Ideals Whose First Two Betti Numbers Are Close

For an ideal I of a Noetherian local ring （R, m, k） one hasβR1（I） -β0R（I） ≥-1. It is demonstrated that some residual intersections of an ideal I for whichβ1R（I） -β0R（I） = -1 or 0 are perfect.

On Graded Betti Numbers of a Class of Graphs

In this paper, Betti numbers are evaluated for several classes of graphs whose complements are bipartite graphs. Relations are established for the general case, and counting formulae are given in several particular cases, including the union of several mutually disjoint complete graphs.

The Betti Numbers of Real Toric Varieties Associated to Weyl Chambers of Type B

The authors compute the(rational) Betti number of real toric varieties associated to Weyl chambers of type B, and furthermore show that their integral cohomology is p-torsion free for all odd primes p.

Curvature, Diameter and Bounded Betti Numbers

In this paper, we introduce the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold （M, g） with Ric （M） ≥ -（n - 1） and Diam （M） ≤D are bounded by a number depending on D and n. We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups （H（M）,Ⅱ·Ⅱ∞） among closed Riemannian manifold （M, g） with K（M） ≥ -1 and Diam （M） ≤ D.

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori,Ⅱ

We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S2× S1 or RP3#P3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni （2014） in an earlier paper.

Betti Numbers of Locally Standard 2-Torus Manifolds

Let Mn be a smooth closed n-manifold with a locally standard （Z2）n-action. This paper deals with the relationship among the rood 2 Betti numbers of Mn, the mod 2 Betti numbers and the h-vector of the orbit space of the action.

Extremal Betti Numbers of Some Cohen-Macaulay Binomial Edge Ideals

We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones,and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals:Cohen-Macaulay bipartite and fan graphs.In addition,we compute the Hilbert-Poincare series of the binomial edge ideals of some Cohen-Macaulaybipartitegraphs.

A Unified Newtonian-Relativistic Quantum Resolution of the Supposedly Missing Dark Energy of the Cosmos and the Constancy of the Speed of Light

Time dilation, space contraction and relativistic mass are combined in a novel fashion using Newtonian dynamics. In this way we can surprisingly retrieve an effective quantum gravity energy-mass equation which gives the accurate experimental value of vacuum density. Furthermore Einstein’s equation of special relativity E = mc2, where m is the mass and c is the velocity of light developed assuming smooth 4D space time is transferred to a rugged Calabi-Yau and K3 fuzzy Kahler manifolds and revised to become E=(mc2)/(22), where the division factor 22 maybe interpreted as the compactified bosonic dimensions of Veneziano-Nambu strings. The result is again an accurate effective quantum gravity energy-mass relation akin to the one found using Newtonian dynamics which correctly predicts that 95.4915028% of the energy in the cosmos is the hypothetical missing dark energy. The agreement with WMAP and supernova measurements is in that respect astounding. In addition different theories are used to check the calculations and all lead to the same quantitative result. Thus the theories of varying speed of light, scale relativity, E-infinity theory, M-theory, Heterotic super strings, quantum field in curved space time, Veneziano’s dual resonance model, Nash Euclidean embedding and super gravity all reinforce, without any reservation, the above mentioned theoretical result which in turn is in total agreement with the most sophisticated cosmological measurements which was deservingly awarded the 2011 Nobel Prize in Physics. Finally and more importantly from certain viewpoints, we reason that the speed of light is constant because it is a definite probabilistic expectation value of a variable velocity in a hierarchical fractal clopen, i.e. closed and open micro space time.

Mohamed El Naschie’s Revision of Albert Einstein’s E=m_(0)c^(2):A Definite Resolution of the Mystery of the Missing Dark Energy of the Cosmos

The Egyptian engineering scientist and theoretical physicist Mohamed El Naschie has found a definite resolution to the missing dark energy of the cosmos based on a revision of the theory of Relativity. Einstein’s equation of special relativity E = m0c2, where m0 is the controversial rest mass and c is the velocity of light developed in smooth 4D space-time was transferred by El Naschie to a rugged Calabi-Yau and K3 fuzzy Kahler manifold. The result is an accurate, effective quantum gravity energy-mass relation which correctly predicts that 95.4915028% of the energy in the cosmos is the missing hypothetical dark energy. The agreement with WMAP and supernova measurements is astounding. Different theories are used by El Naschie to check the calculations and all lead to the same quantitative result. Thus the theories of varying speed of light, scale relativity, E-infinity theory, M-theory, Heterotic super strings, quantum field in curved space-time, Veneziano’s dual resonance model and Nash’s Euclidean embedding all reinforce, without any reservation, the above mentioned theoretical result of El Naschie which in turn is in total agreement with the most sophisticated cosmological measurement. Incidentally these experimental measurements and analysis were awarded the 2011 Nobel Prize in Physics to Adam Riess, Brian Schmidt, and Saul Perlmutter.

