conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

the nirenberg problem and its generalizations a unified approach making use of integral representations we develop a unified approach to establish blow up profiles compactness and existence of positive solutions of the conformally invariant equations p sigma v kv frac n 2 sigma n 2 sigma on the standard unit sphere mathbb s n for all sigma in 0 n 2 where p sigma is the intertwining operator of order 2 sigma finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures when sigma 1 it is the prescribing scalar curvature problem or the nirenberg problem and when sigma 2 it is the prescribing q curvature problem

conformal deformation to constant negative scalar curvature on noncompact riemannian manifolds si m g est une variete de riemann complete de courbure scalaire non positive s satisfaisant s x e 0 pour x m m 0 ou m 0 est un ensemble compact alors il y a une metrique conforme complete g de courbure scalaire s 1

meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature abstract we give a detailed description of the schwartz kernel of the resolvent of the laplacian on a certain class of complete riemannian manifolds with negative sectional curvature near infinity these manifolds have compactifications as c manifolds with boundary on which the metric is conformai to a c metric up to the boundary with conformal factor 2 where is a defining function for the boundary the modified resolvent r s 2 n 1 defined and analytic in r n is shown to extend to be meromorphic in the whole complex plane when s 2 is equal to the sectional curvature at infinity for hyperbolic spaces obtained as quotients hn 1 where is a discrete group of fractional linear transformations having a geometrically finite fundamental domain without cusps at infinity s is constant we then deduce from the existence of this meromorphic extension of the resolvent the existence of a similar extension for the eisenstein series the basic method used to produce a parametrix for the laplacian is part of a more general construction by g mendoza and the second author but since this is not yet available the present paper has been made essentially self contained

a necessary and sufficient condition for approachability in the framework of two person repeated games with vector payoffs blackwell defined the property of approachability and introduced a sufficient condition for it he also proved that this condition is necessary in the case of convex sets we generalize this result a set is approachable if and only if it contains a set that satisfies blackwell s condition

pseudo marginal metropolis hastings sampling using averages of unbiased estimators we consider a pseudo marginal metropolis hastings kernel pm that is constructed using an average of m exchangeable random variables and an analogous kernel p s that averages s m of these same random variables using an embedding technique to facilitate comparisons we provide a lower bound for the asymptotic variance of any ergodic average associated with pm in terms of the asymptotic variance of the corresponding ergodic average associated with p s we show that the bound is tight and disprove a conjecture that when the random variables to be averaged are independent the asymptotic variance under p m is never less than s m times the variance under p s the conjecture does however hold when considering continuous time markov chains these results imply that if the computational cost of the algorithm is proportional to m it is often better to set m 1 we provide intuition as to why these findings differ so markedly from recent results for pseudo marginal kernels employing particle filter approximations our results are exemplified through two simulation studies in the first the computational cost is effectively proportional to m and in the second there is a considerable start up cost at each iteration

blow up phenomena for scalar flat metrics on manifolds with boundary abstract let m n g be a compact riemannian manifold with boundary m this article is concerned with the set of scalar flat metrics on m which are in the conformal class of g and have m as a constant mean curvature hypersurface we construct examples of metrics on the unit ball b n in dimensions n 25 for which this set is non compact these manifolds have umbilic boundary but they are not conformally equivalent to b n

fractional laplacian in conformal geometry abstract in this note we study the connection between the fractional laplacian operator that appeared in the recent work of caffarelli and silvestre and a class of conformally covariant operators in conformal geometry

opportunistic approachability and generalized no regret problems blackwell s theory of approachability introduced in 1956 has since proved a useful tool in the study of a range of repeated multiagent decision problems given a repeated matrix game with vector payoffs a target set s is approachable by a certain player if he can ensure that the average payoff vector converges to that set for any strategy of the opponent in this paper we consider the case where a set need not be approachable in general but may be approached if the opponent played favorably in some sense in particular we consider nonconvex sets that satisfy blackwell s dual condition namely can be approached when the opponent plays a stationary strategy whereas the convex hull of such a set is approachable this is not generally the case for the original nonconvex set itself we start by defining a sense of restricted play of the opponent with stationary strategies being a special case and then formulate appropriate goals for an opportunistic approachability algorithm that can take advantage

non compactness and multiplicity results for the yamabe problem on sn the aim of this paper is to show the existence of metrics ge on sn where ge is a perturbation of the standard metric g0 for which the yamabe problem possesses a sequence of solutions unbounded in l sn the metrics ge that we find are of class ck on sn with k n 34 we also prove some new multiplicity results

