Topological relations between spherical spatial regions with holes

There is growing interest in globally modelling the entire planet.Although topological relations between spherical simple regions and topological relations between regions with holes in the plane have been investigated,few studies have focused on the topological relations between spherical spatial regions with holes.The 16-intersection model(16IM)is proposed to describe the topological relations between spatial regions with holes.A total of 25 negative conditions are proposed to eliminate the impossible topological relations between spherical spatial regions with holes.The results show that(1)3 disjoint relations,3 meet relations,66 overlap relations,7 cover relations,3 contain relations,1 equal relation,7 coveredBy relations,3 inside relations,1 attach relation,52 entwined relations,and 28 embrace relations can be distinguished by the 16IM and that(2)the formalisms of attach,entwined,and embrace relations between the spherical spatial regions without holes based on the 9IM and that between the spherical spatial regions with holes based on the simplified 16IM are different,whereas the formalisms of other types of relations between spherical spatial regions without holes based on the 9IM and that between the spherical spatial regions with holes based on a simplified 16IM are the same.

The Existence and Application of Strongly Idempotent Self-orthogonal Row Latin Magic Arrays

Let N = {0, 1, ···, n-1}. A strongly idempotent self-orthogonal row Latin magic array of order n（SISORLMA（n） for short） based on N is an n × n array M satisfying the following properties：（1） each row of M is a permutation of N, and at least one column is not a permutation of N;（2） the sums of the n numbers in every row and every column are the same;（3） M is orthogonal to its transpose;（4） the main diagonal and the back diagonal of M are 0, 1, ···, n-1 from left to right. In this paper, it is proved that an SISORLMA（n）exists if and only if n ？ {2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n ？ {2, 3}, and a rational diagonally ordered magic square exists if and only if n ≠2.