57,691 research outputs found
On String Graph Limits and the Structure of a Typical String Graph
We study limits of convergent sequences of string graphs, that is, graphs
with an intersection representation consisting of curves in the plane. We use
these results to study the limiting behavior of a sequence of random string
graphs. We also prove similar results for several related graph classes.Comment: 18 page
Multi-loop open string amplitudes and their field theory limit
JHEP is an open-access journal funded by SCOAP3 and licensed under CC BY 4.0This work
was supported by STFC (Grant ST/J000469/1, ‘String theory, gauge theory & duality’)
and by MIUR (Italy) under contracts 2006020509 004 and 2010YJ2NYW 00
On the Typical Structure of Graphs in a Monotone Property
Given a graph property , it is interesting to determine the
typical structure of graphs that satisfy . In this paper, we
consider monotone properties, that is, properties that are closed under taking
subgraphs. Using results from the theory of graph limits, we show that if
is a monotone property and is the largest integer for which
every -colorable graph satisfies , then almost every graph with
is close to being a balanced -partite graph.Comment: 5 page
Comments on worldsheet theories dual to free large N gauge theories
We continue to investigate properties of the worldsheet conformal field
theories (CFTs) which are conjectured to be dual to free large N gauge
theories, using the mapping of Feynman diagrams to the worldsheet suggested in
hep-th/0504229. The modular invariance of these CFTs is shown to be built into
the formalism. We show that correlation functions in these CFTs which are
localized on subspaces of the moduli space may be interpreted as delta-function
distributions, and that this can be consistent with a local worldsheet
description given some constraints on the operator product expansion
coefficients. We illustrate these features by a detailed analysis of a specific
four-point function diagram. To reliably compute this correlator we use a novel
perturbation scheme which involves an expansion in the large dimension of some
operators.Comment: 43 pages, 16 figures, JHEP format. v2: added reference
GPU-Accelerated BWT Construction for Large Collection of Short Reads
Advances in DNA sequencing technology have stimulated the development of
algorithms and tools for processing very large collections of short strings
(reads). Short-read alignment and assembly are among the most well-studied
problems. Many state-of-the-art aligners, at their core, have used the
Burrows-Wheeler transform (BWT) as a main-memory index of a reference genome
(typical example, NCBI human genome). Recently, BWT has also found its use in
string-graph assembly, for indexing the reads (i.e., raw data from DNA
sequencers). In a typical data set, the volume of reads is tens of times of the
sequenced genome and can be up to 100 Gigabases. Note that a reference genome
is relatively stable and computing the index is not a frequent task. For reads,
the index has to computed from scratch for each given input. The ability of
efficient BWT construction becomes a much bigger concern than before. In this
paper, we present a practical method called CX1 for constructing the BWT of
very large string collections. CX1 is the first tool that can take advantage of
the parallelism given by a graphics processing unit (GPU, a relative cheap
device providing a thousand or more primitive cores), as well as simultaneously
the parallelism from a multi-core CPU and more interestingly, from a cluster of
GPU-enabled nodes. Using CX1, the BWT of a short-read collection of up to 100
Gigabases can be constructed in less than 2 hours using a machine equipped with
a quad-core CPU and a GPU, or in about 43 minutes using a cluster with 4 such
machines (the speedup is almost linear after excluding the first 16 minutes for
loading the reads from the hard disk). The previously fastest tool BRC is
measured to take 12 hours to process 100 Gigabases on one machine; it is
non-trivial how BRC can be parallelized to take advantage a cluster of
machines, let alone GPUs.Comment: 11 page
Graph properties, graph limits and entropy
We study the relation between the growth rate of a graph property and the
entropy of the graph limits that arise from graphs with that property. In
particular, for hereditary classes we obtain a new description of the colouring
number, which by well-known results describes the rate of growth.
We study also random graphs and their entropies. We show, for example, that
if a hereditary property has a unique limiting graphon with maximal entropy,
then a random graph with this property, selected uniformly at random from all
such graphs with a given order, converges to this maximizing graphon as the
order tends to infinity.Comment: 24 page
The phantom menace in representation theory
Our principal goal in this overview is to explain and motivate the concept of
a phantom in the representation theory of a finite dimensional algebra
. In particular, we exhibit the key role of phantoms towards
understanding how a full subcategory of the category
of all finitely generated left -modules is
embedded into , in terms of maps leaving or entering .
Contents: 1. Introduction and prerequisites; 2. Contravariant finiteness and
first examples; 3. Homological importance of contravariant finiteness and a
model application of the theory; 4. Phantoms. Definitions, existence, and basic
properties; 5. An application: Phantoms over string algebras
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