16,100 research outputs found

    Lines in Euclidean Ramsey theory

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    Let m\ell_m be a sequence of mm points on a line with consecutive points of distance one. For every natural number nn, we prove the existence of a red/blue-coloring of En\mathbb{E}^n containing no red copy of 2\ell_2 and no blue copy of m\ell_m for any m2cnm \geq 2^{cn}. This is best possible up to the constant cc in the exponent. It also answers a question of Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every natural number nn, there is a set KE1K \subset \mathbb{E}^1 and a red/blue-coloring of En\mathbb{E}^n containing no red copy of 2\ell_2 and no blue copy of KK.Comment: 7 page

    Graph removal lemmas

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    The graph removal lemma states that any graph on n vertices with o(n^{v(H)}) copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.Comment: 35 page

    Understanding al-Shabaab : clan, Islam and insurgency in Kenya

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    Harakat al-Shabaab al-Mujahideen has proven itself to be a highly adaptable organisation. Their most recent evolution has seen them transform from an overt, military and governmental force in southern Somalia to a covert, insurgent and anarchic force in Kenya. This article indicates how al-Shabaab has reinvented itself in Kenya. Both ‘clan’ and ‘Islam’ are often thought of as immutable factors in al-Shabaab's make-up, but here we show that the organisation is pragmatic in its handling of clan relations and of Islamic theology. The movement is now able to exploit the social and economic exclusion of Kenyan Muslim communities in order to draw them into insurgency, recruiting Kenyans to its banner. Recent al-Shabaab attacks in Kenya, launched since June 2014, indicate how potent and dangerous their insurgency has become in the borderlands and coastal districts where Kenya's Islamic population predominates

    Learning Representations in Model-Free Hierarchical Reinforcement Learning

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    Common approaches to Reinforcement Learning (RL) are seriously challenged by large-scale applications involving huge state spaces and sparse delayed reward feedback. Hierarchical Reinforcement Learning (HRL) methods attempt to address this scalability issue by learning action selection policies at multiple levels of temporal abstraction. Abstraction can be had by identifying a relatively small set of states that are likely to be useful as subgoals, in concert with the learning of corresponding skill policies to achieve those subgoals. Many approaches to subgoal discovery in HRL depend on the analysis of a model of the environment, but the need to learn such a model introduces its own problems of scale. Once subgoals are identified, skills may be learned through intrinsic motivation, introducing an internal reward signal marking subgoal attainment. In this paper, we present a novel model-free method for subgoal discovery using incremental unsupervised learning over a small memory of the most recent experiences (trajectories) of the agent. When combined with an intrinsic motivation learning mechanism, this method learns both subgoals and skills, based on experiences in the environment. Thus, we offer an original approach to HRL that does not require the acquisition of a model of the environment, suitable for large-scale applications. We demonstrate the efficiency of our method on two RL problems with sparse delayed feedback: a variant of the rooms environment and the first screen of the ATARI 2600 Montezuma's Revenge game

    The Minimal Modal Interpretation of Quantum Theory

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    We introduce a realist, unextravagant interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes, but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory asserts that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the case of open quantum systems as well. We provide independent justification for the partial-trace operation for density matrices, reformulate wave-function collapse in terms of an underlying interpolating dynamics, derive the Born rule from deeper principles, resolve several open questions regarding ontological stability and dynamics, address a number of familiar no-go theorems, and argue that our interpretation is ultimately compatible with Lorentz invariance. Along the way, we also investigate a number of unexplored features of quantum theory, including an interesting geometrical structure---which we call subsystem space---that we believe merits further study. We include an appendix that briefly reviews the traditional Copenhagen interpretation and the measurement problem of quantum theory, as well as the instrumentalist approach and a collection of foundational theorems not otherwise discussed in the main text.Comment: 73 pages + references, 9 figures; cosmetic changes, added figure, updated references, generalized conditional probabilities with attendant changes to the sections on the EPR-Bohm thought experiment and Lorentz invariance; for a concise summary, see the companion letter at arXiv:1405.675

    Electronic structure and transport properties of atomic NiO spinvalves

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    Ab-initio quantum transport calculations show that short NiO chains suspended in Ni nanocontacts present a very strong spin-polarization of the conductance. The generalized gradient approximation we use here predicts a similiar polarization of the conductance as the one previously computed with non-local exchange, confirming the robustness of the result. Their use as nanoscopic spinvalves is proposed.Comment: 2 pages, 1 figure; accepted in JMMM (Proceedings of ICM'06, Kyoto

    Primer for the algebraic geometry of sandpiles

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    The Abelian Sandpile Model (ASM) is a game played on a graph realizing the dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of this primer is to apply the theory of lattice ideals from algebraic geometry to the Laplacian matrix, drawing out connections with the ASM. An extended summary of the ASM and of the required algebraic geometry is provided. New results include a characterization of graphs whose Laplacian lattice ideals are complete intersection ideals; a new construction of arithmetically Gorenstein ideals; a generalization to directed multigraphs of a duality theorem between elements of the sandpile group of a graph and the graph's superstable configurations (parking functions); and a characterization of the top Betti number of the minimal free resolution of the Laplacian lattice ideal as the number of elements of the sandpile group of least degree. A characterization of all the Betti numbers is conjectured.Comment: 45 pages, 14 figures. v2: corrected typo
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