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| '''John's equation''' is an [[ultrahyperbolic partial differential equation]] satisfied by the [[X-ray transform]] of a function. It is named after [[Fritz John]].
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| Given a function <math>f\colon\mathbb{R}^n \rightarrow \mathbb{R}</math> with compact support the ''X-ray transform'' is the integral over all lines in <math>\mathbb{R}^n</math>. We will parameterise the lines by pairs of points <math>x,y \in \mathbb{R}^n</math>, <math>x \ne y </math> on each line and define ''<math>u</math>'' as the ray transform where
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| :<math> u(x,y) = \int\limits_{-\infty}^{\infty} f( x + t(y-x) ) dt. </math>
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| Such functions ''<math>u</math>'' are characterized by John's equations
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| :<math> \frac{\partial^2u}{\partial x_i \partial y_j} - \frac{\partial^2u}{\partial y_i \partial x_j}=0 </math>
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| which is proved by [[Fritz John]] for dimension three and by [[:hu:Kurusa Árpád|Kurusa]] for higher dimensions.
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| In three dimensional x-ray [[computerized tomography]] John's equation can be solved to fill in missing data, for example where the data is obtained from a point source traversing a curve, typically a helix.
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| More generally an ''ultrahyperbolic'' partial differential equation (a term coined by [[Richard Courant]]) is a second order partial differential equation of the form
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| :<math> \sum\limits_{i,j=1}^{2n} a_{ij}\frac{\partial^2 u}{\partial x_i \partial x_j} +
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| \sum\limits_{i=1}^{2n} b_i\frac{\partial u}{\partial x_i} + cu =0</math>
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| where <math>n \ge 2</math>, such that the [[quadratic form]]
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| :<math> \sum\limits_{i,j=1}^{2n} a_{ij} \xi_i \xi_j</math> | |
| can be reduced by a linear change of variables to the form
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| :<math> \sum\limits_{i=1}^{n} \xi_i^2 - \sum\limits_{i=n+1}^{2n} \xi_i^2. </math> | |
| It is not possible to arbitrarily specify the value of the solution on a non-characteristic hypersurface. John's paper however does give examples of manifolds on which an arbitrary specification of ''u'' can be extended to a solution.
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| ==References==
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| *{{Citation | last1=John | first1=Fritz | url=http://projecteuclid.org/euclid.dmj/1077490637|title=The ultrahyperbolic differential equation with four independent variables| doi=10.1215/S0012-7094-38-00423-5 | mr=1546052 | zbl = 0019.02404 | year=1938 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=4 | issue=2 | pages=300–322}}
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| * Á. Kurusa, A characterization of the Radon transform's range by a system of PDEs, J. Math. Anal. Appl., 161(1991), 218--226. {{doi|10.1016/0022-247X(91)90371-6}}
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| * S K Patch, Consistency conditions upon 3D CT data and the wave equation, Phys. Med. Biol. 47 No 15 (7 August 2002) 2637-2650 {{doi|10.1088/0031-9155/47/15/306}}
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| [[Category:Partial differential equations]]
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Oscar is how he's known as and he completely loves this name. My family life in Minnesota and my family loves it. Doing ceramics is what adore doing. Bookkeeping is my profession.
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