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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 Janet basis ： ウィキペディア英語版 Janet basis In mathematics, a Janet basis is a normal form for systems of linear homogeneous partial differential equations (PDEs) that removes the inherent arbitrariness of any such system. It was introduced in 1920 by Maurice Janet.〔M. Janet, Les systemes d'equations aux derivees partielles, Journal de mathematiques 83 (1920), pages 65–123.〕 It was first called the Janet basis by F. Schwarz in 1998.〔F. Schwarz, "Janet Bases for Symmetry Groups", in: ''Groebner Bases and Applications; Lecture Notes Series'' 251, London Mathematical Society, pages 221–234 (1998); B. Buchberger and F. Winkler, Edts.〕 The left hand sides of such systems of equations may be considered as differential polynomials of a ring, and Janet's normal form as a special basis of the ideal that they generate. By abuse of language, this terminology will be applied both to the original system and the ideal of differential polynomials generated by the left hand sides. A Janet basis is the predecessor of a Groebner basis introduced by Bruno Buchberger〔B. Buchberger, Ein algorithmisches Kriterium fuer die Loesbarkeit eines algebraischen Gleichungssystems, Aequ. Math. 4, 374–383(1970).〕 for polynomial ideals. In order to generate a Janet basis for any given system of linear pde's a ranking of its derivatives must be provided; then the corresponding Janet basis is unique. If a system of linear pde's is given in terms of a Janet basis its differential dimension may easily be determined; it is a measure for the degree of indeterminacy of its general solution. In order to generate a Loewy decomposition of a system of linear pde's its Janet basis must be determined first. == Generating a Janet basis == Any system of linear homogeneous pde's is highly non-unique, e.g. an arbitrary linear combination of its elements may be added to the system without changing its solution set. A priori it is not known whether it has any nontrivial solutions. More generally, the degree of arbitrariness of its general solution is not known, i.e. how many undetermined constants or functions it may contain. These questions were the starting point of Janet's work; he considered systems of linear pde's in any number of dependent and independent variables and generated a normal form for them. Here mainly linear pde's in the plane with the coordinates $x$ and $y$ will be considered; the number of unknown functions is one or two. Most results described here may be generalized in an obvious way to any number of variables or functions.〔F. Schwarz, Algorithmic Lie Theory for Solving Linear Ordinary Differential Equations, Chapman & Hall/CRC, 2007 Chapter 2.〕〔W. Plesken, D. Robertz, Janet's approach to presentations and resolutions for polynomials and linear pdes, Archiv der Mathematik 84, page 22–37, 2005.〕〔T. Oaku, T. Shimoyama, A Groebner Basis Method for Modules over Rings of Differential Operators, Journal of Symbolic Computation 18, page 223–248, 1994.〕 In order to generate a unique representation for a given system of linear pde's, at first a ranking of its derivatives must be defined. Definition A ranking of derivatives is a total ordering such that for any two derivatives $\backslash delta$, $\backslash delta\_1$ and $\backslash delta\_2$, and any derivation operator $\backslash theta$ the relations $\backslash delta\backslash leq\backslash theta\backslash delta$ and $\backslash delta\_1\backslash leq\backslash delta\_2\backslash rightarrow\backslash delta\backslash delta\_1\backslash leq\backslash delta\backslash delta\_2$ are valid. A derivative $\backslash delta\_2$ is called ''higher'' than $\backslash delta\_1$ if $\backslash delta\_2>\backslash delta\_1$. The highest derivative in an equation is called its ''leading derivative''. For the derivatives up to order two of a single function $z$ depending on $x$ and $y$ with $x>y$ two possible order are : the $LEX$ order $z\_>z\_>z\_x>z\_>z\_y>z$ and the $GRLEX$ order $z\_>z\_>z\_>z\_x>z\_y>z$. Here the usual notation $\backslash partial\_xz=z\_x,\; \backslash partial\_yz=z\_y,\backslash ldots$ is used. If the number of functions is higher than one, these orderings have to be generalized appropriately, e.g. the orderings $TOP$ or $POT$ may be applied.〔W. Adams, P. Loustaunau, An introduction to Groebner bases, American Mathematical Society, Providence, 1994.