''C'''<sup>''n''</sup> and the ''A<sub>i</sub>'' are constant ''n''×''n'' matrices. By placing ''n'' independent column solutions into a [[fundamental matrix (linear differential equation)|fundamental matrix]] we can regard ''Y'' as taking values in GL(''n'', '''C'''). Solutions to this equation have simple poles at ''x'' = λ<sub>''i''</sub>. For simplicity, we shall assume that there is no further pole at infinity which amounts to the condition that
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| :<math>\sum_{i=1}^{n}A_i=0.</math>
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| ===Monodromy data===
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| Now, fix a basepoint ''b'' on the Riemann sphere away from the poles. [[Analytic continuation]] of the solution ''Y'' around any pole λ<sub>''i''</sub> and back to the basepoint will produce a new solution ''Y′''. The new and old solutions are linked by the [[monodromy]] matrix ''M<sub>i</sub>'' as follows:
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| :<math>Y'=YM_i.</math>
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| We therefore have the [[Riemann–Hilbert]] [[homomorphism]] from the [[fundamental group]] of the punctured sphere to the monodromy representation:
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| :<math>\pi_1 \left (\mathbf{P}^1(\mathbf{C}) - \{\lambda_1,\dots,\lambda_n\} \right )\to GL(n,\mathbf{C}).</math>
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| A change of basepoint merely results in a (simultaneous) conjugation of all the monodromy matrices. The monodromy matrices modulo simultaneous conjugation define the '''monodromy data''' of the Fuchsian system.
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| ===Hilbert's twenty-first problem===
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| Now, with given monodromy data, can we find a Fuchsian system which exhibits this monodromy? This is one form of [[Hilbert's twenty-first problem]]. We do not distinguish between coordinates ''x'' and <math>\hat{x}</math> which are related by [[Möbius transformation]]s, and we do not distinguish between gauge equivalent Fuchsian systems - this means that we regard ''A'' and
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| :<math>g^{-1}(x)Ag(x)-g^{-1}(x)\frac{dg(x)}{dx}</math>
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| as being equivalent for any holomorphic [[gauge transformation]] ''g''(''x''). (It is thus most natural to regard a Fuchsian system geometrically, as a [[Connection (mathematics)|connection]] with simple poles on a trivial rank ''n'' [[vector bundle]] over the Riemann sphere).
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| For generic monodromy data, the answer to Hilbert's twenty-first problem is 'yes' - as was first proved by [[Josip Plemelj]]. However, Plemelj neglected certain degenerate cases, and it was shown in 1989 by [[Andrei Bolibrukh]] that there are cases when the answer is 'no'. Here, we focus entirely on the generic case.
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| ===Schlesinger's equations===
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| There are (generically) many Fuchsian systems with the same monodromy data. Thus, given any such Fuchsian system with specified monodromy data, we can perform '''isomonodromic deformations''' of it. We are therefore led to study '''families''' of Fuchsian systems, and allow the matrices ''A<sub>i</sub>'' to depend on the positions of the poles.
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| In 1912 (following earlier incorrect attempts) [[Ludwig Schlesinger]] proved that in general, the deformations which preserve the monodromy data of a (generic) Fuchsian system are governed by the [[integrable]] [[holonomic system]] of [[partial differential equations]] which now bear his name:
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| :<math>\begin{align}
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| \frac{\partial A_i}{\partial \lambda_j} &= \frac{[A_i,A_j]}{\lambda_i-\lambda_j} \qquad \qquad j\neq i \\
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| \frac{\partial A_i}{\partial \lambda_i} &= -\sum_{j\neq i}\frac{[A_i,A_j]}{\lambda_i-\lambda_j}.
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| \end{align}</math>
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| These are therefore the '''isomonodromy equations''' for (generic) Fuchsian systems. It should be noted that the natural interpretation of these equations is as the flatness of a natural connection on a vector bundle over the 'deformation parameter space' which consists of the possible pole positions. For non-generic isomonodromic deformations, there will still be an integrable isomonodromy equation, but it will no longer be Schlesinger.
