In addition to the known method given in [1], authors provide other three methods to the enumeration of one-vertex maps with face partition on the plane. Correspondingly, there are four functional equations in the enu...In addition to the known method given in [1], authors provide other three methods to the enumeration of one-vertex maps with face partition on the plane. Correspondingly, there are four functional equations in the enufuntion. It is shown that the four equations are equivalent. Moreover, an explicit expression of the solution is found by expanding the powers of the matrix of infinite order directly. This is a new complement of what appeared in [1].展开更多
We investigate the Hill differential equation ?where A(t), B(t), and D(t) are trigonometric polynomials. We are interested in solutions that are even or odd, and have period π or semi-period π. The above equation wi...We investigate the Hill differential equation ?where A(t), B(t), and D(t) are trigonometric polynomials. We are interested in solutions that are even or odd, and have period π or semi-period π. The above equation with one of the above conditions constitutes a regular Sturm-Liouville eigenvalue problem. We investigate the representation of the four Sturm-Liouville operators by infinite banded matrices.展开更多
文摘In addition to the known method given in [1], authors provide other three methods to the enumeration of one-vertex maps with face partition on the plane. Correspondingly, there are four functional equations in the enufuntion. It is shown that the four equations are equivalent. Moreover, an explicit expression of the solution is found by expanding the powers of the matrix of infinite order directly. This is a new complement of what appeared in [1].
文摘We investigate the Hill differential equation ?where A(t), B(t), and D(t) are trigonometric polynomials. We are interested in solutions that are even or odd, and have period π or semi-period π. The above equation with one of the above conditions constitutes a regular Sturm-Liouville eigenvalue problem. We investigate the representation of the four Sturm-Liouville operators by infinite banded matrices.
