scholarly journals A Combinatorial Proof of a Formula for Betti Numbers of a Stacked Polytope

10.37236/281 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Suyoung Choi ◽  
Jang Soo Kim
Keyword(s):  
Simplicial Complex ◽  
Betti Number ◽  
Betti Numbers ◽  
Combinatorial Proof ◽  
Boundary Complex ◽  
Combinatorial Interpretation ◽  
Stacked Polytope

For a simplicial complex $\Delta$, the graded Betti number $\beta_{i,j}({\bf k}[\Delta])$ of the Stanley-Reisner ring ${\bf k}[\Delta]$ over a field ${\bf k}$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $\Delta$ is the boundary complex of a $d$-dimensional stacked polytope with $n$ vertices for $d\geq3$, then $\beta_{k-1,k}({\bf k}[\Delta])=(k-1){n-d\choose k}$. We prove this combinatorially.

scholarly journals Combinatorics of Tableau Inversions

10.37236/4932 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Jonathan E. Beagley ◽  
Paul Drube
Keyword(s):  
Betti Number ◽  
Young Tableau ◽  
Betti Numbers ◽  
Young Tableaux ◽  
Standard Tableau ◽  
Specific Standard ◽  
Different Shapes ◽  
Springer Fibers ◽  
Standard Young Tableau ◽  
Fixed Standard

A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard.  An $i$-inverted Young tableau is a row-standard tableau along with precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableau that standardize a specific standard Young tableau, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different shapes. Finally, we share some the results of a computer program developed to calculate tableaux inversions.

scholarly journals APPROXIMATING THE FIRST L2-BETTI NUMBER OF RESIDUALLY FINITE GROUPS

2011 ◽  
Vol 03 (02) ◽  
pp. 153-160 ◽  
Author(s):  
W. LÜCK ◽  
D. OSIN
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Betti Number ◽  
Betti Numbers ◽  
Torsion Group ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups

We show that the first L2-betti number of a finitely generated residually finite group can be estimated from below by using ordinary first betti numbers of finite index normal subgroups. As an application, we construct a finitely generated infinite residually finite torsion group with positive first L2-betti number.

SOME RESULTS ON FUNDAMENTAL GROUPS AND BETTI NUMBERS OF FINSLER MANIFOLDS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250063 ◽  
Author(s):  
YIBING SHEN ◽  
WEI ZHAO
Keyword(s):  
Ricci Curvature ◽  
Betti Number ◽  
Betti Numbers ◽  
Finsler Manifold ◽  
Fundamental Groups ◽  
Finsler Manifolds ◽  
Nonnegative Ricci Curvature ◽  
Finiteness Theorems ◽  
The Relationship

In this paper the relationship between the Ricci curvature and the fundamental groups of Finsler manifolds are studied. We give an estimate of the first Betti number of a compact Finsler manifold. Two finiteness theorems for fundamental groups of compact Finsler manifolds are proved. Moreover, the growth of fundamental groups of Finsler manifolds with almost-nonnegative Ricci curvature are considered.

Evolution Characteristics of Existing Bridge Safety Based on Algebraic Topology and Image Processing

2014 ◽  
Vol 501-504 ◽  
pp. 1210-1213
Author(s):  
Ji Guang Han ◽  
Jian Xin Xu ◽  
Ze Min Xu
Keyword(s):  
Internal Structure ◽  
Betti Number ◽  
Algebraic Topology ◽  
Betti Numbers ◽  
Bridge Pier ◽  
Bridge Safety ◽  
Smooth Change ◽  
Evolution Characteristics ◽  
Number Curve ◽  
Safe Condition

This paper investigates the evolution characteristics of existing bridge safety based on algebraic topology and image analysis. Through the calculation of betti numbers of cross section in bridge pier binarization images, evolution curve of betti numbers time series is observed, which reflects the changes in internal structure of bridge piers, due to the variation of external environment. The analysis results show that when the evolution trend of the betti number appears a smooth change, the bridge pier is in safe condition, and when betti number curve appears sudden fluctuations, the internal structure of piers presents some changes. This study has positive significance for long-term monitoring of bridge safety.

