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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 Colombeau algebra ： ウィキペディア英語版 Colombeau algebra In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this. Such a multiplication of distributions has long been believed to be impossible because of L. Schwartz' impossibility result, which basically states that there cannot be a differential algebra containing the space of distributions and preserving the product of continuous functions. However, if one only wants to preserve the product of smooth functions instead such a construction becomes possible as demonstrated first by Colombeau. As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, geophysics, microlocal analysis and general relativity so far. == Schwartz' Impossibility Result == Attempting to embed the space $\backslash mathcal\text{'}(\backslash mathbb)$ of distributions on $\backslash mathbb$ into an associative algebra $(A(\backslash mathbb),\; \backslash circ,\; +)$, the following requirements seem to be natural: # $\backslash mathcal\text{'}(\backslash mathbb)$ is linearly embedded into $A(\backslash mathbb)$ such that the constant function $1$ becomes the unity in $A(\backslash mathbb)$, # There is a partial derivative operator $\backslash partial$ on $A(\backslash mathbb)$ which is linear and satisfies the Leibnitz rule, # the restriction of $\backslash partial$ to $\backslash mathcal\text{'}(\backslash mathbb)$ coincides with the usual partial derivative, # the restriction of $\backslash circ$ to $C(\backslash mathbb)\; \backslash times\; C(\backslash mathbb)$ coincides with the pointwise product. However, L. Schwartz' result〔L. Schwartz, 1954, "Sur l'impossibilité de la multiplication des distributions", ''Comptes Rendus de L'Académie des Sciences'' 239, pp. 847-848 ()〕 implies that these requirements cannot hold simultaneously. The same is true even if, in 4., one replaces $C(\backslash mathbb)$ by $C^k(\backslash mathbb)$, the space of $k$ times continuously differentiable functions. While this result has often been interpreted as saying that a general multiplication of distributions is not possible, in fact it only states that one cannot unrestrictedly combine differentiation, multiplication of continuous functions and the presence of singular objects like the Dirac delta. Colombeau algebras are constructed to satisfy conditions 1.-3. and a condition like 4., but with $C(\backslash mathbb)\; \backslash times\; C(\backslash mathbb)$ replaced by $C^\backslash infty(\backslash mathbb)\; \backslash times\; C^\backslash infty(\backslash mathbb)$, i.e., they preserve the product of smooth (infinitely differentiable) functions only. 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「Colombeau algebra」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.