the brezis nirenberg result for the fractional laplacian the aim of this paper is to deal with the non local fractional counterpart of the laplace equation involving critical non linearities studied in the famous paper of brezis and nirenberg 1983 namely our model is the equation su u u 2 2u in u 0 in rn where s is the fractional laplace operator s 0 1 is an open bounded set of rn n 2s with lipschitz boundary 0 is a real parameter and 2 2n n 2s is a fractional critical sobolev exponent in this paper we first study the problem in a general framework indeed we consider the equation lku u u 2 2u f x u 0 in u 0 in rn where lk is a general non local integrodifferential operator of order s and f is a lower order perturbation of the critical power u 2 2u in this setting we prove an existence result through variational techniques then as a concrete example we derive a brezis nirenberg type result for our model equation that is we show that if 1 s is the first eigenvalue of the non local operator s with homogeneous dirichlet boundary datum then for any 0 1 s there exists a non trivial solution of the above model equation provided n 4s in this sense the present work may be seen as the extension of the classical brezis nirenberg result to the case of non local fractional operators

a non compactness result on the fractional yamabe problem in large dimensions abstract let x n 1 g be an n 1 dimensional asymptotically hyperbolic manifold with conformal infinity m n h the fractional yamabe problem addresses to solve p g h u c u n 2 n 2 u 0 on m where c r and p g h is the fractional conformal laplacian whose principal symbol is the laplace beltrami operator on m in this paper we construct a metric on the half space x r n 1 which is conformally equivalent to the unit ball for which the solution set of the fractional yamabe equation is non compact provided that n 24 for 0 and n 25 for 1 where 0 1 is a certain transition exponent the value of turns out to be approximately 0 940197

online learning with sample path constraints we study online learning where a decision maker interacts with nature with the objective of maximizing her long term average reward subject to some sample path average constraints we define the reward in hindsight as the highest reward the decision maker could have achieved while satisfying the constraints had she known nature s choices in advance we show that in general the reward in hindsight is not attainable the convex hull of the reward in hindsight function is however attainable for the important case of a single constraint the convex hull turns out to be the highest attainable function using a calibrated forecasting rule we provide an explicit strategy that attains this convex hull we also measure the performance of heuristic methods based on non calibrated forecasters in experiments involving a cpu power management problem

response based approachability with applications to generalized no regret problems blackwell s theory of approachability provides fundamental results for repeated games with vector valued payoffs which have been usefully applied in the theory of learning in games and in devising online learning algorithms in the adversarial setup a target set s is approachable by a player the agent in such a game if he can ensure that the average payoff vector converges to s no matter what the opponent does blackwell provided two equivalent conditions for a convex set to be approachable standard approachability algorithms rely on the primal condition which is a geometric separation condition and essentially require to compute at each stage a projection direction from a certain point to s here we introduce an approachability algorithm that relies on blackwell s dual condition which requires the agent to have a feasible response to each mixed action of the opponent namely a mixed action such that the expected payoff vector belongs to s thus rather than projections the proposed algorithm relies on computing the response to a certain action of the opponent at each stage we demonstrate the utility of the proposed approach by applying it to certain generalizations of the classical regret minimization problem which incorporate side constraints reward to cost criteria and so called global cost functions in these extensions computation of the projection is generally complex while the response is readily obtainable

on the regularity of solutions to the yamabe equation and the existence of smooth hyperboloidal initial data for einstein s field equations the regularity of the solutions to the yamabe problem is considered in the case of conformally compact manifolds and negative scalar curvature the existence of smooth hyperboloidal initial data for einstein s field equations is demonstrated

complete conformal metrics with negative scalar curvature in compact riemannian manifolds soit m g une variete de riemann compacte de dimension n 3 et soit une sous variete lisse fermee de dimension d alors il y a une metrique conforme complete sur m m a courbure scalaire negative constante si et seulement si d n 2 2

nonlinear equations for fractional laplacians ii existence uniqueness and qualitative properties of solutions this paper which is the follow up to part i concerns the equation delta s v g v 0 in mathbb r n with s in 0 1 where delta s stands for the fractional laplacian the infinitesimal generator of a l evy process when n 1 we prove that there exists a layer solution of the equation i e an increasing solution with limits pm 1 at pm infty if and only if the potential g has only two absolute minima in 1 1 located at pm 1 and satisfying g 1 g 1 0 under the additional hypothesis g 1 0 and g 1 0 we also establish its uniqueness and asymptotic behavior at infinity furthermore we provide with a concrete almost explicit example of layer solution for n geq 1 we prove some results related to the one dimensional symmetry of certain solutions in the spirit of a well known conjecture of de giorgi for the standard laplacian

