1,301 research outputs found
On the Combinatorics of Locally Repairable Codes via Matroid Theory
This paper provides a link between matroid theory and locally repairable
codes (LRCs) that are either linear or more generally almost affine. Using this
link, new results on both LRCs and matroid theory are derived. The parameters
of LRCs are generalized to matroids, and the matroid
analogue of the generalized Singleton bound in [P. Gopalan et al., "On the
locality of codeword symbols," IEEE Trans. Inf. Theory] for linear LRCs is
given for matroids. It is shown that the given bound is not tight for certain
classes of parameters, implying a nonexistence result for the corresponding
locally repairable almost affine codes, that are coined perfect in this paper.
Constructions of classes of matroids with a large span of the parameters
and the corresponding local repair sets are given. Using
these matroid constructions, new LRCs are constructed with prescribed
parameters. The existence results on linear LRCs and the nonexistence results
on almost affine LRCs given in this paper strengthen the nonexistence and
existence results on perfect linear LRCs given in [W. Song et al., "Optimal
locally repairable codes," IEEE J. Sel. Areas Comm.].Comment: 48 pages. Submitted for publication. In this version: The text has
been edited to improve the readability. Parameter d for matroids is now
defined by the use of the rank function instead of the dual matroid. Typos
are corrected. Section III is divided into two parts, and some numberings of
theorems etc. have been change
Noncommutative geometry on trees and buildings
We describe the construction of theta summable and finitely summable spectral
triples associated to Mumford curves and some classes of higher dimensional
buildings. The finitely summable case is constructed by considering the
stabilization of the algebra of the dual graph of the special fiber of the
Mumford curve and a variant of the Antonescu-Christensen spectral geometries
for AF algebras. The information on the Schottky uniformization is encoded in
the spectral geometry through the Patterson-Sullivan measure on the limit set.
Some higher rank cases are obtained by adapting the construction for trees.Comment: 23 pages, LaTeX, 2 eps figures, contributed to a proceedings volum
Computational Geometric and Algebraic Topology
Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity.
At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field
DANIEL: Towards Automated Bug Discovery By Black Box Test Case Generation & Recommendation
Finding and documenting bugs in software systems is an essential component of the software development process. A bug is defined as a series of steps that produces behavior which differs from the software specification and requirements. Finding steps to produce such behavior requires expert knowledge of the possible operations of the software in development as well as intuition and creativity. This thesis proposes the Directed Action Node Input Execution Language (DANIEL), a language that represents test cases as directed graphs, where each node represents an action, and possible input arguments for each action are represented along the incoming directed edges. With this representation, it is possible to form a union of all recorded test cases, making a combined directed graph which represents all of the paths of interaction with the developing software. This thesis demonstrates how DANIEL can generate prioritized test cases for a web form application, while also preserving workflow context. Using a graph built on Selenium test cases, we evaluate a random walk, a weighted walk, and model-weighted walks integrating logistic regression and XGBoost to compute the relevant probabilities. We find that the weighted walk discovers the most bugs while the model-weighted walk provides the most meaningful coverage
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