Commutative Artinian Algebras

1.高度为3的完全交的一般元的行为我们的主要结果表述如下：定理1。设R=k [x, y， z]是特征为0的域k上的多项式环。设I是由齐次元素f_1， f_2， f_3 * R分别为d_1， d_2， d_3生成的R的完全交理想，其中我们假设2<小于等于>d_1<小于等于>d_2<小于等于>d_3。那么下列条件是等价的。(i) mu(i +lR/lR)=3对于任何一般线性形式l * r (ii) d_3<或等于>d_1+d_2-2。定理2。与上面相同的符号和假设，我们有(i) d_3<小于或等于>d_1+d_2-2<小于或等于>d_3* i: l由3个元素生成。（ii） d_3<或等于>d_1+d_2-2*I: 1由5个元素生成。因此，我们可以证明Hard Lefschetz定理在环R/I上对以下情况成立：(I) d_1<小于或等于>,d_2<小于或等于>3,*d_3, (ii) d_1<小于或等于>4,d_2<小于或…More = >4, *d_3*4, (iii) d_3<大于或等于>d_1+d_2-3. ii。带移动边界的振动弦的行为我们详细地研究了带移动边界的振动弦的行为。最一般的结果如下。研究了具有时周期边界条件和时周期边界函数的一维波动方程的初边值问题。这是具有两端的振动弦的数学模型它描述了利萨诸图。如果由边界函数定义的复合函数的旋转数与上述时间段满足某些Diophantine不等式，则每个解都是时间准周期的。由此可以得出，对于“几乎所有”边界函数，其解都是准周期的。进一步将解推广到整个R^2平面上满足波动方程的空间准周期函数，解的奇异性沿反射特性传播。我们的研究表明，解析数论的几个基本性质在解的行为中起着重要的作用。少

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