Related papers: Betti numbers of hypergraphs
To a simple graph we associate a so-called graph series, which can be viewed as the Hilbert--Poincar\'e series of a certain infinite jet scheme. We study new $q$-representations and examine modular properties of several examples including…
The generating function of the Betti numbers of the Heisenberg Lie algebra over a field of characteristic 2 is calculated using discrete Morse theory.
We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the…
Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the…
We construct a CW decomposition $C_n$ of the $n$-dimensional half cube in a manner compatible with its structure as a polytope. For each $3 \leq k \leq n$, the complex $C_n$ has a subcomplex $C_{n, k}$, which coincides with the clique…
We describe an inner product on the diagrams on which the Temperley-Lieb algebra can be represented. We exhibit several constructions which are in natural combinatorial bijection with these diagrams, which are generalizations of various…
Diagram semigroups are interesting algebraic and combinatorial objects, several types of them originating from questions in computer science and in physics. Here we describe diagram semigroups in a general framework and extend our…
We study the computational model of polygraphs. For that, we consider polygraphic programs, a subclass of these objects, as a formal description of first-order functional programs. We explain their semantics and prove that they form a…
Let $\R$ be a real closed field, $ {\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], $ with $ \deg_{Y}(Q) \leq 2, \deg_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m$, and $ {\mathcal P} \subset \R[X_1,...,X_k] $ with $\deg_{X}(P)…
The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a Fibonacci graph which gives a generic example of non-series-parallel graphs. Our intention in this paper is to simplify the…
We study the maximal values of Betti numbers of tropical subvarieties of a given dimension and degree in $\mathbb{TP}^n$. We provide a lower estimate for the maximal value of the top Betti number, which naturally depends on the dimension…
We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and…
Polygon spaces like $M_\ell=\{(u_1,...,u_n)\in S^1\times... S^1 ;\ \sum_{i=1}^n l_iu_i=0\}/SO(2)$ or they three dimensional analogues $N_\ell$ play an important r\^ole in geometry and topology, and are also of interest in robotics where the…
For a hypersurface defined by a complex analytic function, we obtain a chain complex of free abelian groups, with ranks given in terms of relative polar multiplicities, which has cohomology isomorphic to the reduced cohomology of the real…
We use Boij-S\"oderberg theory to provide some order of magnitude bounds on algebraic Betti numbers.
We give a sufficient condition for a monomial ideal to have a nonzero Betti number in each multidegree. In the case of facet ideals of simplicial forests, this condition becomes a necessary one and it allows us to characterize Betti…
In this paper, we introduce a new concept namely degree polynomial for vertices of a simple graph. This notion leads to a concept namely degree polynomial sequence which is stronger than the concept of degree sequence. After obtaining the…
We present the relation between the genus in cosmology and the Betti numbers for excursion sets of three- and two-dimensional smooth Gaussian random fields, and numerically investigate the Betti numbers as a function of threshold level.…
We investigate the set of partial partitions of a finite set, ordered by inclusion. With this ordering the set of partial partitions can be studied as an abstract simplicial complex. We use the theory of shellable nonpure complexes to find…
In this paper we study underlying graphs corresponding to a set of halving lines. We establish many properties of such graphs. In addition, we tighten the upper bound for the number of halving lines.