球面中具有平行平均曲率向量的2—调和等距浸入

本文讨论了单位球面中具平行平均曲率向量的２－调和等距浸入，获得其成为极小浸入的充分条件和关于第二基本形式的Ｐｉｎｃｈｉｎｇ定理．     

拟遗传代数的诱导模与广义Betti数

本文引入了模的广义Betti数,给出了经典Betti数与广义Betti数之间的关系,证明了具有纯粹强正合Borel子代数的零关系拟遗传代数的诱导模的极小投射分解可通过其正合Borel子代数的相应模的极小投射分解的诱导给出,从而两者具有相同的广义Betti数.     

单位球面中极小子流形的曲率拼挤

本文证明了单位球面中极小子流形的一些拼挤定理,特别注意到单位球面中的极小超曲面、给出了截曲率的拼挤常数,我们也改进了由Ｎ．Ｅｊｉｒｉ得到的Ｒｉｃｃｉ曲率拼挤常数。     

图的直径与Betti亏数

对于任意的正数Ｍ以及正整数ｄ≥４,存在直径为ｄ的ｉ－边连通无环图Ｇ使得ζ（Ｇ）≥Ｍ,其中ζ（Ｇ）是Ｇ的Ｂｅｔｔｉ亏数,ｉ＝１,２,３。     

一个Chern—Kuiper型的不可浸入性定理

本文建立了一个新的Chern-Kuiper型的不可浸入性定理.假设M是一个n-维紧致的黎曼流形,M的Ricci和数量曲率R满足Ric+R≥0andR＞n(n-1)λ-2,则M不能等距地浸入在欧氏空间Rn+1的半径为λ的闭球中.从而推广了Deshmukh和Al-Gwaiz的结果.     

图的生成树, 基本圈与Betti亏数

G为图且T是G的一棵生成树. 记号ξ(G, T)表示G＼E(T)中边数为奇数的连通分支个数. 文献［2］称ξ(G)=min[DD(X]T[DD)]ξ(G, T)为图G的Betti亏数, 这里min取遍G的所有生成树T. 由文献［2］知, 确定一个图G的最大亏格主要确定这个图的Betii亏数ξ(G).该文研究与Betti亏数有关的图的特征结构, 得到了关于图的最大亏格的若干结果.     

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), || · ||∞) among closed Riemannian manifold (M, g) with K(M) ≥-1 and Diam (M) ≤ D.     

Betti Numbers of Hypergraphs

In this article, we study some algebraic properties of hypergraphs, in particular their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.     

Shifting Operations and Graded Betti Numbers

The behaviour of graded Betti numbers under exterior and symmetric algebraic shifting is studied. It is shown that the extremal Betti numbers are stable under these operations. Moreover, the possible sequences of super extremal Betti numbers for a graded ideal with given Hilbert function are characterized. Finally it is shown that over a field of characteristic 0, the graded Betti numbers of a squarefree monomial ideal are bounded by those of the corresponding squarefree lexsegment ideal.     

Estimates for the Betti Numbers of Rationally Elliptic Spaces

We give some upper bounds for the Betti numbers of rationally elliptic spaces.     

Betti Numbers of Path Ideals of Trees

Let R = k[x₁,…, x_n], where k is a field. The path ideal (of length t) of a directed graph G is the monomial ideal, denoted by I_t(G), whose generators correspond to the directed paths of length t in G. We determine all the graded Betti numbers of the path ideal of a directed rooted tree with respect to some graphical terms.     

Betti Numbers of Toric Varieties and Eulerian Polynomials

It is well known that the Eulerian polynomials, which count permutations in S _n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the S _n -orbit for a generic weight in the weight lattice of S _n . Therefore, the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra.

In this article we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a nongeneric weight, namely, a weight fixed by only the simple reflections J = {s _n , s _n?1, s _n?2,…, s _n?k+2, s _n?k+1} for some k with respect to the A _n root lattice. Furthermore, they give rise to certain rationally smooth toric varieties X(J) that come naturally from the theory of algebraic monoids. Using effectively the theory of reductive algebraic monoids and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincaré polynomial of X(J) in terms of the Eulerian polynomials.     

On the Betti Numbers of Chessboard Complexes

In this paper we study the Betti numbers of a type of simplicial complex known as a chessboard complex. We obtain a formula for their Betti numbers as a sum of terms involving partitions. This formula allows us to determine which is the first nonvanishing Betti number (aside from the 0-th Betti number). We can therefore settle certain cases of a conjecture of Björner, Lovász, Vreica, and ivaljevi in [2]. Our formula also shows that all eigenvalues of the Laplacians of the simplicial complexes are integers, and it gives a formula (involving partitions) for the multiplicities of the eigenvalues.     

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-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs.     

Betti Numbers of Locally Standard 2-Torus Manifolds

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