Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userWord posets, with applications to Coxeter groups
149
0
0.0
(
0
)
Added by
EPTCS
Publication date
2011
fields
Informatics Engineering
and research's language is
English
Authors
Matthew J. Samuel
Ask ChatGPT about the research
No Arabic abstract
We discuss the theory of certain partially ordered sets that capture the structure of commutation classes of words in monoids. As a first application, it follows readily that counting words in commutation classes is #P-complete. We then apply the partially ordered sets to Coxeter groups. Some results are a proof that enumerating the reduced words of elements of Coxeter groups is #P-complete, a recursive formula for computing the number of commutation classes of reduced words, as well as stronger bounds on the maximum number of commutation classes than were previously known. This also allows us to improve the known bounds on the number of primitive sorting networks.
rate research
Read More
The canonical tree-decomposition theorem, given by Robertson and Seymour in their seminal graph minors series, turns out to be one of the most important tool in structural and algorithmic graph theory. In this paper, we provide the canonical tree decomposition theorem for digraphs. More precisely, we construct directed tree-decompositions of digraphs that distinguish all their tangles of order $k$, for any fixed integer $k$, in polynomial time. As an application of this canonical tree-decomposition theorem, we provide the following result for the directed disjoint paths problem: For every fixed $k$ there is a polynomial-time algorithm which, on input $G$, and source and terminal vertices $(s_1, t_1), dots, (s_k, t_k)$, either 1. determines that there is no set of pairwise vertex-disjoint paths connecting each source $s_i$ to its terminal $t_i$, or 2.finds a half-integral solution, i.e., outputs paths $P_1, dots, P_k$ such that $P_i$ links $s_i$ to $t_i$, so that every vertex of the graph is contained in at most two paths. Given known hardness results for the directed disjoint paths problem, our result cannot be improved for general digraphs, neither to fixed-parameter tractability nor to fully vertex-disjoint directed paths. As far as we are aware, this is the first time to obtain a tractable result for the $k$-disjoint paths problem for general digraphs. We expect more applications of our canonical tree-decomposition for directed results.
Handelman (J. Operator Theory, 1981) proved that if the spectral radius of a matrix $A$ is a simple root of the characteristic polynomial and is strictly greater than the modulus of any other root, then $A$ is conjugate to a matrix $Z$ some power of which is positive. In this article, we provide an explicit conjugate matrix $Z$, and prove that the spectral radius of $A$ is a simple and dominant eigenvalue of $A$ if and only if $Z$ is eventually positive. For $ntimes n$ real matrices with each row-sum equal to $1$, this criterion can be declined into checking that each entry of some power is strictly larger than the average of the entries of the same column minus $frac{1}{n}$. We apply the criterion to elements of irreducible infinite nonaffine Coxeter groups to provide evidences for the dominance of the spectral radius, which is still unknown.
It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper we prove that the set of all solutions of a twisted word equation is an EDT0L language whose specification can be computed in $mathsf{PSPACE}$. Within the same complexity bound we can decide whether the solution set is empty, finite, or infinite. In the second part of the paper we apply the results for twisted equations to obtain in $mathsf{PSPACE}$ an EDT0L description of the solution set of equations with rational constraints for finitely generated virtually free groups in standard normal forms with respect to a natural set of generators. If the rational constraints are given by a homomorphism into a fixed (or small enough) finite monoid, then our algorithms can be implemented in $mathsf{NSPACE}(n^2log n)$, that is, in quasi-quadratic nondeterministic space. Our results generalize the work by Lohrey and Senizergues (ICALP 2006) and Dahmani and Guirardel (J. of Topology 2010) with respect to both complexity and expressive power. Neither paper gave any concrete complexity bound and the results in these papers are stated for subsets of solutions only, whereas our results concern all solutions.
This paper deals with the theory and application of 2-Dimensional, nine-neighborhood, null- boundary, uniform as well as hybrid Cellular Automata (2D CA) linear rules in image processing. These rules are classified into nine groups depending upon the number of neighboring cells influences the cell under consideration. All the Uniform rules have been found to be rendering multiple copies of a given image depending on the groups to which they belong where as Hybrid rules are also shown to be characterizing the phenomena of zooming in, zooming out, thickening and thinning of a given image. Further, using hybrid CA rules a new searching algorithm is developed called Sweepers algorithm which is found to be applicable to simulate many inter disciplinary research areas like migration of organisms towards a single point destination, Single Attractor and Multiple Attractor Cellular Automata Theory, Pattern Classification and Clustering Problem, Image compression, Encryption and Decryption problems, Density Classification problem etc.
We use probabilistic methods to prove that many Coxeter groups are incoherent. In particular, this holds for Coxeter groups of uniform exponent > 2 with sufficiently many generators.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
