ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

integer points on spheres and their orthogonal lattices linnik proved in the late 1950 s the equidistribution of integer points on large spheres under a congruence condition the congruence condition was lifted in 1988 by duke building on a break through by iwaniec using completely different techniques we conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement we use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition

entropy of quantum limits in this paper we show that any measure arising as a weak limit of microlocal lifts of eigenfunctions of the laplacian on certain arithmetic manifolds have dimension at least 11 9 and in particular all ergodic components of this measure with respect to the geodesic flow have positive entropy

traces of cm values of modular functions zagier proved that the traces of singular moduli i e the sums of the values of the classical j invariant over quadratic irrationalities are the fourier coefficients of a modular form of weight 3 2 with poles at the cusps using the theta correspondence we generalize this result to traces of cm values of weakly holomorphic modular functions on modular curves of arbitrary genus we also study the theta lift for the weight 0 eisenstein series for sl2 and realize a certain generating series of arithmetic intersection numbers as the derivative of zagier s eisenstein series of weight 3 2 this recovers a result of kudla rapoport and yang

extracting volatility signal using maximum a posteriori estimation abstract this paper outlines a methodology to estimate a denoised volatility signal for foreign exchange rates using a hidden markov model hmm for this purpose a maximum a posteriori map estimation is performed a double exponential prior is used for the state variable the log volatility in order to allow sharp jumps in realizations and then log returns marginal distributions with heavy tails we consider two routes to choose the regularization and we compare our map estimate to realized volatility measure for three exchange rates

gauss class number problem for imaginary quadratic fields a result first proved by heilbronn h in 1934 the disquisitiones also contains tables of binary quadratic forms with small class numbers actually tables of imaginary quadratic fields of small class number with even discriminant which is a much easier problem to deal with and gauss conjectured that his tables were complete in modern parlance we can rewrite gauss tables we are including both even and odd discriminants in the following form

on moments of twisted l functions we study the average of the product of the central values of two l functions of modular forms f and g twisted by dirichlet characters to a large prime modulus q as our principal tools we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q and we use the theory of deligne and katz to prove new bounds on bilinear forms in kloosterman sums with power savings when both variables are near the square root of q when at least one of the forms f and g is non cuspidal we obtain an asymptotic formula for the mixed second moment of twisted l functions with a power saving error term in particular when both are non cuspidal this gives a significant improvement on m young s asymptotic evaluation of the fourth moment of dirichlet l functions in the general case the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in kloosterman sums

general aspects of pt symmetric and p self adjoint quantum theory in a krein space in our previous work we proposed a mathematical framework for symmetric quantum theory and in particular constructed a krein space in which symmetric operators would naturally act in this work we explore and discuss various general consequences and aspects of the theory defined in the krein space not only spectral properties and symmetry breaking but also several issues crucial for the theory to be physically acceptable such as time evolution of state vectors probability interpretation uncertainty relation classical quantum correspondence completeness existence of a basis and so on in particular we show that for a given real classical system we can always construct the corresponding symmetric quantum system which indicates that symmetric theory in the krein space is another quantization scheme rather than a generalization of the traditional hermitian one in the hilbert space we propose a postulate for an operator to be a physical observable in this framework

recent progress on the quantum unique ergodicity conjecture we report on some recent striking advances on the quantum unique ergodicity or que conjecture concerning the distribution of large frequency eigenfunctions of the laplacian on a negatively curved manifold the account falls naturally into two categories the first concerns the general conjecture where the tools are more or less limited to micro local analysis and the dynamics of the geodesic flow the second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue the recent account by zelditch zel4 covers some of the same material and his and our discussion complement each other nicely as he goes into more detail with the first category and we with the second this note is not meant to be a survey of these topics and the discussion is not chronological our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context recent progress on que 2 1 the general que conjecture in figure 1 the domains e s and b are an ellipse a stadium and a barnett billiard table respectively superimposed on these are the densities of a consecutive sequence of high frequency eigenfunctions states modes of the laplacian that is they are solutions to 4 j j j 0 in 0 dirichlet boundary conditions j dxdy 1 0 here 4 divgrad 2 x2 2 y2 1 2 3 are the eigenvalues and the eigenfunctions are normalized to have unit l norm the sequences are of 12 consecutive modes around the 5600 eigenvalue they are ordered from left to right and then down and the grayscale represents the probability density with zero white and larger values darker

