This paper presents a region merging process controlled by topological features on regions in 3D images. Betti numbers, a well-known topological invariant, are used as criteria. Classical and incremental algorithms to compute Betti numbers using information represented by the topological map of an image are provided. The region merging algorithm, which merges of any number of connected componen...

There are notions of L2-Betti numbers for discrete groups (CheegerGromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück’s dimension theory, Gaboriau’s definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition o...

Let R = k[x, y] denote the polynomial ring in two variables over an infinite field k. We study the Betti strata of the family G(H) parametrizing graded Artinian quotients of R = k[x, y] having given Hilbert function H . The Betti stratum Gβ(H) parametrizes all quotients A of having the graded Betti numbers determined by H and the minimal relation degrees β, with βi = dimk Tor R 1 (I, k)i. We re...

We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of...

We derive type II supergravity solutions corresponding to space-filling regular and fractional Dp branes on (9− p)-dimensional conical transverse spaces. Fractional Dp-branes are wrapped D(p + 2)-branes; therefore, our solutions exist only if the base of the cone has a non-vanishing Betti number b2. We also consider 11-dimensional SUGRA solutions corresponding to regular and fractional M2 brane...

In this paper we study complete linear series on a hyperelliptic curve C of arithmetic genus g. Let A be the unique line bundle on C such that |A| is a g 2 , and let L be a line bundle on C of degree d. Then L can be factorized as L = A ⊗ B where m is the largest integer satisfying H(C,L⊗A) 6= 0. Let b = deg(B). We say that the factorization type of L is (m, b). Our main results in this paper a...

We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds M with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of M , for there to exist a family of Zn covering spaces, Mn, in which β1(Mn) increases linearly with n....

We give results on the L2-Betti numbers and Novikov-Shubin invariants of compact manifolds, especially 3-manifolds. We first study the Betti numbers and Novikov-Shubin invariants of a chain complex of Hilbert modules over a finite von Neumann algebra. We establish inequalities among the Novikov-Shubin invariants of the terms in a short exact sequence of chain complexes. Our algebraic results, a...

We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire’s splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bi...

We use Klee’s Dehn-Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f -vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k an...