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We extend the Poincare inequality to functions of Sobolev type on a stratified set. The integrability exponents in these analogs depend on the geometric characteristics of the stratified set which show to what extent their strata are connected with each other and the boundary. We apply the results to proving the solvability of boundary value problems for the p-Laplacian with boundary conditions of Neumann or Wentzel type.

A transport network is a complex non-stationary open environment with multiple heterogeneous stochastic agents such as intersection controllers and road users. Changes in such a network may occur due to random demand fluctuations, supply degradation, or actions of agents. Thus, it makes the task of adaptive Traffic Light Control (TLC) extremely challenging.This paper is our attempt to analyze the complexity of TLC on a single isolated intersection by considering it from the perspective of supervised learning. We explicitly split the problem into two parts: demand estimation and traffic light control itself. For each task we train offline a model (Neural Network) on synthesized data.Our experiments show that even Shallow Neural Network (SNN) models can estimate and control traffic well on a single intersection. The only bottleneck of this method resides in the process of data generation and annotation - the number of simulations needed for data generation polynomially grows with the increase of number of intersections. This might prevent the current approach from being applied to transport networks with multiple intersections.

This paper presents an online model-free adaptive traffic signal controller for an isolated intersection using a Reinforcement Learning (RL) approach. We base our solution on the Q-learning algorithm with action-value approximation. In contrast with other studies in the field, we use the queue length in adddition to the average delay as a measure of performance. Also, the number of queuing vehicles and the green phase duration in four directions are aggregated to represent a state. The duration of phases is a precise value for the nonconflicting directions. Therefore, cycle length is non-fixed. Finally, we analyze and update the equilibrium and queue reduction terms in our previous equation of an immediate reward. Also, the delay based reward is tested in the given control system. The performance of the proposed method is compared with an optimal symmetric fixed signal plan.

A vector field on Riemannian manifold is called conformal Killing if it generates oneparameter group of conformal transformation. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric and physical problems. In this paper, we prove that a trace-free conformal Killing tensor field is identically zero if it vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a compact manifold with boundary to a sum of three fields of special types. We also establish triviality of the space of trace-free conformal Killing tensor fields on some closed manifolds.

We develop a method for reconstructing the conformal factor of a Riemannian metric and the magnetic field on a surface from the scattering relation associated to the corresponding magnetic flow. The scattering relation maps a starting point and direction of a magnetic geodesic into its end point and direction. The key point in the reconstruction is the interplay between the magnetic ray transform, the fiberwise Hilbert transform on the circle bundle of the surface, and the Laplace-Beltrami operator of the underlying Riemannian metric.

We consider an optical hypersurface Σ in the cotangent bundle τ:T*M→M of a closed manifold M endowed with a twisted symplectic structure. We show that if the characteristic foliation of Σ is Anosov, then a smooth 1-form θ on M is exact if and only if τ*θ has zero integral over every closed characteristic of Σ. This result is derived from a related theorem about magnetic flows which generalizes our previous work [N. S. Dairbekov and G. P. Paternain. Longitudinal KAM cocycles and action spectra of magnetic flows. Math. Res. Lett.12 (2005), 719–729]. Other rigidity issues are also discussed.

For a compact Riemannian manifold with boundary, endowed with a magnetic potential α , we consider the problem of restoring the metric g and the magnetic potential α from the values of the Mane action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this case, knowledge of the Mane action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction. This problem can only be solved up to an isometry and a gauge transformation of α . For the linearized problem, we show injectivity, up to the natural obstruction, under explicit bounds on the curvature and on α . We also show injectivity and stability for g and α in a generic class G including real analytic ones. For the nonlinear problem, we show rigidity for real analytic simple ( g , α ) , rigidity for metrics in a given conformal class, and locally, near any ( g , α ) ∈ G . We also show that simple magnetic systems on two-dimensional manifolds are always rigid.

We consider the integral geometry problem of restoring a tensor field on a manifold with boundary from its integrals over geodesics running between boundary points. For nontrapping manifolds with a certain upper curvature bound, we prove that a tensor field, integrating to zero over geodesics between boundary points, is potential. For functions and 1-forms, this is shown to be true for arbitrary nontrapping manifolds with no conjugate points. As a consequence, we also establish deformation boundary rigidity for strongly geodesic minimizing manifolds with a certain upper curvature bound.