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| If we limit ourselves to the case when the ''A<sub>i</sub>'' take values in the Lie algebra <math>\mathfrak{sl}(2,\mathbf{C})</math>, we obtain the so-called '''Garnier systems'''.If we specialize further to the case when there are only four poles, then the Schlesinger/Garnier equations can be reduced to the famous sixth [[Painlevé equation]]. | |
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| ==Irregular singularities==
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| Motivated by the appearance of [[Painlevé transcendents]] in [[correlation functions]] in the theory of [[Bose gases]], Michio Jimbo, Tetsuji Miwa and Kimio Ueno extended the notion of isomonodromic deformation to the case of arbitrary pole structure. The linear system we study is now of the form
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| :<math>\frac{dY}{dx}=AY=\sum_{i=1}^{n}\sum_{j=1}^{r_i+1}\frac{A^{(i)}_j}{(x-\lambda_i)^j}Y,</math>
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| with ''n'' poles, with the pole at λ<sub>''i''</sub> of order <math>(r_i+1)</math>. The <math>A^{(i)}_j</math> are constant matrices.
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| ===Extended monodromy data=== | |
| As well as the monodromy representation described in the Fuchsian setting, deformations of irregular systems of linear ordinary differential equations are required to preserve ''extended'' monodromy data. Roughly speaking, monodromy data is now regarded as data which glues together canonical solutions near the singularities. If we take <math>x_i = x - \lambda_i</math> as a local coordinate near a pole λ<sub>''i''</sub>of [[Degree of a polynomial|order]] <math>r_i+1</math>, then we can solve term-by-term for a holomorphic gauge transformation ''g'' such that locally, the system looks like
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| :<math>\frac{d(g_i^{-1}Z_i)}{dx_i} = \left(\sum_{j=1}^{r_i} \frac{(-j)T^{(i)}_j}{x_i^{j+1}}+\frac{M^{(i)}}{x_i}\right)(g_i^{-1}Z_i)</math>
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| where <math>M^{(i)}</math> and the <math>T^{(i)}_j</math> are '''diagonal''' matrices. If this were valid, it would be extremely useful, because then (at least locally), we have decoupled the system into ''n'' scalar differential equations which we can easily solve to find that (locally):
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| :<math>Z_i = g_i \exp\left(M^{(i)} \log(x_i)+\sum_{j=1}^{r_i}\frac{T^{(i)}_j}{x_i^{j}}\right).</math>
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| However, this does not work - because the power series we have solved term-for-term for ''g'' will not, in general, converge.
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| It was the great insight of Jimbo, Miwa and Ueno to realize that nevertheless, this approach provides canonical solutions near the singularities, and can therefore be gainfully employed to define extended monodromy data. This is because of a theorem of [[George Birkhoff]] which states that given such a formal series, there is a unique '''convergent''' function ''G<sub>i</sub>'' such that in any particular sufficiently large sector around the pole, ''G<sub>i</sub>'' is [[asymptotic]] to ''g<sub>i</sub>'', and
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| :<math>Y = G_i \exp\left(M^{(i)} \log(x_i)+\sum_{j=1}^{r_i}\frac{T^{(i)}_j}{x_i^{j}}\right).</math>
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| is a true solution of the differential equation. We therefore have a canonical solution in each such sector near each pole. The extended monodromy data consists of
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| * the data from the monodromy representation as for the Fuchsian case;
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| * [[Stokes' matrices]] which connect canonical solutions between adjacent sectors at the same pole;
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| * connection matrices which connect canonical solutions between sectors at different poles.
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| ===General isomonodromic deformations===
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| As before, we now consider families of systems of linear differential equations, all with the same singularity structure. We therefore allow the matrices <math>A^{(i)}_j</math> to depend on parameters. We allow ourselves to vary the positions of the poles λ<sub>''i''</sub>, but now, in addition, we also vary the entries of the diagonal matrices <math>T^{(i)}_j</math> which appear in the canonical solution near each pole.
