maass cusp forms abstract it is shown that under certain standard assumptions such as extended riemann hypotheses the scattering matrix s for generic sl 2 r is unexpectedly of order 2 this leads to the conjecture that the generic cofinite has very few maass cusp forms

nonvanishing of l values and the weyl law this work is concerned with the validity of weyl law for hyperbolic surfaces on the asymptotic counting of the laplace eigenvalues following phillipssarnak we show that weyl law is false for generic hyperbolic surfaces under the standard multiplicity assumption by establishing that a positive proportion of certain critical values of rankin selberg l functions do not vanish

analytical level crossing rates and average fade durations for diversity techniques in nakagami fading channels the level crossing rates lcrs and average fade durations afds of a fading channel find diverse applications in the evaluation and design of wireless communication systems analytical expressions for these quantities are available in the literature for certain diversity reception techniques but are generally limited to the rayleigh fading channel with few exceptions moreover the methods employed are usually specific to a certain channel diversity pair and thus cannot be applied to all cases of interest using a unified methodology we derive analytical expressions for the lcrs and afds for three diversity reception techniques and a general nakagami 1960 fading channel we provide novel analytical expressions for selection combining sc and equal gain combining egc and rederive in a more general manner the case of maximal ratio combining mrc it is shown that our general results reduce to some specific cases previously published these results are used to examine the effects of the diversity technique the number of receiving branches and severity of the fading on the concerned quantities it is observed that as the nakagami m parameter and the diversity order increase the behavior of the combined received envelope for egc follows closely the one for mrc and distances itself from sc

spectral simplicity and asymptotic separation of variables we describe a method for comparing the spectra of two real analytic families at and qt of quadratic forms that both degenerate as a positive parameter t tends to zero we suppose that the family at is amenable to separation of variables and that each eigenspace of at is 1 dimensional for some t we show that if qt is asymptotic to at at first order as t 0 then the eigenspaces of qt are also 1 dimensional for all but countably many t as an application we prove that for the generic triangle simplex in euclidean space constant curvature space form each eigenspace of the laplacian acting on dirichlet functions is 1 dimensional

disappearance of cusp forms in special families in his 1954 g6ttingen lectures atle selberg sb posed the question of the asymptotic density of the poles of the eisenstein series for r a cofinite non cocompact subgroup of psl 2 r the question is known to be equivalent to the asymptotic count of the number of eigenvalues for the laplacebeltrami operator d acting in l2 h r where h is the upper halfplane with hyperbolic metric in general the spectral decomposition of d consists of the continuous band 1 4 oo with multiplicity the number of cusps of h r and a discrete component possibly in 1 4 0o spanned by eigenfunctions w1i wp2 see hj1 kb lxp for the eigenvalues aj dpj ajpj o consider the spectral counting function n r ij v n 0 see dips ps2 cor 2 3 luo lu recently established the unconditional result his argument does not involve the lindel6f hypothesis deshouillers et

persistence of embedded eigenvalues we consider conditions under which an embedded eigenvalue of a self adjoint operator remains embedded under small perturbations in the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m infinity we show that in favorable situations the set of small perturbations of a suitable banach space which do not remove the eigenvalue form a smooth submanifold of codimension in we also have results regarding the cases when the eigenvalue is degenerate or when the multiplicity of the continuous spectrum is infinite

the time dependent maxwell system with measurable coefficients in lipschitz domains abstract semigroup theory is employed to study inhomogeneous boundary conditions for the time dependent maxwell equations with anisotropic material parameters in a non smooth domain only boundedness and measurability of the coefficients are assumed and the boundary data is assumed to be time independent when the material parameters are positive definite a higher order sobolev theory is developed for the time dependent maxwell system

the second moment of gl 4 gl 2 l functions at special points abstract in this paper we obtain upper bounds for the second moment of l u j 1 2 i t j where is a hecke maass form for s l 4 z and u j is taken from an orthonormal basis of hecke maass forms on s l 2 z with eigenvalue 1 4 t j 2 the bounds are consistent with the lindelof hypothesis previously these types of upper bounds are available for only g l n g l 2 where n 3

