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

random rotations of the wiener path let w h be an abstract wiener space and letr w be a strongly measurable random variable with values in the set of isometries onh suppose that rh is smooth in the sobolev sense and that it is a quasi nilpotent operator onh for everyh h it is shown that r w h is again a gaussian 0 h 2 random variable consequently if e i i w is a complete orthonormal basis ofh then tilde w sum nolimits i delta r w e i e i defines a measure preserving transformation a rotation onw it is also shown that if for some strongly measurable operator valued onh random variabler r w k h is 0 h 2 gaussian for allk h h thenr is an isometry and rh is quasi nilpotent for allh h the relation between the stochastic calculi for these wiener pathsw and tilde w as well as the conditions of the inverbibility of the map w to tilde w are discussed and the problem of the absolute continuity of the image of the wiener measure under euclidean motion on the wiener space i e w to tilde w composed with a shift is studied

an effective method for solving the hyper singular integral equations in 3 d acoustics the application of the boundary integral methods to the problem of acoustics exterior to a three dimensional surface suffers in general from the nonexistence of nonuniqueness of its solutions at frequencies that are characteristic of the associated interior problem the formulation that is most suitable for numerical implementation still appears to be that proposed by burton and miller proc r soc london ser a 323 201 210 1971 however the hypersingular kernels present in such a formulation render it computationally unattractive previous attempts to regularize such hypersingular kernels involved the use of double surface integrals or implicit use of tangential operators or closed form evaluations of hypersingular integrals the method of double surface integrals is computationally highly inefficient even though it allows higher order interpolation schemes on the surface the other two approaches are more conducive to the assumption of a constant value for each of the variables pressure a

numerical techniques for three dimensional steady state fluid structure interaction the application of a coupled finite element boundary integral approach to the analysis of vibrating elastic arbitrary shaped three dimensional structures in an infinite media is presented a feature of the methodologies implemented is the utilization of the isoparametric concept from finite element theory the same shape functions are used both to model the surface of the structure and to interpolate the acoustic variables over the radiating surface the boundary integral formulation used for the exterior acoustic field is a formulation that ensures uniqueness and existence of solutions for the entire frequency range an assessment of the accuracy was made after comparing the solutions attained for a vibrating submerged sphere for various excitation loads and frequencies with the exact solution

girsanov identities for poisson measures under quasi nilpotent transformations we prove a girsanov identity on the poisson space for anticipating transformations that satisfy a strong quasi nilpotence condition applications are given to the girsanov theorem and to the invariance of poisson measures under random transformations the proofs use combinatorial identities for the central moments of poisson stochastic integrals

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

the vanishing of l2 harmonic one forms on based path spaces abstract we prove the triviality of the first l 2 cohomology class of based path spaces of riemannian manifolds furnished with brownian motion measure and the consequent vanishing of l 2 harmonic one forms we give explicit formulae for closed and co closed one forms expressed as differentials of functions and co differentials of l 2 two forms respectively these are considered as extended clark ocone formulae a feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one forms to the exterior differentiation operator used to construct the de rham complex and the self adjoint laplacian on l 2 one forms this laplacian is shown to have a spectral gap

tangent processes on wiener space abstract this paper deals with the study of the malliavin calculus of euclidean motions on wiener space i e transformations induced by general measure preserving transformations called rotations and h valued shifts and the associated flows on abstract wiener spaces

on the numerical implementation of a cauchy principal value integral to insure a unique solution for acoustic radiation and scattering in this paper the boundary integral formulation proposed by burton and miller proc r soc london ser a 323 201 210 1971 to insure a unique solution for all frequencies is implemented in an isoparametric element environment a regularized normal derivative integral equation originally derived by maue z phys 126 601 618 1949 is used in the formulation to form a linear combination with the conventional helmholtz integral equation this regularized normal derivative integral equation converges in the cauchy principal value sense rather than only in the finite part sense the cauchy principal value integral can be further transformed into an integral that converges in the normal sense the c continuous isoparametric elements are used in the formulation collocation points are placed inside each element to insure a unique normal direction and continuity of tangential derivatives of the acoustic pressure through a systematic collocation point generation scheme the number of collocation point

