Let k1, k2 be nonzero integers with（k1, k2） = 1 and k1k2≠-1. Let Rk1,k2（A, n）be the number of solutions of n = k1a1 ＋ k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A Z such that Rk1,k2（A...Let k1, k2 be nonzero integers with（k1, k2） = 1 and k1k2≠-1. Let Rk1,k2（A, n）be the number of solutions of n = k1a1 ＋ k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A Z such that Rk1,k2（A, n） = 1 for all n ∈ Z. Let f ： Z-→ N0∪ {∞} be a function such that f-1（0） is finite. In this paper, we generalize Xiong＇s result and prove that there exist uncountably many sets A Z such that Rk1,k2（A, n） = f（n） for all n ∈ Z.展开更多
Assume that B is a separable real Banach space and X(t) is a diffusion process on B. In thispaper, we will establish the representation theorem of martingale additive functionals of X(t).
基金Supported by the National Natural Science Foundation of China(Grant No.11471017)
文摘Let k1, k2 be nonzero integers with（k1, k2） = 1 and k1k2≠-1. Let Rk1,k2（A, n）be the number of solutions of n = k1a1 ＋ k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A Z such that Rk1,k2（A, n） = 1 for all n ∈ Z. Let f ： Z-→ N0∪ {∞} be a function such that f-1（0） is finite. In this paper, we generalize Xiong＇s result and prove that there exist uncountably many sets A Z such that Rk1,k2（A, n） = f（n） for all n ∈ Z.
文摘Assume that B is a separable real Banach space and X(t) is a diffusion process on B. In thispaper, we will establish the representation theorem of martingale additive functionals of X(t).
