heights of heegner cycles and derivatives of l series 0 introduction 99 1 intersections and heights 103 2 kuga sato varieties and cm cycles 117 3 heights of cm cycles 124 4 heights of heegner cycles 137 5 proof of the main identity and the consequences 147 references 151

quantum unique ergodicity for sl2 z h we eliminate the possibility of escape of mass for hecke maass forms of large eigenvalue for the modular group combined with the work of lindenstrauss this establishes the quantum unique ergodicity conjecture of rudnick and sarnak for hecke maass forms on the modular surface sl 2

zeros of l functions attached to maass forms this paper considers psl 2 automorphic cusp forms the maass forms and their associated dirichlet series there are two major results the first uses real variable techniques to give an explicit reconstruction of the cusp form in terms of the dirichlet series this representation is valid in the entire upper half plane the second uses this representation to show that the dirichlet series has many zeros on its critical line this is the analogue of the hardy littlewood result for the riemann zeta function

network identifiability from intrinsic noise this paper considers the problem of inferring an unknown network of dynamical systems driven by unknown intrinsic noise inputs equivalently we seek to identify direct causal dependencies among manifest variables only from observations of these variables for linear time invariant systems of minimal order we characterise under what conditions this problem is well posed we first show that if the transfer matrix from the inputs to manifest states is minimum phase this problem has a unique solution irrespective of the network topology this is equivalent to there being only one valid spectral factor up to a choice of signs of the inputs of the output spectral density if the assumption of phase minimality is relaxed we show that the problem is characterised by a single algebraic riccati equation are of dimension determined by the number of latent states the number of solutions to this are is an upper bound on the number of solutions for the network we give necessary and sufficient conditions for any two dynamical networks to have equal output spectral density which can be used to construct all equivalent networks extensive simulations quantify the number of solutions for a range of problem sizes for a slightly simpler case we also provide an algorithm to construct all equivalent networks from the output spectral density

equidistribution in shrinking sets and l 4 norm bounds for automorphic forms we study two closely related problems stemming from the random wave conjecture for maas forms the first problem is bounding the l 4 norm of a maas form in the large eigenvalue limit we complete the work of spinu to show that the l 4 norm of an eisenstein series e z 1 2 it g restricted to compact sets is bounded by sqrt log t g the second problem is quantum unique ergodicity in shrinking sets we show that by averaging over the centre of hyperbolic balls in gamma backslash mathbb h quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the planck scale this result is conditional on the generalised lindelof hypothesis for hecke maas eigenforms but is unconditional for eisenstein series we also show that equidistribution for hecke maas eigenforms need not hold at or below the planck scale finally we prove similar equidistribution results in shrinking sets for heegner points and closed geodesics associated to ideal classes of quadratic fields

non vanishing of high derivatives of automorphic l functions at the center of the critical strip keywords modular l functions mollification nonvanishing results central value of high derivatives analytic rank jacobian variety reference tan article 2000 004doi 10 1515 crll 2000 074 record created on 2008 11 14 modified on 2017 05 12

l infty norms of eigenfunctions of arithmetic surfaces here and elsewhere a l1o0 as a xc is a basic problem of quantum chaos s in any case almost nothing beyond 0 2 is known about 110 joc when the curvature is negative one can push the standard techniques and replace a1 4 by a1 4 log a in this case in this paper we use arithmetic techniques in particular modular correspondences to obtain the first improvement in the exponent 0 2 we also obtain lower bounds on the l norms in these arithmetic cases in more detail let a b be a quaternion division algebra over q a is linearly generated by 1 w q wq over q and w2 a q2 b wq qw 0 here a b e z are square free and we will assume that a 0 as usual the norm and trace are defined by n a a d tr a a oy where if a xo x1w x2q x3wq

planck scale mass equidistribution of toral laplace eigenfunctions we study the small scale distribution of the l 2 mass of eigenfunctions of the laplacian on the two dimensional flat torus given an orthonormal basis of eigenfunctions lester and rudnick commun math phys 350 1 279 300 2017 showed the existence of a density one subsequence whose l 2 mass equidistributes more or less down to the planck scale we give a more precise version of their result showing equidistribution holds down to a small power of log above planck scale and also showing that the l 2 mass fails to equidistribute at a slightly smaller power of log above the planck scale this article rests on a number of results about the proximity of lattice points on circles much of it based on foundational work of javier cilleruelo

