Partial Groups, Simplicial K(G, 1)’s and Kan Complexes

In our paper Simplicial K(G, 1)’s we constructed a sub-complex of the nerve of a group G determined by a partial group structure, and we proved, under a generalized associativity condition called regularity, that the sub-complex realizes as a K(G, 1). This type of sub-complex appears naturally in several topological and algebraic contexts. In this note we prove that regularity of a partial group implies that the Kan extension condition is satisfied on its nerve in dimensions greater than one, and in dimension one a weaker version of the extension condition holds.

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.

FINITE ELEMENT APPROXIMATIONS OF SYMMETRIC TENSORS ON SIMPLICIAL GRIDS IN R^n： THE HIGHER ORDER CASE

The design of mixed finite element methods in linear elasticity with symmetric stress approximations has been a longstanding open problem until Arnold and Winther designed the first family of mixed finite elements where the discrete stress space is the space of H（div,Ω;S）-Pk＋1 tensors whose divergence is a Pk-1 polynomial on each triangle for k ≥ 2. Such a two dimensional family was extended, by Arnold, Awanou and Winther, to a three dimensional family of mixed elements where the discrete stress space is the space of H（div, Ω; S）-Pk＋2 tensors, whose divergence is a Pk-1 polynomial on each tetrahedron for k ≥ 2. In this paper, we are able to construct, in a unified fashion, mixed finite element methods with symmetric stress approximations on an arbitrary simplex in R^n for any space dimension. On the contrary, the discrete stress space here is the space of H（div,Ω; S）-Pk tensors, and the discrete displacement space here is the space of L^2（Ω; R^n）-Pk＋1 vectors for k ≥ n＋ 1. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and can be regarded as extensions to any dimension of those in two and three dimensions by Hu and Zhang.

On the Construction of Well-Conditioned Hierarchical Bases for H(div)-Conforming Rn Simplicial Elements

Hierarchical bases of arbitrary order for H(div)-conforming triangular and tetrahedral elements are constructedwith the goal of improving the conditioning of the mass and stiffness matrices.For the basis with the triangular element,it is found numerically that the conditioning is acceptable up to the approximation of order four,and is better than a corresponding basis in the dissertation by Sabine Zaglmayr[High Order Finite Element Methods for Electromagnetic Field Computation,Johannes Kepler Universit¨at,Linz,2006].The sparsity of the mass matrices from the newly constructed basis and from the one by Zaglmayr is similar for approximations up to order four.The stiffness matrix with the new basis is much sparser than that with the basis by Zaglmayr for approximations up to order four.For the tetrahedral element,it is identified numerically that the conditioning is acceptable only up to the approximation of order three.Compared with the newly constructed basis for the triangular element,the sparsity of the massmatrices fromthe basis for the tetrahedral element is relatively sparser.

Spanning Simplicial Complex of Wheel Graph Wn

In this paper, we explore the spanning simplicial complex of wheel graph Wn on vertex set [n]. Combinatorial properties of the spanning simplicial complex of wheel graph are discussed, which are then used to compute the f-vector and Hilbert series of face ring k[Δs(Wn)] for the spanning simplicial complex Δs(Wn). Moreover, the associated primes of the facet ideal IF(Δs(Wn)) are also computed.

The Twisted Homology of Simplicial Set

In this article,we give a generalization of δ-twisted homology introduced by Jingyan Li,Vladimir Vershinin and Jie Wu,called Δ-twisted homology,which enriches the theory of δ-(co)homology introduced by Alexander Grigor’yan,Yuri Muranov and Shing-Tung Yau.We show that the Mayer-Vietoris sequence theorem holds for Δ-twisted homology.Applying the Δ-twisted ideas to Cartesian products,we introduce the notion of Δ-twisted Cartesian product on simplicial sets,which generalizes the classical work of Barratt,Gugenheim and Moore on twisted Cartesian products of simplicial sets.Under certain hypothesis,we show that the coordinate projection of Δ-twisted Cartesian product admits a fibre bundle structure.

