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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク John's equation ： ウィキペディア英語版 John's equation John's equation is an ultrahyperbolic partial differential equation satisfied by the X-ray transform of a function. It is named after Fritz John. Given a function $f\backslash colon\backslash mathbb^n\; \backslash rightarrow\; \backslash mathbb$ with compact support the ''X-ray transform'' is the integral over all lines in $\backslash mathbb^n$. We will parameterise the lines by pairs of points $x,y\; \backslash in\; \backslash mathbb^n$, $x\; \backslash ne\; y$ on each line and define ''$u$'' as the ray transform where :$u(x,y)\; =\; \backslash int\backslash limits\_^\; f(\; x\; +\; t(y-x)\; )\; dt.$ Such functions ''$u$'' are characterized by John's equations :$\backslash frac\; -\; \backslash frac=0$ which is proved by Fritz John for dimension three and by Kurusa for higher dimensions. 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. More generally an ''ultrahyperbolic'' partial differential equation (a term coined by Richard Courant) is a second order partial differential equation of the form :$\backslash sum\backslash limits\_^\; a\_\backslash frac\; +$ \sum\limits_^ b_i\frac + cu =0 where $n\; \backslash ge\; 2$, such that the quadratic form :$\backslash sum\backslash limits\_^\; a\_\; \backslash xi\_i\; \backslash xi\_j$ can be reduced by a linear change of variables to the form :$\backslash sum\backslash limits\_^\; \backslash xi\_i^2\; -\; \backslash sum\backslash limits\_^\; \backslash xi\_i^2.$ 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. ==References== * * Á. Kurusa, A characterization of the Radon transform's range by a system of PDEs, J. Math. Anal. Appl., 161(1991), 218--226. * S K Patch, Consistency conditions upon 3D CT data and the wave equation, Phys. Med. Biol. 47 No 15 (7 August 2002) 2637-2650 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「John's equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.