the quantum unique ergodicity conjecture for thin sets abstract we consider some analogs of the quantum unique ergodicity conjecture for geodesics and shrinking families of sets in particular we prove the analog of the que conjecture for eisenstein series restricted to the infinite geodesic connecting 0 and inside the modular surface

estimates for rankin selberg l functions and quantum unique ergodicity abstract subconvex bounds in the weight aspect for rankin selberg l functions associated to two cusp forms are established these bounds are applied to prove the equidistribution of mass conjecture for cm forms

error bounds for finite step approximations for solving infinite horizon controlled markov set chains this note considers finite step approximations for solving an infinite horizon controlled markov set chain problem with finite state and action spaces we develop a value iteration type algorithm based on the optimality equation developed by kurano et al and analyze an error bound relative to the optimal value that satisfies the optimality equation from the successive approximation we further analyze an error bound of the approximate control policy defined from a finite step approximate value by applying the value iteration type algorithm

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

equidistribution of eisenstein series in the level aspect we prove an equidistribution property of the eisenstein series for congruence subgroups as the level goes to infinity this is an analogy of the phenomenon called quantum ergodicity

errors in variables identification in dynamic networks consistency results for an instrumental variable approach in this paper we consider the identification of a linear module that is embedded in a dynamic network using noisy measurements of the internal variables of the network this is an extension of the errors in variables eiv identification framework to the case of dynamic networks the consequence of measuring the variables with sensor noise is that some prediction error identification methods no longer result in consistent estimates the method developed in this paper is based on a combination of the instrumental variable philosophy and closed loop prediction error identification methods and leads to consistent estimates of modules in a dynamic network we consider a flexible choice of which internal variables need to be measured in order to identify the module of interest this allows for a flexible sensor placement scheme we also present a method that can be used to validate the identified model

l p norms nodal sets and quantum ergodicity abstract for small range of p 2 we improve the l p bounds of eigenfunctions of the laplacian on negatively curved manifolds our improvement is by a power of logarithm for a full density sequence of eigenfunctions we also derive improvements on the size of the nodal sets our proof is based on a quantum ergodicity property of independent interest which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate

quantum ergodicity and l p norms of restrictions of eigenfunctions we prove an analogue of sogge s local l p estimates for l p norms of restrictions of eigenfunctions to submanifolds and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of burq gerard tzvetkov hu and chen sogge the improvements are logarithmic on negatively curved manifolds without boundary and by o 1 for manifolds with or without boundary with ergodic geodesic flows in the case of ergodic billiards with piecewise smooth boundary we get o 1 improvements on l infty estimates of cauchy data away from a shrinking neighborhood of the corners and as a result using the methods of ghosh et al jung and zelditch jung and zelditch we get that the number of nodal domains of 2 dimensional ergodic billiards tends to infinity as lambda to infty these results work only for a full density subsequence of any given orthonormal basis of eigenfunctions we also present an extension of the l p estimates of burq gerard tzvetkov hu chen sogge for the restrictions of dirichlet and neumann eigenfunctions to compact submanifolds of the interior of manifolds with piecewise smooth boundary this part does not assume ergodicity on the manifolds

a quantitative condition for the well posedness of interconnected dynamical systems a sufficient condition for the well posedness of interconnected dynamical systems is reported the condition is stated in terms of the uniform instantaneous gains of subsystems and interconnections and includes vidyasagar s condition 8 as a special case the result is useful for testing the validity of the dynamical model of a large scale system which is obtained by combining the models of subsystems

identification of systems with unknown inputs using indirect input measurements abstracta common issue with many system identification problems is that the true input to the system is unknown this paper extends a previously presented indirect modelling framework that deals with identification of systems where the input is partially or fully unknown in this framework unknown inputs are eliminated by using additional measurements that directly or indirectly contain information about the unknown inputs the resulting indirect predictor model is only dependent on known and measured signals and can be used to estimate the desired dynamics or properties since the input of the indirect model contains both known inputs and measurements that could all be correlated with the same disturbances as the output estimation of the indirect model has similar challenges as a closed loop estimation problem in fact due to the generality of the indirect modelling framework it unifies a number of already existing system identification problems that are contained as special cases for completeness

correction to equidistribution of eisenstein series in the level aspect the second author formulated quantum unique ergodicity for eisenstein series in the prime level aspect in equidistribution of eisenstein series in the level aspect commun math phys 289 3 1150 2009 we point out errors and correct the proofs with partially weakened claims