A SIMPLICIAL ALGORITHM FOR COMPUTING AN INTEGER ZERO POINT OF A MAPPING WITH THE DIRECTION PRESERVING PROPERTY

A mapping f ： Z^n → Rn is said to possess the direction preserving property if fi（x） 〉 0 implies fi（y） ≥ 0 for any integer points x and y with ｜｜x - y｜｜∞≤ 1. In this paper, a simplicial algorithm is developed for computing an integer zero point of a mapping with the direction preserving property. We assume that there is an integer point x^0 with c ≤ x^0≤d satisfying that maxl≤i≤（xi - xi^0）fi（x） 〉 0 for any integer point x with f（x） ≠ 0 on the boundary of H = {x ∈R^n [c-e ≤ x〈d＋e},wherecanddaretwo finite integer points with c 〈 d and e = （1, 1,... , 1）^T E R^n. This assumption is implied by one of two conditions for the existence of an integer zero point of a mapping with the preserving property in van der Laan et al. （2004）. Under this assumption, starting at x^0, the algorithm follows a finite simplicial path and terminates at an integer zero point of the mapping. This result has applications in general economic equilibrium models with indivisible commodities.

Moment-angle manifolds and connected sums of simplicial spheres

The connected sum is a fundamental operation in geometric topology and combinatorics.In this paper,we study the connection between connected sums of simplicial spheres and the algebraic topology of their corresponding moment-angle manifolds.The cohomology rings of moment-angle manifolds corresponding to connected sums of simplicial spheres are computed,which leads to a conjecture on the topology of such moment-angle manifolds.

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.

Relations between Simplicial Groups, 3-Crossed Modules and 2-Quadratic Modules

In this paper, we define two-quadratic module and explore the relations among two-quadratic modules, three-crossed modules and simplicial groups.

A SIMPLICIAL HOMOTOPY ALGORITHM FOR COMPUTING ZERO POINTS ON POLYTOPES

In this paper a triangulation of continuous and arbitrary refinement of grid sizes is proposed for simplicial homotopy algorithms to compute zero points on a polytope P. The proposed algorithm generates a piecewise linear path in P × [1,∞) from any chosen interior point x0 of P on level {1} to a solution of the underlying problem. The path is followed by making linear programming pivot steps in a linear system and replacement steps in the triangnlation.The starting point x0 is left in a direction to one vertex of P. The direction in which x0 leaves depends on the function value at x0 and the polytope P. Moreover, we also give a new equivalent form of the Brouwer fixed point theorem on polytopes. This form has many important applications in mathematical programming and the theory of differential equations.

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.

基于SimpliciTI的大棚温湿度无线监测系统设计

针对目前温室大棚温湿度监测大多采用人工监测或有线监测方式,设计了一种基于Simplici TI的大棚温湿度无线监测系统。该系统以遵循Simplici TI协议的CC2500无线收发器组成星型无线传感器网络,传感器节点采用HTU21D数字温湿度传感器测量各节点温湿度值,由OLED12832模块现场显示,中央节点把收到的数据经串口上传PC机,并用由VC++6.0制作的监视软件进行显示和记录。测试表明,该系统具有成本低、组网灵活、测量精度高和实用性强等优点。

基于MSP430和CC2500的USB无线数据采集系统

设计基于MSP430单片机和CC2500无线收发器的USB无线数据采集系统,通过USB调试端口在IDE或CCE开发环境下编写、下载和调试应用程序,其目标板可作为一个独立的具有或没有外部传感器的系统。介绍基于MSP430和CC2500无线开发工具的系统结构,控制模块的硬件结构,无线收发部分和数据采集传输电路的设计,以及软件开发、SimpliciTI协议的引用。

基于Simplici Ti的无线火灾报警系统设计

现有火灾报警主要采用有线方式,耗费材料且难以实现互联报警。采用无线方式能够很好地解决互联报警、降低成本。利用MSP430F227和CC2500搭载Simplici Ti实现节点互联,数据中心节点连接至PC机,PC机实时监控节点信息,在火灾到来时能够及时准确实现本地报警和远程报警。火灾节点功耗低,休眠时只有不到1μA电流。

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.

New Simple Semigrorps

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基于GPRS网络和CC1101模块的无线抄表系统的研究

文章针对楼宇的抄表系统中存在的布线困难、线路维护复杂等弊端,提出了基于CC1101模块的无线抄表技术方案。系统选用穿透能力较强的433MHz频段模块组成楼内无线网络,并利用GPRS公用数据网络实现抄表数据的远传。方案中针对现有电能表已有的数据通信方式,设计终端节点直接从电能表中读取数据,再利用CC1101模块组成的无线网络把采集的电能表数据上传楼内网。

基于433MHz无线自组网的火灾监控报警系统

针对复杂建筑结构、受地理条件限制的场所布设通信线有困难的问题,研制了一种基于433 MHz无线自组网GPRS的火灾监控报警系统。以射频芯片CC1110为无线自组网核心,搭载Simplici TI网络协议,实现各节点设备无线连接。数据中心节点接入GPRS向消防中心传输信息。系统已应用于实际,实现了本地报警和远程监控。

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.