numerical modeling of acoustic radiation and scattering from thin bodies using a cauchy principal integral equation in this paper the helmholtz integral equation and its normal derivative are used for the numerical solution of acoustic radiation and scattering from thin bodies a regularized form of the normal derivative integral equation originally derived by maue is adopted to calculate the pressure jump across the thin body this regularized normal derivative integral equation converges in the cauchy principal value sense rather than only in the finite part sense the cauchy principal value integral can be further transformed into an integral that converges in the normal sense the c0 continuous isoparametric elements are used in the formulation so that the numerical model is compatible with other analysis tools such as the finite element method collocation points are placed inside each element to insure a unique normal direction and the c1 continuity condition the pressure jump is forced to be zero at the knife edge where the 1 r 1 2 singularity of the tangential velocities is modeled by the quarter point tec

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

a weighted residual formulation for the chief method in acoustics in this article the combined helmholtz integral equation formulation chief proposed by schenck is greatly improved so that it will alleviate the annoying nodal surface problem the constraint provided by each chief equation and its first order derivatives is enforced in a weighted residual sense over a small interior region rather than only at a single point consequently chief blocks rather than chief points are used in the numerical implementation although the corresponding interior problem may have nodal surfaces at characteristic frequencies it has no such things as nodal blocks therefore the chief block chief b method is expected to work even if all of the chief blocks fall on the intersection of several nodal surfaces this new formulation is implemented in an isoparametric element environment numerical examples in spherical radiation and scattering at intermediate frequencies are given to test the new formulation

absolutely continuous flows generated by sobolev class vector fields in finite and infinite dimensions abstract we prove the existence of the global flow ut generated by a vector field a from a sobolev class w1 1 on a finite or infinite dimensional space x with a measure provided is sufficiently smooth and that a a and a where a is the divergence with respect to are exponentially integrable in addition the measure is shown to be quasi invariant under ut in the case x r n and p dx where p w1 1loc r n is a locally uniformly positive probability density a sufficient condition is exp c a exp c a p p l1 for all c in the infinite dimensional case we get analogous results for measures differentiable along sufficiently many directions examples of measures which fit our framework important for applications are symmetric invariant measures of infinite dimensional diffusions and gibbs measures typically in both cases such measures are essentially non gaussian our result in infinite dimensions significantly extends previously studied cases where was a gaussian measure finally we study flows generated by vector fields whose values are not necessarily in the cameron martin space

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

bounds for rankin selberg integrals and quantum unique ergodicity for powerful levels reference epfl article 202508doi 10 1090 s0894 0347 2013 00779 1view record in web of science record created on 2014 10 23 modified on 2017 12 10

iterative solution of high order boundary element method for acoustic impedance boundary value problems abstract a high order boundary element method for time harmonic acoustic impedance boundary value problems is presented the method is based on the galerkin type formulation of the burton miller integral equation bmie with high order polynomial basis and testing functions this formulation has several important advantages it is free of the interior resonance problem the hypersingular integral operator of the traditional bmie formulation can be avoided it leads to faster convergence in terms of the number of unknowns than the low order methods and it shows a good performance with iterative solvers to avoid the numerical difficulties associated to the implementation of the singular surface integral equations the singularity extraction technique is applied to evaluate the integrals in the singular and near singular cases the resulting matrix equation is solved iteratively with the generalized minimal residual method gmres and a simple preconditioner based on the incomplete lu factorization is applied to expedite the convergence numerical results indicate that the bmie formulation with galerkin method and high order basis functions has good convergence properties for various geometries and boundary conditions on a wide frequency range when the gmres method is applied to solve the matrix equation iteratively this in turn indicates that the formulation is well suited for an efficient application of fast solution procedures such as the fast multipole method

a comparison of techniques for overcoming non uniqueness of boundary integral equations for the collocation partition of unity method in two dimensional acoustic scattering the partition of unity method has become an attractive approach for extending the allowable frequency range for wave simulations beyond that available using piecewise polynomial elements the non uniqueness of solution obtained from the conventional boundary integral equation cbie is well known the cbie derived through green s identities suffers from a problem of non uniqueness at certain characteristic frequencies two of the standard methods of overcoming this problem are the so called combined helmholtz integral equation formulation chief method and that of burton and miller the latter method introduces a hypersingular integral which may be treated in various ways in this paper we present the collocation partition of unity boundary element method pubem for the helmholtz problem and compare the performance of chief against a burton miller formulation regularised using the approach of li and huang

