147 research outputs found

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    On the expected efficiency of branch and bound for the asymmetric TSP

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    Let the costs C(i,j)C(i,j) for an instance of the asymmetric traveling salesperson problem be independent uniform [0,1][0,1] random variables. We consider the efficiency of branch and bound algorithms that use the assignment relaxation as a lower bound. We show that w.h.p. the number of steps taken in any such branch and bound algorithm is eΩ(na)e^{\Omega(n^a)} for some small absolute constant a>0a>0

    Persistence of Rademacher-type and Sobolev-to-Lipschitz properties

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    We consider the Rademacher- and Sobolev-to-Lipschitz-type properties for arbitrary quasi-regular strongly local Dirichlet spaces. We discuss the persistence of these properties under localization, globalization, transfer to weighted spaces, tensorization, and direct integration. As byproducts we obtain: necessary and sufficient conditions to identify a quasi-regular strongly local Dirichlet form on an extended metric topological σ\sigma-finite possibly non-Radon measure space with the Cheeger energy of the space; the tensorization of intrinsic distances; the tensorization of the Varadhan short-time asymptotics.Comment: 40 pages, 2 figure

    Sobolev spaces in extended metric-measure spaces

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    Nonlinear Cone Separation Theorems in Real Topological Linear Spaces

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    The separation of two sets (or more specific of two cones) plays an important role in different fields of mathematics such as variational analysis, convex analysis, convex geometry, optimization. In the paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone / conical surface in real (topological) linear spaces. We follow basically the separation approach by Kasimbeyli (2010, SIAM J. Optim. 20) based on augmented dual cones and normlinear separation functions. Classical separation theorems for convex sets will be the key tool for proving our main nonlinear cone separation theorems. Also in the setting of a real reflexive Banach space, we are able to extend the cone separation result derived by Kasimbeyli

    Injectivity of Lipschitz operators

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    Any Lipschitz map f ⁣:MNf\colon M \to N between metric spaces can be "linearised" in such a way that it becomes a bounded linear operator f^ ⁣:F(M)F(N)\widehat{f}\colon \mathcal F(M) \to \mathcal F(N) between the Lipschitz-free spaces over MM and NN. The purpose of this note is to explore the connections between the injectivity of ff and the injectivity of f^\widehat{f}. While it is obvious that if f^\widehat{f} is injective then so is ff, the converse is less clear. Indeed, we pin down some cases where this implication does not hold but we also prove that, for some classes of metric spaces MM, any injective Lipschitz map f ⁣:MNf\colon M \to N (for any NN) admits an injective linearisation. Along our way, we study how Lipschitz maps carry the support of elements in free spaces and also we provide stronger conditions on ff which ensure that f^\widehat{f} is injective

    Three Risky Decades: A Time for Econophysics?

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    Our Special Issue we publish at a turning point, which we have not dealt with since World War II. The interconnected long-term global shocks such as the coronavirus pandemic, the war in Ukraine, and catastrophic climate change have imposed significant humanitary, socio-economic, political, and environmental restrictions on the globalization process and all aspects of economic and social life including the existence of individual people. The planet is trapped—the current situation seems to be the prelude to an apocalypse whose long-term effects we will have for decades. Therefore, it urgently requires a concept of the planet's survival to be built—only on this basis can the conditions for its development be created. The Special Issue gives evidence of the state of econophysics before the current situation. Therefore, it can provide excellent econophysics or an inter-and cross-disciplinary starting point of a rational approach to a new era
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