Discrete Differential Calculus on Simplicial Complexes and Constrained Homology

Let V be a finite set.Let K be a simplicial complex with its vertices in V.In this paper,the author discusses some differential calculus on V.He constructs some constrained homology groups of K by using the differential calculus on V.Moreover,he defines an independence hyper graph to be the complement of a simplicial complex in the complete hypergraph on V.Let L be an independence hypergraph with its vertices in V.He constructs some constrained cohomology groups of L by using the differential calculus on V.

The NF-Number of a Simplicial Complex

Let△bea simplicial complex on[n].TheNF-complex of△is the simplicial complexδ_(NF)(△)on[n]for which the facet ideal of△is equal to the Stanley-Reisner ideal ofδ_(NF)(△).Furthermore,for each k=2,3,....,we introduce the kth NF-complexδ_(NF)^(k)(△),which is inductively defined asδ_(NF)(k)(△)=δ_(NF)(δ_(NF)^(k-1)(△))by settingδ_(NF)^(1)(△)=δ_(NF)(△).One canδ_(NF)^(0)(△)=△.The NF-number of△is the smallest integer k>0 for whichδ_(NF)^(k)(△)■△.In the present paper we are especilly interested in the NF-number of a finite graph,which can be regraded as a simplicial complex of dimension one.It is shown that the NF-number of the finite graph K_(n)■K_(m)on[n+m],which is the disjoint union of the complete graphs K_(n)on[n]and K_(m)on[m]for n≥2 and m≥2 with(n,m)≠(2,2),is equal to n+m+2.As a corollary,we find that the NF-number of the complete bipartite graph K_(n,m)on[n+m]is also equal to n+m+2.

有向图的加权解析挠率

In 2020,Alexander Grigor'yan,Yong Lin and Shing-Tung Yau[6]introduced the Reidemeister torsion and the analytic torsion for digraphs by means of the path complex and the path homology theory.Based on the analytic torsion for digraphs introduced in[6],we consider the notion of weighted analytic torsion for vertex-weighted digraphs.For any non-vanishing real functions f and g on the vertex set,we consider the vertex-weighted digraphs with the weights(f;g).We calculate the(f;g)-weighted analytic torsion by examples and prove that the(f;g)-weighted analytic torsion only depend on the ratio f=g.In particular,if the weight is of the diagonal form(f;f),then the weighted analytic torsion equals to the usual(un-weighted)torsion.

Granular Computing on Partitions, Coverings and Neighborhood Systems

Granular Computing on partitions(RST),coverings(GrCC) and neighborhood systems(LNS) are examined: (1) The order of generality is RST, GrCC, and then LNS. (2) The quotient structure: In RST, it is called quotient set. In GrCC, it is a simplical complex, called the nerve of the covering in combinatorial topology. For LNS, the structure has no known description. (3) The approximation space of RST is a topological space generated by a partition, called a clopen space. For LNS, it is a generalized/pretopological space which is more general than topological space. For GrCC,there are two possibilities. One is a special case of LNS,which is the topological space generated by the covering. There is another topological space, the topology generated by the finite intersections of the members of a covering The first one treats covering as a base, the second one as a subbase. (4) Knowledge representations in RST are symbol-valued systems. In GrCC, they are expression-valued systems. In LNS, they are multivalued system; reported in 1998 . (5) RST and GRCC representation theories are complete in the sense that granular models can be recaptured fully from the knowledge representations.

Algebraic Characterization of SSC of Uni-Cyclic Multigraphs

We introduce first the spanning simplicial complex(SSC)of a multigraph g,which gives a generalization of the SSC associated with a simple graph G.Combinatorial properties are discussed for the SSC of a family of uni-cyclic multigraphs U_(n)^(r),m with n edges including r multiple edges within and outside the cycle of length m,which are then used to compute the f-vector and Hilbert series of face ring k[△s(U_(n)^(r),m)]for the SSC △s(U_(n)^(r),m)(un,m).Moreover,we find the associated primes of the facet ideal I_(F)(△s(U_(n)^(r),m).Finally,we device a formula for homology groups of △s(U_(n)^(r),m) prove that the SsC of a family of uni-cyclic multigraphs is Cohen-Macaulay.

Sequentially Cohen-Macaulay Mixed Product Ideals

We classify the ideals of mixed products that are sequentially Cohen-Macaulay.