joint quasimodes positive entropy and quantum unique ergodicity we study joint quasimodes of the laplacian and one hecke operator on compact congruence surfaces and give conditions on the orders of the quasimodes that guarantee positive entropy on almost every ergodic component of the corresponding semiclassical measures together with the measure classification result of lindenstrauss ann math 2 163 1 165 219 2006 this implies quantum unique ergodicity for such functions our result is optimal with respect to the dimension of the space from which the quasi mode is constructed we also study equidistribution for sequences of joint quasimodes of the two partial laplacians on compact irreducible quotients of mathbb h times mathbb h

j self adjoint operators with mathcal c symmetries an extension theory approach a well known tool in conventional von neumann quantum mechanics is the self adjoint extension technique for symmetric operators it is used e g for the construction of dirac hermitian hamiltonians with point interaction potentials here we reshape this technique to allow for the construction of pseudo hermitian j self adjoint hamiltonians with complex point interactions we demonstrate that the resulting hamiltonians are bijectively related to the so called hypermaximal neutral subspaces of the defect krein space of the symmetric operator this symmetric operator is allowed to have arbitrary but equal deficiency indices n n general properties of the operators for these hamiltonians are derived a detailed study of operator parametrizations and krein type resolvent formulae is provided for j self adjoint extensions of symmetric operators with deficiency indices 2 2 the technique is exemplified on 1d pseudo hermitian schr dinger and dirac hamiltonians with complex point interaction potentials

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

maass forms and the mock theta function f q let f q 1 sum n 1 infty alpha n q n be the well known third order mock theta of ramanujan in 1964 george andrews proved an asymptotic formula of the form begin aligned alpha n sum c le sqrt n psi n o epsilon left n epsilon right end aligned where psi n is an expression involving generalized kloosterman sums and the i bessel function andrews conjectured that the series converges to alpha n when extended to infinity and that it does not converge absolutely bringmann and ono proved the first of these conjectures here we obtain a power savings bound for the error in andrews formula and we also prove the second of these conjectures our methods depend on the spectral theory of maass forms of half integral weight and in particular on an average estimate for the fourier coefficients of such forms which gives a power savings in the spectral parameter as a further application of this result we derive a formula which expresses alpha n with small error as a sum of exponential terms over imaginary quadratic points this is similar in spirit to a recent result of masri we also obtain a bound for the size of the error term incurred by truncating rademacher s analytic formula for the ordinary partition function which improves a result of the first author and andersen when 24n 23 is squarefree

algebraic trace functions over the primes we study sums over primes of trace functions of l adic sheaves using an extension of our earlier results on algebraic twists of modular forms to the case of eisenstein series and bounds for type ii sums based on similar applications of the riemann hypothesis over finite fields we prove general estimates with power saving for such sums we then derive various concrete applications

one dimensional schrodinger operators with symmetric zero range potentials non hermitian hamiltonians appearing as operator realizations of symmetric symmetric in physical literature zero range singular perturbations of one dimensional schrodinger operators are studied in particular hamiltonians with a real spectrum are described in terms of parameters of singular perturbations and moreover it is shown that only part of them are similar to hermitian ones in this case they can be used as exactly solvable models of symmetric quantum mechanics

spectral points of type π and π of self adjoint operators in krein spaces via approximative eigensequences we introduce the notion of spectral points of type and for self adjoint operators in krein spaces they are stable under compact perturbations for real spectral points of type and which are not in the interior of the spectrum we prove that the growth of the resolvent in some neighbourhood of them is of finite order there exists a local spectral function with singularities it turns out that all spectral subspaces corresponding to sufficiently small neighbourhoods of points of type or type are pontryagin spaces

