翻訳と辞書 Words near each other ・ Piercer (Dungeons & Dragons) ・ Pierces Chapel, Texas ・ Pierces Corner, Virginia ・ Pierces Creek ・ Pierces Creek, Australian Capital Territory ・ Pierceton Historic District ・ Pierceton, Indiana ・ Piercetown, County Westmeath (civil parish) ・ Pierceville ・ Pierceville Township, Finney County, Kansas ・ Pierceville, Indiana ・ Pierceville, Kansas ・ Pierceville, Wisconsin ・ Piercey ・ Piercey Islands ・ Pierce–Birkhoff conjecture ・ Pierce–Hichborn House ・ Piercia ・ Piercing (disambiguation) ・ Piercing (novel) ・ Piercing I ・ Piercing migration ・ Piercing point ・ Piercing saw ・ Piercing the corporate veil ・ Piercing the Darkness ・ Piercing the Veil ・ Piercolias ・ Piercy ・ Piercy Farmstead Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Pierce–Birkhoff conjecture ： ウィキペディア英語版 Pierce–Birkhoff conjecture In abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a maximum of finite minima of finite collections of polynomials. It was first stated, albeit in non-rigorous and vague wording, in the 1956 paper of Garrett Birkhoff and Richard S. Pierce in which they first introduced f-rings. The modern, rigorous statement of the conjecture was formulated by Melvin Henriksen and John R. Isbell, who worked on the problem in the early 1960s in connection with their work on f-rings. Their formulation is as follows: :for every real piecewise-polynomial function $f\; :\; R^n\; \backslash rightarrow\; R$, there exists a finite set of polynomials $g\_\; \backslash in\; R(\backslash ldots,\; x\_n)$ such that $f\; =\; \backslash sup\_i\; \backslash inf\_j\; (\; g\_\; )$. Isbell is likely the source of the name ''Pierce–Birkhoff conjecture'', and popularized the problem in the 1980s by discussing it with several mathematicians interested in real algebraic geometry.〔 The conjecture was proved true for ''n'' = 1 and 2 by Louis Mahé.〔(【引用サイトリンク】 publisher = Atlas Conferences, Inc. )〕 == Local Pierce–Birkhoff conjecture == In 1989, James J. Madden provided an equivalent statement that is in terms of the real spectrum of $A\; =\; R(\backslash ldots,\; x\_n)$ and the novel concepts of local polynomial representatives and separating ideals. Denoting the real spectrum of ''A'' by $\backslash mathrm\; A$, the ''separating ideal of ''α'' and ''β in $\backslash mathrm\; A$ is the ideal of ''A'' generated by all polynomials $g\; \backslash in\; A$ that change sign on ''α'' and ''β'', ''ie.'' $g(\backslash alpha)\; \backslash ge\; 0$ and $g(\backslash beta)\; \backslash le\; 0$. Any finite covering $R^n\; =\; \backslash cup\_i\; P\_i$ of closed, semi-algebraic sets induces a corresponding covering $\backslash mathrm\; A\; =\; \backslash cup\_i\backslash tilde\_i$, so, in particular, when ''f'' is piecewise polynomial, there is a polynomial $f\_i$ for every $\backslash alpha\backslash in\backslash mathrm\; A$ such that $f|\_\; =\; f\_i|\_$ and $\backslash alpha\backslash in\backslash tilde\_i$. This $f\_i$ is termed the ''local polynomial representative of ''f'' at ''α. Madden's so-called ''local Pierce–Birkhoff conjecture at ''α'' and ''β, which is equivalent to the Pierce–Birkhoff conjecture, is as follows: : Let ''α'', ''β'' be in $\backslash mathrm\; A$ and ''f'' be piecewise-polynomial. It is conjectured that for every local representative of ''f'' at ''α'', $f\_\backslash alpha$, and local representative of ''f'' at ''β'', $f\_\backslash beta$, $f\_\backslash alpha\; -\; f\_\backslash beta$ is in the separating ideal of ''α'' and ''β''.〔 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「Pierce–Birkhoff conjecture」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.