scholarly journals Average Betti Numbers of Induced Subcomplexes in Triangulations of Manifolds

10.37236/8564 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Giulia Codenotti ◽  
Jonathan Spreer ◽  
Francisco Santos
Keyword(s):  
Simplicial Complex ◽  
Lower Bounds ◽  
Commutative Algebra ◽  
Betti Numbers ◽  
Upper Bounds ◽  
Weighted Sums ◽  
Weighted Averages ◽  
Connected Sums ◽  
Graded Betti Numbers ◽  
Strongly Connected

We study a variation of Bagchi and Datta's $\sigma$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of $C$. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sphere maximizes these invariants among triangulated spheres with a given $f$-vector. For the first entry of $\sigma$, we extend this bound to the class of strongly connected pure complexes. As an application, we show how upper bounds on $\sigma$ can be used to obtain lower bounds on the $f$-vector of triangulated $4$-manifolds with transitive symmetry on vertices and prescribed vector of Betti numbers.

scholarly journals Componentwise linear ideals

1999 ◽  
Vol 153 ◽  
pp. 141-153 ◽  
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Homogeneous Polynomials ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Alexander Dual ◽  
Componentwise Linear Ideals ◽  
The Ideal

AbstractA componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal IΔ arising from a simplicial complex Δ is componentwise linear if and only if the Alexander dual of Δ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

scholarly journals Graded Betti Numbers of Balanced Simplicial Complexes

2021 ◽  
Author(s):  
Martina Juhnke-Kubitzke ◽  
Lorenzo Venturello
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Simplicial Complexes ◽  
Upper Bounds ◽  
Combinatorial Type ◽  
Homogeneous Ideal ◽  
Graded Rings ◽  
Number Of Generators ◽  
Graded Betti Numbers ◽  
Linear Syzygies

AbstractWe prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings S/I, where S is a polynomial ring and $I\subseteq S$ I ⊆ S is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.

scholarly journals Homological invariants of the Stanley–Reisner ring of a k-decomposable simplicial complex

2017 ◽  
Vol 10 (03) ◽  
pp. 1750061
Author(s):  
Somayeh Moradi
Keyword(s):  
Simplicial Complex ◽  
Upper Bound ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Projective Dimension ◽  
Simplicial Complexes ◽  
Monomial Ideals ◽  
Edge Ideals ◽  
Graded Betti Numbers ◽  
Shellable Simplicial Complex

In this paper, we study the regularity and the projective dimension of the Stanley–Reisner ring of a [Formula: see text]-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of decomposable monomial ideals which is the dual concept for [Formula: see text]-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a shellable simplicial complex [Formula: see text], a formula for the regularity of the Stanley–Reisner ring of [Formula: see text] is presented. Finally, for a chordal clutter [Formula: see text], an upper bound for [Formula: see text] is given in terms of the regularities of edge ideals of some chordal clutters which are minors of [Formula: see text].

scholarly journals On Vertex Decomposable Simplicial Complexes and Their Alexander Duals

2016 ◽  
Vol 118 (1) ◽  
pp. 43 ◽  
Author(s):  
Somayeh Moradi ◽  
Fahimeh Khosh-Ahang
Keyword(s):  
Simplicial Complex ◽  
Vertex Cover ◽  
Betti Numbers ◽  
Recursive Formula ◽  
Projective Dimension ◽  
Graded Betti Numbers ◽  
Alexander Dual ◽  
Special Cases ◽  
Vertex Decomposable Simplicial Complex ◽  
Vertex Decomposable

In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex $\Delta$ is vertex decomposable if and only if $I_{\Delta^{\vee}}$ is a vertex splittable ideal. Moreover, the properties of vertex splittable ideals are studied. As the main result, it is proved that any vertex splittable ideal has a Betti splitting and the graded Betti numbers of such ideals are explained with a recursive formula. As a corollary, recursive formulas for the regularity and projective dimension of $R/I_{\Delta}$, when $\Delta$ is a vertex decomposable simplicial complex, are given. Moreover, for a vertex decomposable graph $G$, a recursive formula for the graded Betti numbers of its vertex cover ideal is presented. In special cases, this formula is explained, when $G$ is chordal or a sequentially Cohen-Macaulay bipartite graph. Finally, among the other things, it is shown that an edge ideal of a graph is vertex splittable if and only if it has linear resolution.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

scholarly journals Sharp upper bounds for the Betti numbers of a given Hilbert polynomial

2013 ◽  
Vol 7 (5) ◽  
pp. 1019-1064 ◽  
Author(s):  
Giulio Caviglia ◽  
Satoshi Murai
Keyword(s):  
Betti Numbers ◽  
Upper Bounds ◽  
Hilbert Polynomial

scholarly journals Bounding the Size and Probability of Epidemics on Networks

2008 ◽  
Vol 45 (2) ◽  
pp. 498-512 ◽  
Author(s):  
Joel C. Miller
Keyword(s):  
Infectious Disease ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Marginal Probability ◽  
Transmission Properties ◽  
Disease Spreading ◽  
Special Case

We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find the conditions which give the upper (or lower) bounds on the size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on the size and probability. The distributions leading to these bounds are network independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general, the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.

Weakly Polymatroidal Ideals

2006 ◽  
Vol 13 (04) ◽  
pp. 711-720 ◽  
Author(s):  
Masako Kokubo ◽  
Takayuki Hibi
Keyword(s):  
Betti Numbers ◽  
Partially Ordered Sets ◽  
Finite Sequence ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Fundamental Fact ◽  
Polymatroidal Ideals ◽  
Polymatroidal Ideal

The concept of the weakly polymatroidal ideal, which generalizes both the polymatroidal ideal and the prestable ideal, is introduced. A fundamental fact is that every weakly polymatroidal ideal has a linear resolution. One of the typical examples of weakly polymatroidal ideals arises from finite partially ordered sets. We associate each weakly polymatroidal ideal with a finite sequence, alled the polymatroidal sequence, which will be useful for the computation of graded Betti numbers of weakly polymatroidal ideals as well as for the construction of weakly polymatroidal ideals.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

scholarly journals The regularity of Tor and graded Betti Numbers

2006 ◽  
Vol 128 (3) ◽  
pp. 573-605 ◽  
Author(s):  
David Eisenbud ◽  
C. (Craig) Huneke ◽  
Bernd Ulrich
Keyword(s):  
Betti Numbers ◽  
Graded Betti Numbers
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