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

p mathcal t symmetric non self adjoint operators diagonalizable and non diagonalizable with a real discrete spectrum consider in the operator family h0 a 1a1 a dad d 2 is the quantum harmonic oscillator with rational frequencies w is a symmetric bounded potential and g is a real coupling constant we show that if g being an explicitly determined constant the spectrum of h g is real and discrete moreover we show that the operator has a real discrete spectrum but is not diagonalizable

a nonsplit sum of coefficients of modular forms we introduce and study certain truncated sums of hecke eigenvalues of gl2 automorphic forms along quadratic polynomials a power saving estimate is established and new applications to moments of critical l values associated to quadratic fields are derived an application to the asymptotic behavior of the height of heegner points and singular moduli is discussed in detail

floquet theory for left definite sturm liouville problems abstract we determine the essential spectrum of left definite sturm liouville problems with periodic coefficients and a weight function which changes sign

bilinear forms with kloosterman sums and applications we prove nontrivial bounds for general bilinear forms in hyper kloosterman sums when the sizes of both variables may be below the range controlled by fourier analytic methods polya vinogradov range we then derive applications to the second moment of cusp forms twisted by characters modulo primes and to the distribution in arithmetic progressions to large moduli of certain eisenstein hecke coefficients on gl 3 our main tools are new bounds for certain complete sums in three variables over finite fields proved using methods from algebraic geometry especially l adic cohomology and the riemann hypothesis

the distribution of realized exchange rate volatility using high frequency data on deutschemark and yen returns against the dollar we construct model free estimates of daily exchange rate volatility and correlation that cover an entire decade our estimates termed realized volatilities and correlations are not only model free but also approximately free of measurement error under general conditions which we discuss in detail hence for practical purposes we may treat the exchange rate volatilities and correlations as observed rather than latent we do so and we characterize their joint distribution both unconditionally and conditionally noteworthy results include a simple normality inducing volatility transformation high contemporaneous correlation across volatilities high correlation between correlation and volatilities pronounced and persistent dynamics in volatilities and correlations evidence of long memory dynamics in volatilities and correlations and remarkably precise scaling laws under temporal aggregation

definitizable operators and quasihyperbolic operator polynomials abstract the main concern of this paper is bounded operators a on a hilbert space with inner product which are selfadjoint in an indefinite scalar product say x y gx y and have entirely real spectrum in addition all points of spectrum are required to have determinate type a notion refining earlier ideas of krein langer et al which implies a strong stability property under perturbations the central result states that the spectrum is of this type if and only if the operator in question is uniformly definitizable i e gp a for some polynomial p as a first application characterizations of compact uniformly definitizable operators on pontrjagin spaces are obtained then the basic ideas are extended to selfadjoint monic operator polynomials via their linearizations in particular a new class of quasihyperbolic polynomials qhp with real and determinate spectrum is introduced it is shown that qhp have nontrivial monic factors the special cases of strongly hyperbolic and quadratic polynomials are also discussed in particular a factorization theorem is proved for a class of gyroscopically stabilized quadratic polynomials which originate in recent investigations of problems in mechanics

entropy and the localization of eigenfunctions we study the large eigenvalue limit for the eigenfunctions of the laplacian on a compact manifold of negative curvature in fact we only assume that the geodesic flow has the anosov property in the semi classical limit we prove that the wigner measures associated to eigenfunctions have positive metric entropy in particular they cannot concentrate entirely on closed geodesies 1 introduction statement of results we consider a compact riemannian manifold m of dimension d 2 and assume that the geodesic flow ter acting on the unit tangent bundle of m has a chaotic behaviour this refers to the asymptotic properties of the flow when time t tends to infinity ergodicity mixing hyperbolicity we assume here that the geodesic flow has the anosov property the main example being the case of negatively curved manifolds the words quantum chaos express the intuitive idea that the chaotic features of the geodesic flow should imply certain special features for the corresponding quantum dynamical system that is according to schrodinger the unitary flow exp iht t r acting on the hilbert space l2 m where a stands for the laplacian on m and h is proportional to the planck constant recall that the quantum flow converges in a sense to the classical flow g in the so called semi classical limit h 0 one can imagine that for small values ofh the quantum system will inherit certain qualitative properties of the classical flow one expects for instance a very different behaviour of eigenfunctions of the laplacian or the distribution of its eigenvalues if the geodesic flow is anosov or in the other extreme completely integrable see sa95 the convergence of the quantum flow to the classical flow is stated in the egorov theorem consider one of the usual quantization procedures op which associates an operator oph a acting on l2 m to every smooth compactly supported function a e c t m on the cotangent bundle t m according to the egorov theorem we have for any fixed t