an indirect method for transfer function estimation from closed loop data abstract an indirect method is introduced that is able to estimate consistently the transfer function of a linear plant on the basis of data obtained from closed loop experiments even in the situation where the model of the noise disturbance on the data is not accurate moreover the method allows approximate identification of the open loop plant with an explicit and tunable expression for the bias distribution of the resulting model

identification of dynamic models in complex networks with prediction error methods basic methods for consistent module estimates abstract the problem of identifying dynamical models on the basis of measurement data is usually considered in a classical open loop or closed loop setting in this paper this problem is generalized to dynamical systems that operate in a complex interconnection structure and the objective is to consistently identify the dynamics of a particular module in the network for a known interconnection structure it is shown that the classical prediction error methods for closed loop identification can be generalized to provide consistent model estimates under specified experimental circumstances two classes of methods considered in this paper are the direct method and the joint io method that rely on consistent noise models and indirect methods that rely on external excitation signals like two stage and iv methods graph theoretical tools are presented to verify the topological conditions under which the several methods lead to consistent module estimates

exponential decay in the frequency of analytic ranks of automorphic l functions this note should be seen as a appendum to the work of e kowalski and the second author km1 and deals with the problem of bounding unconditionnally the order of vanishing at the critical point in a family of l functions this problem is illustrated in the particular case of l functions of weight 2 primitive modular forms of prime level recall the notations from km1 for q a prime number let s2 q be the set of primitive forms of weight 2 and level q normalized so that their first fourier coefficient is 1 for

necessary and sufficient conditions for dynamical structure reconstruction of lti networks this paper formulates and solves the network reconstruction problem for linear time invariant systems the problem is motivated from a variety of disciplines but it has recently received considerable attention from the systems biology community in the study of chemical reaction networks here we demonstrate that even when a transfer function can be identified perfectly from input output data not even boolean reconstruction is possible in general without more information about the system we then completely characterize this additional information that is essential for dynamical reconstruction without appeal to ad hoc assumptions about the network such as sparsity or minimality

planck scale number of nodal domains for toral eigenfunctions abstract we study the number of nodal domains in balls shrinking slightly above the planck scale for generic toral eigenfunctions we prove that up to the natural scaling the nodal domains count obeys the same asymptotic law as the global number of nodal domains the proof on one hand uses new arithmetic information to refine bourgain s de randomisation technique at planck scale and on the other hand it requires a planck scale version of yau s conjecture which we believe to be of independent interest

rankin selberg l functions in the level aspect keywords moments rankin selberg convolution level aspect convexity breaking reference tan article 2002 003doi 10 1215 s0012 7094 02 11416 1 record created on 2008 11 14 modified on 2017 05 12

a forecast horizon and a stopping rule for general markov decision processes in this paper we are concerned with stochastic optimal control problems over an infinite horizon when the reward is discounted there is only one optimality criterion and standard dynamic programming can be applied now in the undiscounted case the total reward might be unbounded for any feasible policy and there is not a unique optimahty criterion any more greatest average reward overtaking optimality average overtaking optimality and i optimality are such criteria 4 lo the most used of them is the greatest average reward and several technics have been proposed to get an average optimal policy at least for finite state space markov decision processes mdp i 1 these results are all based on the existence of a fixed point solution to the optimality equations for a relative value function in the case of denumerable state space weak conditions have been recently exhibited under which a fixed point solution exists 3 in the case of finite state space non homogeneous mdp the relative value function is also shown to converge under so called weak ergodicity con ditions s while the latter assumptions are stronger one can easily check them by inspection of the one step transition probability matrices and also

instrumental variable methods for closed loop system identification in this paper several instrumental variable iv and instrumental variable related methods for closed loop system identification are considered and set in an extended iv framework extended iv methods require the appropriate choice of particular design variables as the number and type of instrumental signals data prefiltering and the choice of an appropriate norm of the extended iv criterion the optimal iv estimator achieves minimum variance but requires the exact knowledge of the noise model for the closed loop situation several iv methods are put in an extended iv framework and characterized by different choices of design variables their variance properties are considered and illustrated with a simulation example

