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Register a new userAsymptotically Affine and Asymptotically Conformal Circle Endomorphisms
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We show that every uniformly asymptotically affine circle endomorphism has a uniformly asymptotically conformal extension.
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We prove microlocal estimates at the trapped set of asymptotically Kerr spacetimes: these are spacetimes whose metrics decay inverse polynomially in time to a stationary subextremal Kerr metric. This combines two independent results. The first one is purely dynamical: we show that the stable and unstable manifolds of a decaying perturbation of a time-translation-invariant dynamical system with normally hyperbolic trapping are smooth and decay to their stationary counterparts. The second, independent, result provides microlocal estimates for operators whose null-bicharacteristic flow has a normally hyperbolic invariant manifold, under suitable non-degeneracy conditions on the stable and unstable manifolds; this includes operators on closed manifolds, as well as operators on spacetimes for which the invariant manifold lies at future infinity.
We investigate the relationship between quasisymmetric and convergence groups acting on the circle. We show that the Mobius transformations of the circle form a maximal convergence group. This completes the characterization of the Mobius group as a maximal convergence group acting on the sphere. Previously, Gehring and Martin had shown the maximality of the Mobius group on spheres of dimension greater than one. Maximality of the isometry (conformal) group of the hyperbolic plane as a uniform quasi-isometry group, uniformly quasiconformal group, and as a convergence group in which each element is topologically conjugate to an isometry may be viewed as consequences.
Asynchronous stochastic approximations (SAs) are an important class of model-free algorithms, tools and techniques that are popular in multi-agent and distributed control scenarios. To counter Bellmans curse of dimensionality, such algorithms are coupled with function approximations. Although the learning/ control problem becomes more tractable, function approximations affect stability and convergence. In this paper, we present verifiable sufficient conditions for stability and convergence of asynchronous SAs with biased approximation errors. The theory developed herein is used to analyze Policy Gradient methods and noisy Value Iteration schemes. Specifically, we analyze the asynchronous approximate counterparts of the policy gradient (A2PG) and value iteration (A2VI) schemes. It is shown that the stability of these algorithms is unaffected by biased approximation errors, provided they are asymptotically bounded. With respect to convergence (of A2VI and A2PG), a relationship between the limiting set and the approximation errors is established. Finally, experimental results are presented that support the theory.
We consider a cosmology in which the final stage of the Universe is neither accelerating nor decelerating, but approaches an asymptotic state where the scale factor becomes a constant value. In order to achieve this, we first bring in a scale factor with the desired property and then determine the details of the energy contents as a result of the cosmological evolution equations. We show that such a scenario can be realized if we introduce a generalized quintom model which consists of a scalar field and a phantom with a {it negative} cosmological constant term. The standard cold dark matter with $w_m=0$ is also introduced. This is possible basically due to the balance between the matter and the {it negative} cosmological constant which tend to attract and scalar field and phantom which repel in the asymptotic region. The stability analysis shows that this asymptotic solution is classically stable.
This is an introduction to asymptotically safe quantum gravity, explaining the main idea of asymptotic safety and how it could solve the problem of predictivity in quantum gravity. In the first part, the concept of an asymptotically safe fixed point is discussed within the functional Renormalization Group framework for gravity, which is also briefly reviewed. A concise overview of key results on asymptotically safe gravity is followed by a short discussion of important open questions. The second part highlights how the interplay with matter provides observational consistency tests for all quantum-gravity models, followed by an overview of the state of results on asymptotic safety and its implications in gravity-matter models. Finally, effective asymptotic safety is briefly discussed as a scenario in which asymptotically safe gravity could be connected to other approaches to quantum gravity.
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