Let M beaclosedorientedsurfaceandletΩbeanon-exact2-form.Suppose that the magnetic flow φ of the pair ( g, Ω) is Anosov. We show thatthelongitudinalKAM-cocycleof φ isacoboundaryifandonlyiftheGaussiancurvatureisconstantandΩisaconstantmultipleoftheareaformthusextendingtheresultsin[12]. Wealsoshowinfinitesimalrigidityoftheactionspectrumof φ withrespecttovariationsofΩ. Bothresultsareobtainedbyshowingthatif G : M → R is any smooth function and ω is any smooth 1-form on M suchthat G ( x )+ ω x ( v )integratestozeroalonganyclosedorbitof φ ,then G mustbeidenticallyzeroand ω mustbeexact. 1. IntroductionLet M beaclosedorientedsurfaceendowedwithaRiemannianmetric g andletΩbea2-form.The magneticflow ofthepair( g, Ω)istheflow φ ontheunitspherebundle SM determinedbytheequation Dγ ˙ dt (1) = λ ( γ ) iγ, ˙where i indicatesrotationby π/ 2accordingtotheorientationofthesurfaceand λ isthesmoothfunctionon M uniquelydeterminedbyΩ= λ Ω a ,whereΩ a istheareaformof M .WhenΩvanisheswerecovertheusualgeodesicflowofthesurface.Acurve

This paper considers the boundary rigidity problem for a compact convex Riemannian manifold ( M , g ) (M,g) with boundary ∂ M \partial M whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics g ′ g’ on M M there is a C 3 , α C^{3,\alpha } -neighborhood of g g such that g g is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance — as measured in M M ). More precisely, given any metric g ′ g’ in this neighborhood with the same boundary distance function there is diffeomorphism φ \varphi which is the identity on ∂ M \partial M such that g ′ = φ ∗ g g’=\varphi ^{*}g . There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.

We show a Hopf type rigidity for thermostats without conjugate points on a 2-torus, 9 pages; minor revisions to reflect published version

We consider a general family of curves $\Gamma$ on a compact oriented Finsler surface $(M,F)$ with boundary $\partial M$. Let $\varphi\in C^{\infty}(M)$ and $\omega$ a smooth 1-form on $M$. We show that $$\int_{\gamma(t)}\{\varphi(\gamma(t))+\omega_{\gamma(t)}(\dot{\gamma}(t))\}\,dt=0$$ holds for every $\gamma\in\Gamma$ whose endpoints belong to $\partial M$, $\gamma(a)\in\partial M$, $\gamma(b)\in\partial M$ if and only if $\varphi=0$ and $\omega$ is exact. Similar results were proved when $M$ is closed and some additional conditions on Gaussian curvature are imposed. We also study the cohomological equations of Anosov generelized thermostats on a closed Finsler surface. Finally, we gave conditions when thermostat is of Anosov type., Comment: 25 pages

We consider a magnetic flow without conjugate points on a closed manifold $M$ with generating vector field $\G$. Let $h\in C^{\infty}(M)$ and let $\theta$ be a smooth 1-form on $M$. We show that the cohomological equation \[\G(u)=h\circ \pi+\theta\] has a solution $u\in C^{\infty}(SM)$ only if $h=0$ and $\theta$ is closed. This result was proved in \cite{DP2} under the assumption that the flow of $\G$ is Anosov.

We show that an arbitrary Anosov Gaussian thermostat on a surface is dissipative unless the external field has a global potential.

We consider the question of when an inequality between lengths of corresponding geodesics implies a corresponding inequality between volumes. We prove this in a number of cases for compact manifolds with and without boundary. In particular, we show that for two Riemannian metrics of negative curvature on a compact surface without boundary, an inequality between the marked length spectra implies the same inequality between the areas, with equality implying isometry.

In this paper we prove that on an Anosov manifold the space of symmetric m -tensor fields of vanishing energy is finite dimensional modulo the space of potential tensor fields for an arbitrary m and coincides with the latter for m=0 and m=1 . For m=2 this question relates to the spectral rigidity problem.