Commutative Artinian Algebras

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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