57,691 research outputs found

    On String Graph Limits and the Structure of a Typical String Graph

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    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

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    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

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    Given a graph property P\mathcal{P}, it is interesting to determine the typical structure of graphs that satisfy P\mathcal{P}. 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 P\mathcal{P} is a monotone property and rr is the largest integer for which every rr-colorable graph satisfies P\mathcal{P}, then almost every graph with P\mathcal{P} is close to being a balanced rr-partite graph.Comment: 5 page

    Comments on worldsheet theories dual to free large N gauge theories

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    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

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    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

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    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

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    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 Λ\Lambda. In particular, we exhibit the key role of phantoms towards understanding how a full subcategory A\cal A of the category Λ-mod\Lambda\text{-mod} of all finitely generated left Λ\Lambda-modules is embedded into Λ-mod\Lambda\text{-mod}, in terms of maps leaving or entering A\cal A. 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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