erratum to spectral simplicity and asymptotic separation of variables the statement of lemma a 3 in hlljdg11 is false 1 this lemma is used in the proof of lemma 8 2 which is also incorrect and is used in the proof of proposition 9 1 despite this gap proposition 9 1 is correct and in order to rectify the situation we provide here the correct estimates needed to derive proposition 9 1 as a consequence the statements of the main results of the article hlljdg11 are correct we would like to emphasize that the overall strategy of the proof of proposition 9 1 remains unchanged in particular proposition 9 1 is a statement about concentration properties of quasimodes for the quadratic form a t coming from separation of variables more precisely it relies on the fact that a quasimode of order t for a t at a non critical energy cannot concentrate on the turning point and thus must have some mass in the classically allowed region in the exposition given in hlljdg11 this non concentration was hidden behind lemmas 9 4 and 9 5 we will make it more transparent here by directly using the langer cherry transform and the following estimate for solutions to the semiclassical airy equation 2

deformations of maass forms we describe numerical calculations which examine the phillips sarnak conjecture concerning the disappearance of cusp forms on a noncompact finite volume riemann surface s under deformation of the surface our calculations indicate that if the teichmuller space of s is not trivial then each cusp form has a set of deformations under which either the cusp form remains a cusp form or else it dissolves into a resonance whose constant term is uniformly a factor of 10 8 smaller than a typical fourier coefficient of the form we give explicit examples of those deformations in several cases

on the distribution of eigenvalues of maass forms on certain moonshine groups in this paper we study both analytically and numerically questions involving the distribution of eigen values of maass forms on the moonshine groups 0 n where n 1 is a square free integer after we prove that 0 n has one cusp we compute the constant term of the associated non holomorphic eisenstein series we then derive an average weyl s law for the distribution of eigenvalues of maass forms from which we prove the classical weyl s law as a special case the groups corresponding to n 5 and n 6 have the same signature however our analysis shows that asymptotically there are infinitely more cusp forms for 0 5 than for 0 6 we view this result as being consistent with the phillips sarnak philosophy since we have shown unconditionally the existence of two groups which have different weyl s laws in addition we employ hejhal s algorithm together with recently developed refinements from 32 and numerically determine the first 3557 of 0 5 and the first 12474 eigenvalues of 0 6 with this information we empirically verify some conjectured distributional properties of the eigenvalues

on a structural observation in generalized electromagnetic theory are satisfied equations 0 2 describe the electromagnetic properties of the medium where 0 1 is considered if this medium is confined to a certain region g of r3 then the electromagnetic properties of its complent cc are usually represented via boundary conditions on dg we shall deal here with the boundary condition appearing if cc is assumed to be a perfect conductor i e nxe t o on c7g 13 0 0 3

asymptotics of dirichlet eigenvalues and eigenfunctions of the laplacian on thin domains in rd abstract we consider the laplace operator with dirichlet boundary conditions on a domain in r d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero this generalizes our previous results in two dimensions and as in that case allows us to obtain an approximation for dirichlet eigenvalues for a large class of domains under very mild assumptions as an application we derive a three term asymptotic expansion for the first eigenvalue of d dimensional ellipsoids

deformation of compact clifford klein forms of indefinite riemannian homogeneous manifolds 1 1 let g be a lie group and h a closed subgroup of g if a discrete subgroup of g acts properly discontinuously and freely on g h then the double coset space g h carries naturally a manifold structure such that the quotient map g h g h is locally diffeomorphic the manifold g h is said to be a clifford klein form of g h if it is compact then is said to be a uniform lattice for g h a typical example is a compact riemann surface mg with genus g 2 which is biholomorphic to a compact clifford klein form of the poincare disk g h sl 2 r so 2 by the uniformization theorem it is important from geometric view point that a clifford klein form g h inherits any g invariant local geometric structure on g h such as indefinite riemannian metric complex structure symplectic structure causal structure and so on

computations of eisenstein series on fuchsian groups we present numerical investigations of the value distribution and distribution of fourier coefficients of the eisenstein series e z s on arithmetic and non arithmetic fuchsian groups our numer