transformation of wiener measure under anticipative flows lett f be a transformation from the wiener space to itself with the range off assumed to be in the cameron martin space the absolute continuity and the density function associated witht is considered t is assumed to be embedded in or defined through a parameterizationt t f t andf t is assumed to be differentiable int the paper deals first with the case where the range of thet derivative off t is also in the cameron martin space and new representations for the radon nikodym derivative and the carleman fredholm determinant are derived the case where thet derivative off t is not in the cameron martin space is considered next and results on the absolute continuity and the density function under conditions which are considerably weaker than previously known conditions are presented

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

a method to overcome computational difficulties in the exterior acoustics problem when the boundary integral equation method is applied to exterior acoustics problems singularities occur in the resulting algebraic equations at various frequencies associated with the eigenvalues of an interior problem these frequencies are referred to as forbidden and various methods have been devised to overcome the computational difficulties presented at these frequencies the work presented here is an extension to the chief method in that higher derivatives in addition to the function itself are constrained to be zero at selected points in the interior domain whereas the relative success of either method depends on the quantity and selection of interior points the superchief method requires fewer interior points and is less sensitive to point selection resulting in improved robustness without a significant increase in computational complexity

laplace transform identities and measure preserving transformations on the lie wiener poisson spaces abstract given a divergence operator on a probability space such that the law of h is infinitely divisible with characteristic exponent 0 1 h 1 2 0 h t 2 d t or 0 e i h t i h t 1 d t h l 2 r we derive a family of laplace transform identities for the derivative e e u when u is a non necessarily adapted process these expressions are based on intrinsic geometric tools such as the carleman fredholm determinant of a covariant derivative operator and the characteristic exponent 0 1 in a general framework that includes the wiener space the path space over a lie group and the poisson space we use these expressions for measure characterization and to prove the invariance of transformations having a quasi nilpotent covariant derivative for gaussian and other infinitely divisible distributions

a boundary node method for airfoils based on the dirichlet condition abstract a nonsingular integral equation is developed for thin airfoils with the dirichlet boundary condition both singular and near singular kernels which accompany the original integral equation are flattened out after such regularizations the difficulty in the numerical quadrature is greatly reduced a program with an arbitrary order integral formulation becomes available in the present approach two kinds of airfoils the van de vooren and joukowski airfoils are analyzed for demonstration in this paper in comparison with the results from constant doublet methods the results from the proposed methods are shown to match the analytic solutions excellently

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

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

a novel technique in boundary integral equations for analyzing acoustic radiation from axisymmetric bodies a novel technique is suggested to avoid singular integrals of boundary integral equations bie for exterior sound radiation problems the technique reproduction of diagonal terms rdt utilizes special solutions of some monopole sources which give the distribution of theoretical surface pressure and velocity to reproduce diagonal terms in the bie matrices it is proposed to employ three monopole sources in reproduction of diagonal terms in both pressure and velocity matrices although the interior points are applied to generate monopole solutions the concept is basically different from that of general interior helmholtz equations so the present technique is different from those traditional methods to improve bie method through added interior helmholtz equations several examples are presented to demonstrate the effectiveness of the proposed technique for closed axisymmetric surfaces numerical results show that the technique of rdt can improve the accuracy of the solution up to 10 times compared to the conventional techniques such as shf chief shf

applications of the degree theorem to absolute continuity on wiener space let h p be an abstract wiener space and define a shift on byt f wheref is anh valued random variable we study the absolute continuity of the measurespot 1and f p ot 1 with respect top using the techniques of the degree theory of wiener maps where f det2 1 f exp f 1 2 f 2