Biggs Theorem for Directed Cycles and Topological Invariants of Digraphs

We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By considering two-dimensional CW complex of elementary cycles and deriving formulas for the Betti numbers of the associated cellular homology groups, we extend the list of representation independent topological inavariants measuring the graph structure. We prove the computation of the 2nd Betti number to be sharp #P hard in general and present specific representation invariant sub-fillings yielding efficiently computable homology groups. Finally, we suggest how to use the provided structural measures to shed new light on graph theoretical problems as graph embeddings, discrete Morse theory and graph clustering.

移动传感器网络区域覆盖快速检测的拓扑方法研究

无线传感器网络以获取有用信息为最终目的,而获得的有用信息的多少取决于对监测区域的覆盖程度,因此覆盖算法是其研究的重要问题之一。现有的覆盖算法大多假定传感器节点能够提供精确的部署位置及相对方位。介绍了一种基于代数同调群的快速覆盖检测算法,基于网络拓扑即可快速判断网络的覆盖情况,无需传感器精确位置,降低了传感器设计复杂度。仿真结果表明,该算法可以快速有效地检测移动传感器部署的覆盖盲区。

ON THE(CO)HOMOLOGY OF(QUOTIENTS OF)MOMENT-ANGLE MANIFOLDS OVER POLYGONS

The purpose of this paper is to represent the cohomology ring of a moment-angle manifold over an m-gon explicitly in terms of the quotient of an exterior algebra,and to count the Betti numbers of the cohomology groups of a special class of quotients of moment-angle manifolds.

Homology of Powers of Ideals： Artin-Rees Numbers of Syzygies and the Golod Property

Let Ro be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal ra and residue class field K = Rim. For a graded ideal I in R we show that for k 〉〉 0： （1） the Artin-Rees number of the syzygy modules of Ik as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and （2） R is a polynomial ring over the regular Ro, the ring R/Ik is Golod, its Poincar4-Betti series is rational and the Betti numbers of the free resolution of K over R/I^k are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of I^k for k 〉〉 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/Ik over R.

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.

Index Estimates for Free Boundary f-minimal Hypersurfaces

We prove that the index is bounded from below by a linear function of its first Betti number for any compact free boundary f-minimal hypersurface in certain positively curved weighted manifolds.

A Generalized Discrete Morse-Floer Theory

Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a(simplicial,polyhedral or cellular)complex without closed orbits,where each cell may either have no arrows,receive a single arrow from one of its facets,or conversely,send a single arrow into a cell of which it is a facet.By following arrows,one can then construct a natural Floer-type boundary operator.Here,we develop such a construction for arrow patterns where each cell may support several outgoing or incoming arrows(but not both),again in the absence of closed orbits.Our main technical achievement is the construction of a boundary operator that squares to 0 and therefore recovers the homology of the underlying complex.

Torsions of 3-Dimensional Small Covers

In this paper, it is shown that for a 3-dimensional small cover M over a polytope P, there are only 2-torsions in H_1(M; Z). Moreover, the mod 2 Betti number growth of finite covers of M is studied.

EXTENSIONS ON 2-EDGE CONNECTED 3-REGULAR UP-EMBEDDABLE GRAPHS

It is known[5] that an investigation of the up-embeddability of the 3-regular graphs shows a useful approach to that of the general graph. But as far, very few characterizations of the upembeddability are known on the 3-regular graphs. Let G be a 2-edge connected 3-regular graph.We prove that G is up-embeddable if and only if G can be obtained from the graphs θ, θ or K4by a series of M- or N-extensions. Meanwhile, we also present a new structural characterization of such graph G provided that G is up-embeddable.