Based on the falling shadow theory, the concept of falling fuzzy (implicative) ideals as a generalization of that of a T ∧-fuzzy (implicative) ideal is proposed in MV-algebras. The relationships between falling fuzzy (implicative) ideals and T-fuzzy (implicative) ideals are discussed, and conditions for a falling fuzzy (implicative) ideal to be a T ∧-fuzzy (implicative) ideal are provided. Some...

We use i and i for the falling and rising factorials respectively and we adopt the convention a binomial coeÆcient is zero if either of its parameters is negative. Let Fn;r denote the n-by-n matrix (i 1) j i j 1 r 1 1 i;j n . Note that Fn;r is a block matrix 0 0 Gn;r 0 with Gn;r lower triangular of size n r. Let Hn;r denote the (n r)-by-(n r) lower triangular banded matrix ( 1) j) i 1 j 1 r j ...

OBJECTIVES This study aims to explore the prevalence of depression and anxiety in left-behind children, and to identify patterns of parents' migration and relative factors associated with depression and anxiety risk in this population. SETTING A cross-sectional survey using a school-based sample was conducted in Puyang, Hebei, North China in December 2012. PARTICIPANTS 2283 students aged 10...

We give eecient algorithms for listing equivalence classes of k-ary strings under reversal and permutation of alphabet symbols. As representative of each equivalence class we choose that string which is lexicographically smallest. These algorithms use space O(n) and time O(p kN), where N is the total number of strings generated and n is the length of each string. For k = 2, we obtain a recursiv...

Abstract Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer’s symbol) and the falling factorial. It is a special ...

This classic problem was brought to the author’s attention by Johann Otto. This view seems to be advocated by Routh (1896) in Ex. 4, p. 82 of [13]. Examples of falling chains were often considered at Cambridge U. [11, 12, 13, 14, 16, 18, 19], as recounted in a footnote on p. 80 of [13]: Problems on infinitesimal impulses were solved in the lecture room of the late Mr Hopkins as long ago as 1850...