solving the hypersingular boundary integral equation in three dimensional acoustics using a regularization relationship regularization of the hypersingular integral in the normal derivative of the conventional helmholtz integral equation through a double surface integral method or regularization relationship has been studied by introducing the new concept of discretized operator matrix evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices such a treatment greatly improves the computational efficiency as the number of frequencies to be computed increases the computational cost of solving the composite helmholtz integral equation is comparable to that of solving the conventional helmholtz integral equation in this paper the detailed formulation of the proposed regularization method is presented the computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems the radiation of a pulsating sphere an oscillating sphere and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements it is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied

acoustic scattering by a hard or soft body across a wide frequency range by the helmholtz integral equation method in this paper the acoustic scattering by an obstacle across a wide frequency range of sound waves is investigated on the basis of the helmholtz integral formulation to overcome the nonuniqueness difficulties the methods proposed by burton npl report nac 30 jan 1973 and by burton and miller proc r soc london ser a 323 201 210 1971 are adopted for the dirichlet and neumann problems respectively the aim of this paper is twofold the first is to bring together completely regular formulations of the helmholtz integral equation and its normal derivative the second is to extend these formulations to treat the higher frequency problems the weakly singular integrals are regularized by subtracting out one term and adding it back depending on the problem concerned the additional integral can finally be expressed in an explicit form or results in solving a surface source distribution of the equipotential body the hypersingular kernels are regularized by the method of using some properties of the associated laplace equation originally proposed by chien et al j acoust soc am 88 918 937 1990 the completely regularized integral equations are amenable to computation by direct use of the standard quadrature methods to study the acoustic scattering due to higher frequency waves filon s quadrature method proc r soc edinburgh 49 38 47 1928 is extended to treat the rapidly oscillatory integrands numerical examples consist of acoustic scattering from a hard or soft sphere of radius a across a wide spectrum of wave numbers ka 20 comparisons of the numerical results with the exact solutions demonstrate the validity and efficiency of the implementation

a boundary integral equation method for two dimensional acoustic scattering problems this work formulates the singularity free integral equations to study 2 d acoustic scattering problems to avert the nonuniqueness difficulties burton s and burton and miller s methods are employed to solve the dirichlet and neumann problems respectively the surface helmholtz integral equations and their normal derivative equations in bounded form are derived the weakly singular integrals are desingularized by subtracting a term from the integrand and adding it back with an exact value depending on the relevant problem the additional integral can finally be either expressed in an explicit form or transformed to form a surface source distribution of the related equipotential body the hypersingular kernel is desingularized further using some properties of an interior laplace problem the new formulations are advantageous in that they can be computed by directly using standard quadrature formulas also discussed is the contour a unique feature of 2 d problems instead of dividing the boundary surface into several small elements a parametric representation of a 2 d boundary curve is further proposed to facilitate a global numerical implementation calculations consist of acoustic scattering by a hard and a soft circular cylinder respectively comparing the numerical results with the exact solutions demonstrates the proposed method s effectiveness

trilinear forms and the central values of triple product l functions we give an explicit formula for certain global trilinear forms that appear in jacquet s conjecture in terms of local trilinear forms and the central values of triple product l functions

a mean value of a triple product of l functions luo has proven an optimal upper bound for the l 4 norm of dihedral maass forms of large eigenvalue by bounding a mean value of triple product l functions motivated by this result we study a mean value of l functions having similar shape and obtain for it an asymptotic with power savings our work may be helpful in eventually obtaining an asymptotic for the l 4 norm

hypersingular boundary integral equations for exterior acoustic problems a bounded form of the hypersingular helmholtz integral equation for 3 d acoustic problems is developed in this paper all integrals with singular kernels are regularized within each regular surface element in the sense of kellogg with twice continuous differentiability the standard gaussian quadrature formula is applied without any special treatment for all the computational points the collocation points are chosen to be the gauss legendre nodes and no interpolation function is assumed for acoustic variables

an investigation into integral equation methods involving nearly singular kernels for acoustic scattering this work presents a numerical means of investigating the acoustic scattering by disk shaped bodies on the basis of burton and miller s method the singularity free formulations of helmholtz integral equation and its normal derivative are used to form a composite equation a triangle polar co ordinate transformation method is further applied to treat the nearly singular kernels arising from a situation in which the field points and source points are very close together numerical simulations consist of the acoustic scattering by a short circular cylinder and a thin circular disk respectively for the latter case with zero thickness the corresponding analytical solutions involving angular and radial oblate spheroidal wave functions are evaluated as well comparing the numerical results with the experimental data and analytical solutions demonstrates the effectiveness of the proposed method