lattice points in rational ellipsoids we combine exponential sums character sums and fourier coefficients of automorphic forms to improve the best known upper bound for the lattice error term associated to rational ellipsoids

arithmetic zeros and nodal domains on the sphere we obtain lower bounds for the number of nodal domains of hecke eigenfunctions on the sphere assuming the generalized lindelof hypothesis we prove that the number of nodal domains of any hecke eigenfunction grows with the eigenvalue of the laplacian by a very different method we show unconditionally that the average number of nodal domains of degree l hecke eigenfunctions grows significantly faster than the uniform growth obtained under lindelof

the modulus of continuity for γ0 m semi classical limits we study the behavior of a large eigenvalue limit of eigenfunctions for the hyperbolic laplacian for the modular quotient sl 2 fejer summation and results of s zelditch are used to show that the microlocal lifts of eigenfunctions have large eigenvalue limit a geodesic flow invariant measure for the modular unit cotangent bundle the limit is studied for hecke maass forms joint eigenfunctions of the hecke operators and the hyperbolic laplacian the first modulus of continuity result is presented for the limit the singular concentration set of the limit cannot be a compact union of closed geodesics and measured geodesic laminations

groups of units of zero ternary quadratic forms the groups of units of indefinite ternary quadratic forms with rational integer coefficients contain subgroups of index two which are isomorphic to fuchsian groups and which for zero forms are commensurable with the classical modular group this is used to obtain a family of forms whose groups are representatives of the conjugacy classes of maximal groups associated with zero forms the signatures of the groups of the forms in this family are determined and it is shown that the group associated to any zero form is isomorphic to a subgroup of finite index in the group of one of three particular forms this last result should be compared with the corresponding result by mennicke on non zero forms

a generalized cubic moment and the petersson formula for newforms using a cubic moment we prove a weyl type subconvexity bound for the quadratic twists of a holomorphic newform of square free level trivial nebentypus and arbitrary even weight this generalizes work of conrey and iwaniec in that the newform that is being twisted may have arbitrary square free level and also that the quadratic character may have even conductor one of the new tools developed in this paper is a more general petersson formula for newforms of square free level

on the statistical analysis of smoothing by maximizing dirty markov random field posterior distributions we consider bayesian nonparametric function estimation using a markov random field prior based on the laplace distribution we describe efficient methods for finding the exact maximum a posteriori estimate which handle constraints naturally and avoid the problems posed by nondifferentiability of the posterior distribution the methods also make links to spline and wavelet smoothers and to a dual posterior distribution three automatic smoothing parameter selection procedures are described empirical bayes two fold cross validation and a universal rule for the laplace prior monte carlo simulation with gaussian and poisson responses demonstrates that the new estimator can give better estimates of nonsmooth functions than can a similar prior based on the gaussian distribution or wavelet based competitors applications are given to spectral density estimation and to poisson image denoising

space of state vectors in symmetric quantum mechanics the space of states of symmetric quantum mechanics is examined the requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to a space with an indefinite metric self consistent expressions for the probability amplitude and the average value of operator are suggested further specification of the space of state vectors yields a superselection rule redefining the notion of the superposition principle an expression for the probability current density satisfying the equation of continuity and vanishing for the bound state is proposed

bilinear forms in weyl sums for modular square roots and applications abstract let q be a prime p 1 and let n q p denote the number of rational primes p p that split in the imaginary quadratic field q q the first part of this paper establishes various unconditional and conditional under existence of a siegel zero lower bounds for n q p in the range q 1 4 e p q for any fixed e 0 this improves upon what is implied by work of pollack and benli pollack the second part of this paper is dedicated to proving an estimate for a bilinear form involving weyl sums for modular square roots equivalently salie sums our estimate has a power saving in the so called polya vinogradov range and our methods involve studying an additive energy coming from quadratic residues in f q this bilinear form is inspired by the recent automorphic motivation the second moment for twisted l functions attached to kohnen newforms has recently been computed by the first and fourth authors so the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form these include the equidistribution of quadratic roots of primes products of primes and relaxations of a conjecture of erd s odlyzko sarkozy