on the dirichlet problem for minimal graphs in hyperbolic space soit r n n 2 un domaine borne on etudie le probleme de dirichlet suivant f f i f j f ij 1 df 2 n f 0 dans f 0 dans f 0 sur on considere le comportement a la frontiere de la solution

a multiplicity result for the yamabe problem on sn abstract we prove a multiplicity result for the yamabe problem on the manifold s g where g is a perturbation of the standard metric g 0 of s n solutions are found by variational methods via an abstract perturbation result

guaranteed performance regions in markovian systems with competing decision makers a decision maker is facing a dynamic system which is being controlled by himself herself as well as by other decision makers he she considers a vector of performance measures acceptable performance is defined through a set in the space of performance vectors whether this decision maker can guarantee a time averaged performance vector which approaches this desired set is considered for the worst case scenario in which other decision makers may for selfish reasons try to exclude his her vector from the desired set for a controlled markov model of the system a sufficient condition for approachability is given and appropriate control strategies are constructed under certain recurrence conditions a complete characterization of approachability is then provided for convex sets the mathematical formulation leads to a theory of approachability for stochastic games with vector payoffs a simple queuing example illustrates this approach

compactness results in conformal deformations of riemannian metrics on manifolds with boundaries this paper is devoted to the study of a problem arising from a geometric context namely the conformal deformation of a riemannian metric to a scalar flat one having constant mean curvature on the boundary by means of blow up analysis techniques and the positive mass theorem we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the c2 norm and the total leray schauder degree of all solutions is equal to 1 then we deduce from this compactness result the existence of at least one solution to our problem

the effect of linear perturbations on the yamabe problem in conformal geometry the compactness conjecture asserts that the set of yamabe metrics on a smooth compact aspherical riemannian manifold left m g right is compact established in the locally conformally flat case by schoen lecture notes in mathematics vol 1365 pp 120 154 springer berlin 1989 surveys pure application and mathematics 52 longman science technology pp 311 320 harlow 1991 and for n le 24 by khuri marques schoen j differ geom 81 1 143 196 2009 it has revealed to be generally false for n ge 25 as shown by brendle j am math soc 21 4 951 979 2008 and brendle marques j differ geom 81 2 225 250 2009 a stronger version of it the compactness under perturbations of the yamabe equation is addressed here with respect to the linear geometric potential frac n 2 4 n 1 mathrm scal g mathrm scal g being the scalar curvature of left m g right we show that a priori l infty bounds fail for linear perturbations on all manifolds with n ge 4 as well as a priori gradient l 2 bounds fail for non locally conformally flat manifolds with n ge 6 and for locally conformally flat manifolds with n ge 7 in several situations the results are optimal our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of g

asymptotic behavior of solutions for nonlinear elliptic problems with the fractional laplacian abstract in this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional laplacians and the critical exponents this work can be seen as a nonlocal analog of the results of han 1991 24 and rey 1990 35

comment the impossibility of inductive inference oakes s argument shows that no statistical analysis however complex of sequential data can be guaranteed to provide asymptotically valid forecasts for every possible set of outcomes

the empirical bayes envelope and regret minimization in competitive markov decision processes this paper proposes an extension of the regret minimizing framework from repeated matrix games to stochastic game models under appropriate recurrence conditions a decision maker p1 who wishes to maximize his long term average reward is facing a markovian environment which may also be affected by arbitrary actions of other agents the latter are collectively modeled as a second player p2 whose strategy is arbitrary both states and actions are fully observed by both players while p1 may obviously secure the min max value of the game he may wish to improve on that when the opponent is not playing a worst case strategy for repeated matrix games an achievable goal is presented by the bayes envelope that traces p1 s best response payoff against the observable frequencies of p2 s actions we propose a generalization to the stochastic game framework under recurrence conditions that amount to fixed state reachability the empirical bayes envelope ebe is defined as p1 s best response payoff against the stationary strategies of p2 that agree with the observed state action frequencies because the ebe may not be attainable in general we consider its lower convex hull the convex bayes envelope cbe which is proved to be achievable by p1 the analysis relies on blackwell s approachability theory the cbe is lower bounded by the value of the game and for irreducible games turns out to be strictly above the value whenever p2 s frequencies deviate from a worst case strategy in the special case of single controller games where p2 alone affects the state transitions the ebe itself is shown to be attainable