scholarly journals Random simplicial complexes, duality and the critical dimension

2019 ◽  
pp. 1-31
Author(s):  
Michael Farber ◽  
Lewis Mead ◽  
Tahl Nowik
Keyword(s):  
Simplicial Complex ◽  
Knot Theory ◽  
Betti Numbers ◽  
Critical Dimension ◽  
Simplicial Complexes ◽  
Alexander Duality ◽  
Random Simplicial Complexes

In this paper, we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behavior of the Betti numbers in the lower model is characterized by the notion of critical dimension, which was introduced by Costa and Farber in [Large random simplicial complexes III: The critical dimension, J. Knot Theory Ramifications 26 (2017) 1740010]: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper, we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.

scholarly journals Flow polynomials of a Signed Graph

10.37236/8226 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Xiangyu Ren ◽  
Jianguo Qian
Keyword(s):  
Abelian Group ◽  
Simplicial Complex ◽  
Negative Integer ◽  
Signed Graph ◽  
Combinatorial Interpretation ◽  
Broken Circuit

 For a signed graph $G$ and non-negative integer $d$, it was shown by DeVos et al. that there exists a polynomial $F_d(G,x)$ such that the number of the nowhere-zero $\Gamma$-flows in $G$ equals $F_d(G,x)$ evaluated at $k$ for every Abelian group $\Gamma$ of order $k$ with $\epsilon(\Gamma)=d$, where $\epsilon(\Gamma)$ is the largest integer $d$ for which $\Gamma$ has a subgroup isomorphic to $\mathbb{Z}^d_2$. We define a class of  particular directed circuits in $G$, namely the fundamental directed circuits, and show that all $\Gamma$-flows (not necessarily nowhere-zero) in $G$ can be generated by these circuits. It turns out that all $\Gamma$-flows in $G$ can be evenly partitioned into $2^{\epsilon(\Gamma)}$ classes specified by the elements of order 2 in $\Gamma$, each class of which consists of the same number of flows depending only on the order of  $\Gamma$. Using an extension of  Whitney's broken circuit theorem of Dohmen and Trinks, we give a combinatorial interpretation of the coefficients in $F_d(G,x)$ for $d=0$ in terms of broken bonds. Finally,  we show that the sets of edges  in a signed graph that contain no broken bond form a  homogeneous  simplicial complex.

On the Betti numbers of edge ideals of skew Ferrers graphs

2019 ◽  
Vol 30 (01) ◽  
pp. 125-139
Author(s):  
Do Trong Hoang
Keyword(s):  
Explicit Formula ◽  
Betti Number ◽  
Betti Numbers ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Ferrers Graph ◽  
Ferrers Graphs ◽  
Extremal Betti Number

We prove that [Formula: see text] for any staircase skew Ferrers graph [Formula: see text], where [Formula: see text] and [Formula: see text]. As a consequence, Ene et al. conjecture is confirmed to hold true for the Betti numbers in the last column of the Betti table in a particular case. An explicit formula for the unique extremal Betti number of the binomial edge ideal of some closed graphs is also given.

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.

Characteristics analysis of the side-blown gas jet mixing process

2020 ◽  
pp. 095440622096282
Author(s):  
Fanhan Liu
Keyword(s):  
Betti Number ◽  
Frequency Conversion ◽  
Betti Numbers ◽  
Mixing Process ◽  
Jet Mixing ◽  
Mixing Performance ◽  
Through Flow ◽  
Characteristics Analysis ◽  
Processing Techniques ◽  
Individual Particles

The mixing performances of side-blown gas jets were studied by characterizing their manifold Betti numbers, namely 0-dimensional and 1-dimensional Betti numbers, through flow visualization and image processing techniques. The 0-dimensional Betti number reflects the mixing performance, and the 1-dimensional Betti number reflects the mixing unevenness degree. The mixing performances and flow fields were studied for different modified Froude numbers. As modified Froude numbers increase, the main flow was enhanced, promoting the macro-mixing of individual particles. The mixing performance of the stirring process was improved by a frequency conversion method, which destroyed the original flow pattern and promoted the micro-mixing process by chaotization.

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