147 research outputs found
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
On the expected efficiency of branch and bound for the asymmetric TSP
Let the costs for an instance of the asymmetric traveling
salesperson problem be independent uniform 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 for some small
absolute constant
Persistence of Rademacher-type and Sobolev-to-Lipschitz properties
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 -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
Nonlinear Cone Separation Theorems in Real Topological Linear Spaces
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
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Homogenization of Random Media: Random Walks, Diffusions and Stochastic Interface Models
This thesis concerns homogenization results, in particular scaling limits and heat kernel estimates, for random processes moving in random environments and for stochastic interface models. The first chapter will survey recent research and introduce three models of interest: the random conductance model, the Ginzburg-Landau ∇φ model, and the symmetric diffusion process in a random medium.
In the second chapter we present some novel research on the random conductance model; a random walk on an infinite lattice, usually taken to be Ζ^d with nearest neighbour edges, whose law is determined by random weights on the edges. In the setting of degenerate, ergodic weights and general speed measure, we present a quenched local limit theorem for this model. This states that for almost every instance of the random
environment, the heat kernel, once suitably rescaled, converges to that of Brownian motion with a deterministic, non-degenerate covariance matrix. The quenched local limit theorem is proven under ergodicity and moment conditions on the environment. Under stronger, non-optimal moment conditions, we also prove annealed local limit theorems for the static RCM with general speed measure and for the dynamic RCM. The dynamic
model allows for the random weights, or conductances, to vary with time.
Our focus turns to the Ginzburg-Landau gradient model in the subsequent chapter.
This is a model for a stochastic interface separating two distinct thermodynamic phases, using an infinite system of coupled stochastic differential equations (SDE). Our main assumption is that the potential in the SDE system is strictly convex with second derivative uniformly bounded below. The aforementioned annealed local limit theorem for the dynamic RCM is applied via a coupling relation to prove a scaling limit result for the space-time covariances in the Ginzburg-Landau model. We also show that the associated Gibbs distribution scales to a Gaussian free field.
In the final chapter, we study a symmetric diffusion process in divergence form in a stationary and ergodic random environment. This is a continuum analogue of the random conductance model and similar analytical techniques are applicable here. The coefficients are assumed to be degenerate and unbounded but satisfy a moment condition. We derive upper off-diagonal estimates on the heat kernel of this process for general speed measure. Lower off-diagonal estimates are also proven for a natural choice of speed measure under an additional decorrelation assumption on the environment. Finally, using these estimates, a scaling limit for the Green’s function is derived
Injectivity of Lipschitz operators
Any Lipschitz map between metric spaces can be "linearised"
in such a way that it becomes a bounded linear operator between the Lipschitz-free spaces over and
. The purpose of this note is to explore the connections between the
injectivity of and the injectivity of . While it is obvious
that if is injective then so is , 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 , any injective Lipschitz map
(for any ) 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 which ensure that
is injective
Three Risky Decades: A Time for Econophysics?
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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