Commutative Artinian Algebras

I.高度完全交的一般元素的性质3我们的主要结果如下：定理1.令R=k [x，y，z]是特征为0的域k上的多项式环。设I是R中由d_1，d_2，d_3次齐次元素f_1，f_2，f_3 * R生成的完全交理想，其中设<less than or equal>2d_1d_2d_3<less than or equal><less than or equal>.则以下条件是等价的。(i)对于任意一般线性形式l * R，μ（I+lR/lR）=3。(ii)d_3 <less than or equal>d_1+d_2- 2.定理2.利用与上述相同的符号和假设，我们有（i）<less than or equal>d_3d_1 +d_2- 2d_3 <less than or equal>*I：l由3个元素生成。(ii)d_3d_1 <less than or equal>+d_2-2*I：l是由5个元素生成的环，由此我们可以证明：当（i）<less than or equal>d_13，d_23<less than or equal>，d_3 *，（ii）<less than or equal>d_14，d_2&lt;小于或 关于我们 equal&gt;4，*d_3*4，（iii）d_3d_1 <greater than or equal>+d_2- 3. II.运动边界弦振动的行为我们详细研究了运动边界弦振动的行为。最一般的结果如下。研究了一类具有时间周期边界条件和时间周期边界函数的一维波动方程的初边值问题。这是描述李萨如图形的两端振动弦的数学模型。如果由边界函数和上述时间周期定义的复合函数的旋转数满足丢番图不等式，则每个解都是时间拟周期解。由此可见，对于“几乎所有”边界函数，解都是拟周期的。进一步将解推广到整个R^2平面上满足波动方程的空间拟周期函数，解的奇性沿着反射特征传播。从我们的研究表明，解析数论的几个基本性质在解的行为中起着至关重要的作用。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