DNA Screening, Pooling Design and Simplicial Complex

2003 ◽  
Vol 7 (4) ◽  
pp. 389-394 ◽  
Author(s):  
Haesun Park ◽  
Weili Wu ◽  
Zhen Liu ◽  
Xiaoyu Wu ◽  
Hong G. Zhao
Author(s):  
Benson Farb ◽  
Dan Margalit

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.


10.37236/1245 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Art M. Duval

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.


2013 ◽  
Vol 20 (01) ◽  
pp. 123-140
Author(s):  
Teng Zou ◽  
Bin Zhu

For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of d-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) d-cluster tilted algebras. Moreover, we prove that the generalized d-cluster categories of hereditary algebras of finite representation type provide a category model for the n-repetitive generalized cluster complexes.


2002 ◽  
Vol 109 (2) ◽  
pp. 159-174 ◽  
Author(s):  
Sara Faridi

2016 ◽  
Vol 299 ◽  
pp. 381-395 ◽  
Author(s):  
Art M. Duval ◽  
Bennet Goeckner ◽  
Caroline J. Klivans ◽  
Jeremy L. Martin
Keyword(s):  

2022 ◽  
Vol 417 ◽  
pp. 126773
Author(s):  
Yanyi Nie ◽  
Wenyao Li ◽  
Liming Pan ◽  
Tao Lin ◽  
Wei Wang

10.37236/4894 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mina Bigdeli ◽  
Jürgen Herzog ◽  
Takayuki Hibi ◽  
Antonio Macchia

Let $I\subset K[x_1,\ldots,x_n]$ be  a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that  $\Delta(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $\Delta(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$.


10.37236/1900 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Jakob Jonsson

We consider topological aspects of decision trees on simplicial complexes, concentrating on how to use decision trees as a tool in topological combinatorics. By Robin Forman's discrete Morse theory, the number of evasive faces of a given dimension $i$ with respect to a decision tree on a simplicial complex is greater than or equal to the $i$th reduced Betti number (over any field) of the complex. Under certain favorable circumstances, a simplicial complex admits an "optimal" decision tree such that equality holds for each $i$; we may hence read off the homology directly from the tree. We provide a recursive definition of the class of semi-nonevasive simplicial complexes with this property. A certain generalization turns out to yield the class of semi-collapsible simplicial complexes that admit an optimal discrete Morse function in the analogous sense. In addition, we develop some elementary theory about semi-nonevasive and semi-collapsible complexes. Finally, we provide explicit optimal decision trees for several well-known simplicial complexes.


2020 ◽  
Author(s):  
Kennedy Courtney

Let Γ be a directed graph. The directed forest complex, DF(Γ), is a simplicial complex whose vertices are the edges of Γ and whose simplices are sets of edges that form a directed forest in Γ. We study the directed forest complex of Cayley graphs of finite groups. The homology of DF(Γ) contains information about the graph, Γ and about the group, G. The ultimate goal is to classify DF(Γ) up to homotopy, compute its homology, and interpret the findings in terms of properties of DF(Γ). In this thesis, we present progress made toward this goal.


Author(s):  
Napoleón Vargas Jurado ◽  
Larry A Kuehn ◽  
John W Keele ◽  
Ronald M Lewis

Abstract Despite decreasing genotyping costs, in some cases individually genotyping animals is not economically feasible (e.g., in small ruminants). An alternative is to pool DNA, using the pooled allele frequency (PAF) to garner information on performance. Still, the use of PAF for prediction (estimation of genomic breeding values; GEBV) has been limited. Two potential sources of error on accuracy of GEBV of sires, obtained from PAF of their progeny themselves lacking pedigree information, were tested: i) pool construction error (unequal contribution of DNA from animals in pools), and ii) technical error (variability when reading the array). Pooling design (random, extremes, K-means), pool size (5, 10, 25, 50 and 100 individuals), and selection scenario (random, phenotypic) also were considered. These factors were tested by simulating a sheep population. Accuracy of GEBV—the correlation between true and estimated values—was not substantially affected by pool construction or technical error, or selection scenario. A significant interaction, however, between pool size and design was found. Still, regardless of design, mean accuracy was higher for pools of 10 or less individuals. Mean accuracy of GEBV was 0.174 (SE 0.001) for random pooling, and 0.704 (SE 0.004) and 0.696 (SE 0.004) for extreme and K-means pooling, respectively. Non-random pooling resulted in moderate accuracy of GEBV. Overall, pooled genotypes can be used in conjunction with individual genotypes of sires for moderately accurate predictions of their genetic merit with little effect of pool construction or technical error.


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