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| Jimbo, Miwa and Ueno proved that if we define a one-form on the 'deformation parameter space' by
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| :<math>\Omega = \sum_{i=1}^{n}\left(A d\lambda_i - g_i D \left( \sum_{j=1}^{r_i}T^{(i)}_j \right)g_i^{-1} \right)</math>
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| (where ''D'' denotes [[exterior differentiation]] with respect to the components of the <math>T^{(i)}_j</math> only)
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| then deformations of the meromorphic linear system specified by ''A'' are isomonodromic if and only if
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| :<math>dA + [\Omega,A] + \frac{d\Omega}{dx} = 0.</math>
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| These are the '''general isomonodromy equations'''. As before, these equations can be interpreted as the flatness of a natural connection on the deformation parameter space.
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| ==Properties== | |
| The isomonodromy equations enjoy a number of properties which justify their status as nonlinear [[special functions]].
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| ===Painlevé property===
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| This is perhaps the most important property of a solution to the isomonodromic deformation equations. This means that all [[essential singularities]] of the solutions are fixed, although the positions of poles may move. It was proved by [[Bernard Malgrange]] for the case of Fuchsian systems, and by [[Tetsuji Miwa]] in the general setting.
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| Indeed, suppose we are given a partial differential equation (or a system of them). Then, 'possessing a reduction to an isomonodromy equation' is more or less '''equivalent''' to the [[Painlevé property]], and can therefore be used as a test for [[integrable system|integrability]].
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| ===Transcendence===
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| In general, solutions of the isomonodromy equations cannot be expressed in terms of simpler functions such as solutions of linear differential equations. However, for particular (more precisely, reducible) choices of extended monodromy data, solutions can be expressed in terms of such functions (or at least, in terms of 'simpler' isomonodromy transcendents). The study of precisely what this transcendence means has been largely carried out by the invention of 'nonlinear [[differential Galois theory]]' by [[Hiroshi Umemura]] and [[Bernard Malgrange]].
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| There are also very special solutions which are [[algebraic number|algebraic]]. The study of such algebraic solutions involves examining the [[topology]] of the deformation parameter space (and in particular, its [[mapping class group]]); for the case of simple poles, this amounts to the study of the action of [[braid groups]]. For the particularly important case of the sixth [[Painlevé equation]], there has been a notable contribution by [[Boris Dubrovin]] and [[Marta Mazzocco]], which has been recently extended to larger classes of monodromy data by [[Philip Boalch]].
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| Rational solutions are often associated to special polynomials. Sometimes, as in the case of the sixth [[Painlevé equation]], these are well-known [[orthogonal polynomials]], but there are new classes of polynomials with extremely interesting distribution of zeros and [[interlacing]] properties. The study of such polynomials has largely been carried out by [[Peter Clarkson]] and collaborators.
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| ===Symplectic structure===
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| The isomonodromy equations can be rewritten using [[Hamiltonian mechanics|Hamiltonian]] formulations. This viewpoint was extensively pursued by [[Kazuo Okamoto]] in a series of papers on the [[Painlevé equations]] in the 1980s.
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| They can also be regarding as a natural extension of the Atiyah-Bott symplectic structure on spaces of [[flat connections]] on [[Riemann surfaces]] to the world of meromorphic geometry - a perspective pursued by [[Philip Boalch]]. Indeed, if we fix the positions of the poles, we can even obtain [[complete metric space|complete]] [[hyperkähler manifolds]]; a result proved by [[Oliver Biquard]] and [[Philip Boalch]].
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| There is another description in terms of [[moment maps]] to (central extensions of) [[loop algebras]] - a viewpoint introduced by [[John Harnad]] and extended to the case of general singularity structure by [[Nick Woodhouse]]. This latter perspective is intimately related to a curious [[Laplace transform]] between isomonodromy equations with different pole structure and rank for the underlying equations.
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| ===Twistor structure===
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| The isomonodromy equations arise as (generic) full dimensional reductions of (generalized) anti-self-dual [[Yang-Mills equations]]. By the [[Penrose-Ward transform]] they can therefore be interpreted in terms of holomorphic vector bundles on [[complex manifolds]] called [[twistor]] spaces. This allows the use of powerful techniques from [[algebraic geometry]] in studying the properties of transcendents. This approach has been pursued by [[Nigel Hitchin]], [[Lionel Mason]] and [[Nick Woodhouse]].