exponential sums and the riemann zeta function v a van der corput exponential sum is s sigma exp 2 pi i f m where m has size m the function f x has size t and alpha log m log t 1 there are different bounds for s in different ranges for alpha in the middle range where alpha is near 1 over 2 s o sqrt m t theta epsilon this theta bounds the exponent of growth of the riemann zeta function on its critical line rm re s 1 over 2 van der corput used an iteration which changed alpha at each step the bombieri iwaniec method whilst still based on mean squares introduces number theoretic ideas and problems the second spacing problem is to count the number of resonances between short intervals of the sum when two arcs of the graph of y f x coincide approximately after an automorphism of the integer lattice in the previous paper in this series proc london math soc 3 66 1993 1 40 and the monograph area lattice points and exponential sums we saw that coincidence implies that there is an integer point close to some resonance curve one of a family of curves in some dual space now calculated accurately in the paper resonance curves in the bombieri iwaniec method which is to appear in funct approx comment math we turn the whole bombieri iwaniec method into an axiomatised step an upper bound for the number of integer points close to a plane curve gives a bound in the second spacing problem and a small improvement in the bound for s ends and cusps of resonance curves are treated separately bounds for sums of type s lead to bounds for integer points close to curves and another branching iteration luckily swinnerton dyer s method is stronger we improve theta from 0 156140 in the previous paper and monograph to 0 156098 in fact 32 205 epsilon 269 410 epsilon is an exponent pair for every epsilon 0

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

on the uniform integrability of the radon nikodym densities for wiener measure abstract we give sufficient conditions for the uniform integrability of the radon nikodym derivatives of the images of wiener measure under the non linear transformations of the wiener space these results are then applied to the construction of the functional inverses of these transformations

acoustic isogeometric boundary element analysis an isogeometric boundary element method based on t splines is used to simulate acoustic phenomena we restrict our developments to low frequency problems to establish the fundamental properties of the proposed approach using t splines the computer aided design cad and boundary element analysis are integrated without recourse to geometry clean up or mesh generation a regularized burton miller formulation is used resulting in integrals which are at most weakly singular we employ a collocation based approach to generate the linear system of equations the method is verified against closed form solutions and direct comparisons are made with conventional lagrangian discretizations it is demonstrated that the superior accuracy of the isogeometric approach emanates from the exact geometric description encapsulated in the t spline the method is then applied to a real world application to illustrate the potential for integrated engineering design and analysis

non singular boundary integral formulations for plane interior potential problems in this article a non singular formulation of the boundary integral equation is developed to solve smooth and non smooth interior potential problems in two dimensions the subtracting and adding back technique is used toregularize the singularity of green s function and to simplify the calculation of the normal derivative of green s function after that a global numerical integration is directly applied at the boundary and those integration points are also taken as collocation points to simplify the algorithm of computation the result indicates that this simple method gives the convergence speed of order n 3 in the smooth boundary cases for both dirichlet and mix type problems for the non smooth cases the convergence speed drops at o n 1 2 for the dirichlet problems

transformation of the wiener measure under non invertible shifts in this paper we consider the transformation of measure induced by a not necessarily invertible perturbation of the identity the radon nikodym density for the image of the wiener measure and the associated girsanov type density are derived an application of these results yields an extension of the degree theorem

a boundary integral method for acoustic radiation and scattering in this article a computational method is proposed for computing the helmholtz integral equation for acoustic radiation and scattering problems with a three dimensional body shape unlike the previous boundary element method the integration of acoustic properties is based on the global body surface instead of local elements the approximation function of body geometry and the weighting function of numerical integration are independently chosen when the singular kernels of helmholtz equation are regularized as bounded discontinuous functions the order of integration polynomials for acoustic properties can be arbitrarily chosen therefore the accuracy and efficiency of computation can be increased