error bounds for rolling horizon policies in discrete time markov control processes error bounds are presented for rolling horizon rh policies in general stationary and nonstationary borel markov control problems with both discounted and average reward criteria in each of these cases conditions are given under which the reward of the rolling horizon policy converges geometrically to the optimal reward function uniformly in the initial state as the length of the rolling horizon increases a description of the control model and the general assumptions are given the approach is based on extending the results of j m alden and a r l smith 1988 on nonstationary processes with finite state and action spaces however the proofs presented are simpler this is because when stationary models are analyzed first the error bounds follow more or less directly from well known value iteration results the corresponding error bounds for nonstationary models are obtained by reducing these models to stationary ones

a projection method for closed loop identification a new method for closed loop identification that allows fitting the model to the data with arbitrary frequency weighting is described and analyzed just as the direct method this new method is applicable to systems with arbitrary feedback mechanisms this is in contrast to other methods such as the indirect method and the two stage method that assume linear feedback the finite sample behavior of the proposed method is illustrated in a simulation study

the twisted fourth moment of the riemann zeta function we compute the asymptotics of the fourth moment of the riemann zeta func tion times an arbitrary dirichlet polynomial of length t 1 11 the study of the moments of the riemann zeta function has a long and distinguished history starting with the work of hardy and littlewood in 1918 and continuing to the present day one motivation for understanding moments is that they yield information about the maximum size of the zeta function the lindelof hypothesis another application is to zero density estimates which in turn have consequences for primes in short intervals however they have become an interesting topic in their own right very few rigorous results are known just the second and fourth power moments indeed it is only recently that a believable conjecture for higher powers has been made the twisted moments that is moments of the riemann zeta function times an arbitrary dirichlet polynomial are important too for example levinson s method of detecting zeros of the zeta function lying on the critical line requires knowing the asymptotics of the mollified second moment in a series of papers duke friedlander and iwaniec used estimates for amplified moments of a family of l functions in order to deduce a subconvexity bound for an individual member of the family of course there are far easier methods to give a subconvexity bound for zeta but there are close analogies between different families and it is desirable to understand the structure of these amplified moments in general in this paper we prove an asymptotic formula for the twisted fourth moment of the riemann zeta function where we may take a dirichlet polynomial of length up to t 1 11 e

distribution of mass of holomorphic cusp forms we prove an upper bound for the l4 norm and for the l2 norm restricted to the vertical geodesic of a holomorphic hecke cusp form f of large weight k the method is based on watson s formula and estimating a mean value of certain l functions of degree 6 further applications to restriction problems of siegel modular forms and subconvexity bounds of degree 8 l functions are given

mollification of the fourth moment of automorphic l functions and arithmetic applications we compute the asymptotics of twisted fourth power moments of modular l functions of large prime level near the critical line this allows us to prove some new non vanishing results on the central values of automorphic l functions in particular those obtained by base change from gl 2 q to gl 2 k for k a cyclic field of low degree

moments of the riemann zeta function assuming the riemann hypothesis we obtain an upper bound for the moments of the riemann zeta function on the critical line our bound is nearly as sharp as the conjectured asymptotic formulae for these moments the method extends to moments in other families of l functions

closed loop identification revisited identification of systems operating in closed loop has long been of prime interest in industrial applications the problem offers many possibilities and also some fallacies and a wide variety of approaches have been suggested many quite recently the purpose of the current contribution is to place most of these approaches in a coherent framework thereby showing their connections and display similarities and differences in the asymptotic properties of the resulting estimates the common framework is created by the basic prediction error method and it is shown that most of the common methods correspond to different parameterizations of the dynamics and noise models the so called indirect methods e g are indeed direct methods employing noise models that contain the regulator the asymptotic properties of the estimates then follow from the general theory and take different forms as they are translated to the particular parameterizations we also study a new projection approach to closed loop identification with the advantage of allowing approximation of the open loop dynamics in a given and user chosen frequency domain norm even in the case of an unknown nonlinear regulator

finite step approximation error bounds for solving average reward controlled markov set chains this paper is a sequel to the analysis of finite step approximations in solving controlled markov set chains for infinite horizon discounted reward by the author for average reward controlled markov set chains with finite state and action spaces we develop a value iteration type algorithm and analyze an error bound relative to the optimal average reward that satisfies an optimality equation from the successive approximation under an ergodicity condition we further analyze an error bound of the rolling horizon control policy defined from a finite step approximate value by applying the value iteration type algorithm

