A Spectral Equivalent Condition of the $P$-Polynomial Property for Association Schemes

We give two equivalent conditions of the $P$-polynomial property of a symmetric association scheme. The first equivalent condition shows that the $P$-polynomial property is determined only by the first and second eigenmatrices of the symmetric association scheme. The second equivalent condition is another expression of the first using predistance polynomials.

Construction of Compactly Supported Refinable Componentwise Polynomial Functions inℝ2

We provide a sufficient condition for constructing a class of compactly supported refinable functions with componentwise polynomial property inℝ2. An iteration algorithm is developed to compute the polynomial on each component of the functions' support. Finally, two examples for constructing the symmetric refinable componentwise polynomial functions are given.

dd-SEQUENCES AND PARTIAL EULER–POINCARÉ CHARACTERISTICS OF KOSZUL COMPLEX

The aim of this paper is to introduce a new notion of sequences called dd-sequences and show that this notion may be convenient for studying the polynomial property of partial Euler–Poincaré characteristics of the Koszul complex with respect to the powers of a system of parameters. Some results about the dd-sequences, the partial Euler–Poincaré characteristics and the lengths of local cohomology modules are also presented in the paper. There are also applications of dd-sequences on the structure of sequentially Cohen–Macaulay modules.