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

projective hilbert space structures at exceptional points a non hermitian complex symmetric 2 2 matrix toy model is used to study projective hilbert space structures in the vicinity of exceptional points eps the bi orthogonal eigenvectors of a diagonalizable matrix are puiseuxexpanded in terms of the root vectors at the ep it is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the ep limit can be resolved by projectively extending the original hilbert space the complementary normalization conditions correspond then to two different affine charts of this enlarged projective hilbert space geometric phase and phase jump behaviour are analysed and the usefulness of the phase rigidity as measure for the distance to ep configurations is demonstrated finally ep related aspects of pt symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated

on domains of cal p cal t symmetric operators related to y x 1 nx2ny x in recent years a generalization of hermiticity has been investigated using a complex deformation h p2 x2 ix of the harmonic oscillator hamiltonian where is a real parameter these complex hamiltonians possessing symmetry the product of parity and time reversal can have a real spectrum we will consider the most simple case even in this paper we describe all self adjoint hermitian and at the same time symmetric operators associated with h p2 x2 ix surprisingly it turns out that there is a large class of self adjoint operators associated with h p2 x2 ix which are not symmetric

a krein space related perturbation theory for mhd α2 dynamos and resonant unfolding of diabolical points the spectrum of the spherically symmetric 2 dynamo is studied in the case of idealized boundary conditions starting from the exact analytical solutions of models with constant profiles a perturbation theory and a galerkin technique are developed in a krein space approach with the help of these tools a very pronounced resonance pattern is found in the deformations of the spectral mesh as well as in the unfolding of the diabolical points located at the nodes of this mesh non oscillatory as well as oscillatory dynamo regimes are obtained an estimation technique is developed for obtaining the critical profiles at which the eigenvalues enter the right spectral half plane with non vanishing imaginary components at which overcritical oscillatory dynamo regimes form finally frechet derivative gradient based methods are developed suitable for further numerical investigations of krein space related setups such as mhd 2 dynamos or models of symmetric quantum mechanics

sparse equidistribution problems period bounds and subconvexity we introduce a geometric method to bound periods of automorphic forms the key features of this method are the use of equidistribution results in place of mean value theorems and the systematic use of mixing and the spectral gap applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of cm points to subconvexity of the triple product period in the level aspect over number fields which implies subconvexity for certain standard and rankin selberg l functions and to bounding fourier coefficients of automorphic forms

localization in infinite billiards a comparison between quantum and classical ergodicity consider the non compact billiard in the first quandrant bounded by the positive x semiaxis the positive y semiaxis and the graph of f x x 1 1 2 although the schnirelman theorem holds the quantum average of the position xis finite on any eigenstate while classical ergodicity entails that the classical time average of xis unbounded

an analytic family of cal p cal t symmetric hamiltonians with real eigenvalues we study nonlinear perturbative expansions for symmetric local schrodinger operators the schrodinger operator is a sum of the harmonic oscillator hamiltonian and a local symmetric potential depending in general nonlinearly on the perturbation parameter a specific class of models having real spectrum for any value of the parameter is proposed

perturbation theory of symmetric hamiltonians in the framework of perturbation theory the reality of the perturbed eigenvalues of a class of symmetric hamiltonians is proved using stability techniques we apply this method to symmetric unperturbed hamiltonians perturbed by symmetric additional interactions

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

symmetric quantum theory defined in a krein space we provide a mathematical framework for symmetric quantum theory which is applicable irrespective of whether a system is defined on or a complex contour whether symmetry is unbroken and so on the linear space in which symmetric quantum theory is naturally defined is a krein space constructed by introducing an indefinite metric into a hilbert space composed of square integrable complex functions in a complex contour we show that in this krein space every symmetric operator is hermitian if and only if it has transposition symmetry as well from which the characteristic properties of the symmetric hamiltonians found in the literature follow some possible ways to construct physical theories are discussed within the restriction to the class k h