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

controlled markov set chains under average criteria in this paper applying an interval arithmetic analysis we consider the average case of controlled markov set chains whose process allows for fluctuating transition matrices at each step in time we introduce a v step contractive property for the average case under which a pareto optimal periodic policy is characterized as a maximal solution of optimality equation also in the class of stationary policies the behavior of the expected reward over t horizon as t approaches is investigated and the left and right hand side optimality equations are given by which a pareto optimal stationary policy is found as a numerical example the taxicab problem is considered

integral moments of l functions we give a new heuristic for all of the main terms in the integral moments of various families of primitive l functions the results agree with previous conjectures for the leading order terms our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary orthogonal or symplectic matrices where the moments are defined by the appropriate group averages this lends support to the idea that arithmetical l functions have a spectral interpretation and that their value distributions can be modelled using random matrix theory numerical examples show good agreement with our conjectures

small scale equidistribution for a point scatterer on the torus we study the small scale distribution of the eigenfunctions of a point scatterer the laplacian perturbed by a delta potential on two and three dimensional flat tori in two dimensions we establish small scale equidistribution for the new eigenfunctions holding all the way down to the planck scale in three dimensions small scale equidistribution is established for all of the new eigenfunctions at certain scales

identification and the information matrix how to get just sufficiently rich in prediction error identification the information matrix plays a central role specifically when the system is in the model set the covariance matrix of the parameter estimates converges asymptotically up to a scaling factor to the inverse of the information matrix the existence of a finite covariance matrix thus depends on the positive definiteness of the information matrix and the rate of convergence of the parameter estimate depends on its size the information matrix is also the key tool in the solution of optimal experiment design procedures which have become a focus of recent attention introducing a geometric framework we provide a complete analysis for arbitrary model structures of the minimum degree of richness required to guarantee the nonsingularity of the information matrix we then particularize these results to all commonly used model structures both in open loop and in closed loop in a closed loop setup our results provide an unexpected and precisely quantifiable trade off between controller degree and required degree of external excitation

mass equidistribution for hecke eigenforms in this work we study the mass equidistribution for holomorphic hecke eigenforms and establish by employing incomplete poincare series and the petersson formula sharp equidistribution results when the average is performed over intervals much shorter than before a key feature is the analysis of the off diagonal terms that result from this shortening of intervals 2003 wiley periodicals inc

decentralized learning in finite markov chains revisited the convergence proof in the paper ldquodecentralized learning in finite markov chains rdquo published in the ieee transactions on automatic control vol ac 31 no 6 pp 519 526 1986 is incomplete this note first provides a sufficient condition for the existence of a unique optimal policy for infinite horizon average cost markov decision processes mdps making the convergence result established by wheeler and narendra preserved with the condition we then present a novel simulation based decentralized algorithm called ldquosampled joint strategy fictitious play for mdprdquo for average mdps based on the recent study by garcia of a decentralized approach to discrete optimization via fictitious play applied to games with identical payoffs we establish a stronger almost sure convergence result than wheeler and narendra s showing that the sequence of probability distributions over the policy space for a given mdp generated by the algorithm converges to a unique optimal policy with probability one

the fourth moment of dirichlet l functions we compute the fourth moment of dirichlet l functions at the central point for prime moduli with a power savings in the error term

variance analysis of linear simo models with spatially correlated noise in this paper we address the identification of linear time invariant single input multi output simo systems in particular we assess the performance of the prediction error method by quantifying

equidistribution of eisenstein series on geodesic segments abstract we show an asymptotic formula for the l 2 norm of the eisenstein series e z 1 2 i t restricted to a segment of a geodesic connecting infinity and an arbitrary real for generic geodesics of this form the asymptotic formula shows that the eisenstein series satisfies restricted que on the other hand for rational geodesics the eisenstein series does not satisfy restricted quantum ergodicity as an application we show that the zero set of the eisenstein series intersects every such geodesic segment provided t is large we also make analogous conjectures for the maass cusp forms in particular we predict that cusp forms do not satisfy restricted quantum ergodicity for rational geodesics

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom

mass equidistribution for hecke eigenforms we prove a conjecture of rudnick and sarnak on the mass equidistribution of hecke eigenforms this builds upon independent work of the authors

delay structure conditions for identifiability of closed loop systems abstract when identifying a linear system operating under a linear feedback law specific restrictions have to be imposed on the presence of delays in the plant and or the controller in order to obtain identifiability in this paper the delay structure conditions are investigated under which a least squares direct type prediction error identification method is able to consistently estimate the open loop plant it is shown that the commonly required condition of strictly properness of the loop transfer function can be relaxed to a condition that relates to the absence of algebraic loops in the closed loop plant models