〕 The first basic operation to be applied in generating a Janet basis is the ''reduction'' of an equation $e\_1$ w.r.t. another one $e\_2$. In colloquial terms this means the following: Whenever a derivative of $e\_1$ may be obtained from the leading derivative of $e\_2$ by suitable differentiation, this differentiation is performed and the result is subtracted from $e\_1$. Reduction w.r.t. a system of pde's means reduction w.r.t. all elements of the system. A system of linear pde's is called ''autoreduced'' if all possible reductions have been performed. The second basic operation for generating a Janet basis is the inclusion of ''integrability conditions''. They are obtained as follows: If two equations $e\_1$ and $e\_2$ are such that by suitable differentiations two new equations may be obtained with like leading derivatives, by cross-multiplication with its leading coefficients and subtraction of the resulting equations a new equation is obtained, it is called an integrability condition. If by reduction w.r.t. the remaining equations of the system it does not vanish it is included as a new equation to the system. It may be shown that repeating these operations always terminates after a finite number of steps with a unique answer which is called the Janet basis for the input system. Janet has organized them in terms of the following algorithm. Janet's Algorithm Given a system of linear differential polynomials $S\backslash equiv\backslash$, the Janet basis corresponding to $S$ is returned. : S1: (''Autoreduction'') Assign $S:=\backslash operatorname(S)$ : S2: (''Completion'') Assign $S:=\backslash operatorname(S)$ : S3: (''Integrability conditions'') Find all pairs of leading terms $v\_i$ of $e\_i$ and $v\_j$ of $e\_j$ such that differentiation w.r.t. a nonmultiplier $x\_$ and multipliers $x\_,\backslash ldots,x\_$ leads to ::$\backslash fracv\_j\}$ \cdots \partial x_^} and determine the integrability conditions ::$c\_\; =\backslash operatorname(e\_j)\backslash cdot$ \frac(e_i)\cdot \frac \cdots \partial x_^} : S4: (''Reduction of integrability conditions''). For all $c\_$ assign $c\_:=\backslash operatorname(c\_,S)$ : S5: (''Termination?'') If all $c\_$ are zero return $S$, otherwise make the assignment $S:=S\backslash cup\backslash \backslash neq\; 0\backslash \}$, reorder $S$ properly and goto S1 Here $Autoreduce$ is a subalgorithm that returns its argument with all possible reductions performed, $Completion$ adds certain equations to the system in order to facilitate determining the integrability conditions. To this end the variables are divides into ''multipliers'' and ''non-multipliers''; details may be found in the above references. Upon successful termination a Janet basis for the input system will be returned. Example 1 Let the system $\backslash z\_x-\backslash fracz=0,\; e\_2\backslash equiv\; z\_x+\backslash fracz\_y+xz=0\backslash \}$ be given with ordering $GRLEX$ and $x>y$. Step S1 returns the autoreduced system : $\backslash (xy^3-x^2-y)z\_y-\backslash frac(x^3-x+y)z=0,\; e\_2=z\_x+\backslash fracz\_y+xz=0\backslash \}.$ Steps S3 and S4 generate the integrability condition $c\_\backslash equiv\backslash frac-\backslash frac$ and reduces it to $z=0$, i.e. the Janet basis for the originally given system is $\backslash$ with the trivial solution $z=0$. The next example involves two unknown functions $w$ and $z$, both depending on $x$ and $y$. Example 2 Consider the system :$\backslash -\backslash fracw\_x+\backslash fracw=0,\; f\_2\backslash equiv\; w\_-\backslash fracz\_-\backslash fracw\_y-6x^2z\_x,$ ::$f\_3\backslash equiv\; w\_+4x^2w\_x-8x^2z\_y-8xw=0,\; f\_4\backslash equiv\; z\_+\backslash fracz\_x=0\backslash \}$ in $GRLEX,\; w>z,x>y$ ordering. The system is already autoreduced, i.e. step S1 returns it unchanged. Step S3 generates the two integrability conditions ::$c\_\backslash equiv\backslash frac-\backslash frac\backslash textc\_\backslash equiv\backslash frac-\backslash frac.$ Upon reduction in step S4 they are ::$c\_=z\_-6xz\_x=0,\; c\_=z\_+3x^2z\_-24xz\_y-12w=0.$ In step S5 they are included into the system and the algorithms starts again with step S1 with the extended system. After a few more iterations finally the Janet basis ::$\backslash w=0,z\_x=0,w\_y=0,w\_x-\backslash fracw=0\backslash \}$ is obtained. It yields the general solution $z=C\_1-C\_2x,\; w=2C\_2y$ with two undetermined constants $C\_1$ and $C\_2$. Janet's algorithm has been implemented in Maple ;〔S. Zhang, Z. Li, An Implementation for the Algorithm of Janet bases of Linear Differential Ideals in the Maple System, Acta Mathematicae Applicatae Sinica, English Series, 20, page 605-616 (2004)〕 an implementation is also available on the website www.alltypes.de. 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「Janet basis」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.