Commutative Artinian Algebras

交换阿天尼代数

基本信息

批准号：
06640077
负责人：
YAMAGUCHI Masaru
金额：
$ 1.34万
依托单位：
Tokai University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-06640077/
关键词：
Artinian complete intersection Artinian Gorenstein algebra general element Lefschetz condition one-dimensional wave equation Lissajous boundary condition Diophantine inequality quasiperiodic solution O次元完全交差環 O次元ゴレンスタイン環 Lissajous境界条件

项目摘要

I.Behevior of General Elements of Complete Intersections of Height 3Our main results are stated as follows :Theorem 1. Let R=k [x, y, z] be the polynomial ring over a field k of characteristic 0. Let I be a complete intersection ideal of R generated by homogeneous elements f_1, f_2, f_3 * R of degrees d_1, d_2, d_3 respectively, where we assume that 2<less than or equal>d_1<less than or equal>d_2<less than or equal>d_3. Then the following conditions are equivalent.(i) mu(I+lR/lR)=3 for any general linear form l * R.(ii) d_3<less than or equal>d_1+d_2-2.Theorem 2. With the same notation and assumption as above we have(i) d_3<less than or equal>d_1+d_2-2<less than or equal>d_3*I : l is generated by 3 elements.(ii) d_3<less than or equal>d_1+d_2-2*I : l is generated by 5 elements.As a consequence we can prove that the Hard Lefschetz theorem holds on the the ring R/I for the cases (i) d_1<less than or equal>3, d_2<less than or equal>3, *d_3, (ii) d_1<less than or equal>4, d_2<less than or … More equal>4, *d_3*4, (iii) d_3<greater than or equal>d_1+d_2-3.II.The behavior of the vibrating string with moving boundariesWe studied the behavior of the vibrating string with moving boundaries in detail. The most general results are the following. We are dealt with the initial-boundary value problem for one-dimensional wave equation with time-periodic boundary conditions and time-peridic boundary functions. This is the mathematical model of the vibrating string with the both ends which describe the Lissajous figures. Every solution is time-quasiperiodic if the rotation number of a composed function defined by the boundary functions and the above time-periods satisfy some Diophantine inequality. From this it follows that for 'almost all' boundary functions the solutions are quasiperiodic. Further the solutions are extended to the space-quasiperiodic functions in the whole R^2-plane which satisfy the wave equation and the singularities of the solutions propagate along the reflected characteristics. From our research it is shown that several fundamental properties from the analytic number theory play an essential role in the behavior of the solutions. Less

i.在高度的完整相互作用的一般元素3的一般元素中，主要结果如下：定理1。令r = k [x，y，z]为特征性0的磁场上的多项式环。让我成为一个完整的交点。小于或等于> d_1 <小于或等于> d_2 <小于或等于> d_3。那么以下条件是等效的。（i）Mu（i+lr/lr）= 3对于任何一般线性l * R * R.（ii）d_3 <少于或等于> d_1+d_2-2.theorem 2。具有与上述相同的（i）d_3 <sy> d_1+d_1+d_1+d_2-2-2 <sely> d_2-2-2 <selbiate和假设的符号和假设3 * 3 * 3 * em> d_3 * libe> d_3 * l。 d_3 <少于或等于> d_1+d_2-2 *i：l是由5个元素生成的。结果，我们可以证明，对于（i）d_1 <小于或等于> 3，d_2 <少于或等于> 3， *d_3，（ii）d_3，（ii）d_3，（ii）d_3，（ii）d_3，（ii）d_3，（ii）d_3，ii> 4，d_1 <小于或等于> 3，d_2 <少于或等于> 4 d_2 <小于或…更等于> 4， *d_3 *4，（iii）d_3 <大于或等于> d_1+d_2-3.ii。振动字符串的行为详细研究了振动字符串的行为。最一般的结果是以下。我们正在处理一维波方程的初始有限值问题，并具有时间周期性的边界条件和时间疗法边界函数。这是振动字符串的数学模型，两端都描述了lissajous数字。如果由边界函数定义的组成函数的旋转数，并且上述时间周期满足某些二磷剂不平等，则每个解决方案都是时间Quasiperiodic。由此得出的，对于“几乎所有”边界的功能，解决方案是Quasiperiodic。此外，将溶液扩展到整个R^2平面中的空间Quasiperiodic函数，该函数满足了波动方程，溶液的奇异性沿反射特性传播。从我们的研究中可以看出，分析数理论的几种基本属性在解决方案的行为中起着至关重要的作用。较少的