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| ===Gauss-Manin connections=== | |
| By considering data associated with families of Riemann surfaces branched over the singularities, we can consider the isomonodromy equations as nonhomogenous [[Gauss-Manin connection]]s. This leads to alternative descriptions of the isomonodromy equations in terms of [[abelian function]]s - an approach known to Fuchs and Painlevé, but lost until rediscovery by [[Yuri Manin]] in 1996.
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| ===Asymptotics===
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| Particular transcendents can be characterized by their asymptotic behaviour. The study of such behaviour goes back to the early days of isomonodromy, in work by [[Pierre Boutroux]] and others.
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| ==Applications==
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| Their universality as the simplest genuinely nonlinear integrable systems means that the isomonodromy equations have an extremely diverse range of applications. Perhaps of greatest practical importance is the field of [[random matrix theory]]. Here, the statistical properties of [[eigenvalues]] of large random matrices are described by particular transcendents.
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| The initial impetus for the resurgence of interest in isomonodromy in the 1970s was the appearance of transcendents in [[correlation functions]] in [[Bose gases]].
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| They provide generating functions for [[moduli spaces]] of two-dimensional [[topological quantum field theories]] and are thereby useful in the study of [[quantum cohomology]] and [[Gromov-Witten invariants]]. | |
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| 'Higher-order' isomonodromy equations have recently been used to explain the mechanism and universality properties of shock formation for the [[dispersionless limit]] of the [[Korteweg–de Vries equation]]. | |
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| They are natural reductions of the [[Ernst equation]] and thereby provide solutions to the [[Einstein field equations]] of general relativity; they also give rise to other (quite distinct) solutions of the Einstein equations in terms of [[theta functions]].
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| They have arisen in recent work in [[mirror symmetry (string theory)|mirror symmetry]] - both in the [[geometric Langlands]] programme, and in work on the moduli spaces of [[stability conditions]] on [[derived categories]].
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| ==Generalizations==
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| The isomonodromy equations have been generalized for meromorphic connections on a general [[Riemann surface]].
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| They can also easily be adapted to take values in any [[Lie group]], by replacing the diagonal matrices by the [[maximal torus]], and other similar modifications.
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| There is a burgeoning field studying discrete versions of isomonodromy equations.
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| ==References==
| |
| *{{Citation | last1=Its | first1=Alexander R. | last2=Novokshenov | first2=Victor Yu. | title=The isomonodromic deformation method in the theory of Painlevé equations | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-16483-8 | id={{MathSciNet | id = 851569}} | year=1986 | volume=1191}}
| |
| *{{Citation | last1=Sabbah | first1=Claude | title=Isomonodromic deformations and Frobenius manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Universitext | isbn=978-1-84800-053-7; 978-2-7598-0047-6 | id={{MathSciNet | id = 1933784 }} | year=2007}}
| |
| | |
| {{DEFAULTSORT:Isomonodromic Deformation}}
| |
| [[Category:Ordinary differential equations]]
| |
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Pick normаl water or organiс juice instead of soda. Sugary liquids, fօr example soda, have a sizeable stage hіgh-fructoѕe corn syrup and the right after-outcome will probably be high numbers of sugar taken in. Sodаs likewise have аn ingгedient named citric acid solution, whiсh could erode and weаken your tooth. Thiѕ deterioгation from your citric aсіd is along with the acid which is a by-product of harmful bactеriɑ growth, developing on the toоth on account of ingеsting the сorn syrup.
Eggplant features a spongy textսre that works well in numerous meals, sucҺ as eggplant parmesan and fowl curry. That's not all! Eggplant also provides potassium, mangɑnese, folic acid and herbal antioxidants.
Be sure your daily diet is rich in calcium supplement when expecting. ϒou have to give your chіld some cɑlcium supplements for thеm to have fantastic the teeth and strong bones, of coursе, if they don't get a great deal of calcium it could make you calсium lаcking.
Frosty freѕh vegetables are great to enjoy for their health articleѕ and fit in the freeƶer properly. They can be convenient techniques for getting several of the nutrition that you desire. Findіng yourself in the freezer signifies you won't need to worry about thеm spoiling prior to you've possessed a chance to rely on them.
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