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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 Mingarelli identity ： ウィキペディア英語版 Mingarelli identity In the field of ordinary differential equations, the Mingarelli identity (coined by Philip Hartman) is a theorem that provides criteria for the oscillation and non-oscillation of solutions of some linear differential equations in the real domain. It extends the Picone identity from two to three or more differential equations of the second order. Its most basic form appears here. == The identity == Consider the $n$ solutions of the following (uncoupled) system of second order linear differential equations over the ''t''-interval (). $(p\_i(t)\; x\_i^\backslash prime)^\backslash prime\; +\; q\_i(t)\; x\_i\; =\; 0,\; \backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\; x\_i(a)=1,\backslash ,\backslash ,\; x\_i^\backslash prime(a)=R\_i\backslash ,$ where $i=1,2,\; \backslash ldots,\; n$. Let $\backslash Delta$ denote the forward difference operator, i.e., $\backslash Delta\; x\_i\; =\; x\_-x\_i.$ The second order difference operator is found by iterating the first order operator as in $\backslash Delta^2\; (x\_i)\; =\; \backslash Delta(\backslash Delta\; x\_i)\; =\; x\_-2x\_+x\_$, with a similar definition for the higher iterates. Leaving out the independent variable ''t'' for convenience, and assuming the $x\_i(t)\; \backslash ne\; 0$ on (''a'', ''b''], there holds the identity, : $$ \begin x_^2\Delta^(p_1r_1)]_a^b & = \int_a^b (x^\prime_)^2 \Delta^(p_1) - \int_a^b x_^2 \Delta^(q_1) - \sum_^ C(n-1,k)(-1)^\int_a^b p_ W^2(x_,x_)/x_^2, \end where $r\_i\; =\; x^\backslash prime\_i/x\_i$ is a logarithmic derivative, $W(x\_i,\; x\_j)\; =\; x^\backslash prime\_ix\_j\; -\; x\_ix^\backslash prime\_j$, is a Wronskian and the $C(n-1,k)$ are binomial coefficients. When $n=2$ this reduces to the Picone identity. The above identity leads quickly to the following comparison theorem for three linear differential equations, extending the Sturm–Picone comparison theorem. Let $p\_i,\backslash ,\; q\_i,\backslash ,$ ''i'' = 1, 2, 3 be real-valued continuous functions on the interval () and let #$(p\_1(t)\; x\_1^\backslash prime)^\backslash prime\; +\; q\_1(t)\; x\_1\; =\; 0,\; \backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\; x\_1(a)=1,\backslash ,\backslash ,\; x\_1^\backslash prime(a)=R\_1\backslash ,$ #$(p\_2(t)\; x\_2^\backslash prime)^\backslash prime\; +\; q\_2(t)\; x\_2\; =\; 0,\; \backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\; x\_2(a)=1,\backslash ,\backslash ,\; x\_2^\backslash prime(a)=R\_2\backslash ,$ #$(p\_3(t)\; x\_3^\backslash prime)^\backslash prime\; +\; q\_3(t)\; x\_3\; =\; 0,\; \backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\; x\_3(a)=1,\backslash ,\backslash ,\; x\_3^\backslash prime(a)=R\_3\backslash ,$ be three homogeneous linear second order differential equations in self-adjoint form with :$p\_i(t)>0\backslash ,$ for each i and for all ''t'' in (), and where the $R\_i$ are arbitrary real numbers. Assume that for all ''t'' in () we have, :$\backslash Delta^2(q\_1)\; \backslash ge\; 0$, :$\backslash Delta^2(p\_1)\; \backslash le\; 0$, :$\backslash Delta^2(p\_1(a)R\_1)\; \backslash le\; 0$. If $x\_1(t)\; >\; 0$ on (), and $x\_2(b)=0$, then any solution $x\_3(t)$ has at least one zero in (). 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「Mingarelli identity」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.