on the second order statistics of the multihop rayleigh fading channel second order statistics provides a dynamic representation of a fading channel and plays an important role in the evaluation and design of the wireless communication systems in this paper we present a novel analytical framework for the evaluation of important second order statistical parameters as the level crossing rate lcr and the average fade duration afd of the amplify and forward multihop rayleigh fading channel more specifically motivated by the fact that this channel is a cascaded one and can be modeled as the product of n fading amplitudes we derive novel analytical expressions for the average lcr and the afd of the product of n rayleigh fading envelopes or of the recently so called nrayleigh channel furthermore we derive simple and efficient closed form approximations to the aforementioned parameters using the multivariate laplace approximation theorem it is shown that our general results reduce to the corresponding ones of the specific dual hop case previously published numerical and computer simulation examples verify the accuracy of the presented mathematical analysis and show the tightness of the proposed approximations

perturbation of embedded eigenvalues in the generalized n body problem we discuss the perturbation of continuum eigenvalues without analyticity assumptions among our results we show that generally a small perturbation removes these eigenvalues in accordance with fermi s golden rule thus generically in a baire category sense the schrodinger operator has no embedded non threshold eigenvalues

n ast nakagami a novel stochastic model for cascaded fading channels a generic and novel distribution referred to as nakagami constructed as the product of n statistically independent but not necessarily identically distributed nakagami m random variables rvs is introduced and analyzed the proposed distribution turns out to be a very convenient tool for modelling cascaded nakagami m fading channels and analyzing the performance of digital communications systems operating over such channels the moments generating probability density cumulative distribution and moments functions of the n nakagami distribution are developed in closed form using the meijer s g function using these formulas generic closed form expressions for the outage probability amount of fading and average error probabilities for several binary and multilevel modulation signals of digital communication systems operating over the n nakagami fading and the additive white gaussian noise channel are presented complementary numerical and computer simulation performance evaluation results verify the correctness of the proposed formulation the suitability of the n nakagami fading distribution to approximate the lognormal distribution is also being investigated using kolmogorov smirnov tests the rate of convergence of the central limit theorem as pertaining to the multiplication of nakagami m rvs is quantified

finite element methods with matching and nonmatching meshes for maxwell equations with discontinuous coefficients we investigate the finite element methods for solving time dependent maxwell equations with discontinuous coefficients in general three dimensional lipschitz polyhedral domains both matching and nonmatching finite element meshes on the interfaces are considered and optimal error estimates for both cases are obtained the analysis of the latter case is based on an abstract framework for nested saddle point problems along with a characterization of the trace space for h curl d a new extension theorem for h curl d functions in any lipschitz domain d and a novel compactness argument for deriving discrete inf sup conditions

laplace approximations for hypergeometric functions with matrix argument in this paper we present laplace approximations for two functions of matrix argument the type i confluent hypergeometric function and the gauss hypergeometric function both of these functions play an important role in distribution theory in multivariate analysis but from a practical point of view they have proved challenging and they have acquired a reputation for being difficult to approximate appealing features of the approximations we present are i they are fully explicit and simple to evaluate in practice and ii typically they have excellent numerical accuracy the excellent numerical accuracy is demonstrated in the calculation of noncentral moments of wilks a and the likelihood ratio statistic for testing block independence and in the calculation of the cdf of the noncentral distribution of wilks a via a sequential saddlepoint approximation relative error properties of these approximations are also studied and it is noted that the approximations have uniformly bounded relative errors in important cases

bounds for multihop relayed communications in nakagami m fading we present closed form lower bounds for the performance of multihop transmissions with nonregenerative relays over not necessarily identically distributed nakagami m fading channels the end to end signal to noise ratio is formulated and upper bounded by using an inequality between harmonic and geometric means of positive random variables rvs novel closed form expressions are derived for the moment generating function the probability density function and the cumulative distribution function of the product of rational powers of statistically independent gamma rvs these statistical results are then applied to studying the outage probability and the average bit error probability for phase and frequency modulated signaling numerical examples compare analytical and simulation results verifying the tightness of the proposed bounds

on selberg s small eigenvalue conjecture and residual eigenvalues we show that selberg s eigenvalue conjecture concern ing small eigenvalues of the automorphic laplacian for congruence groups is equivalent to a conjecture about the non existence of residual eigenvalues for a perturbed system we prove this using a combination of methods from asymptotic perturbation theory and number theory

deformation of γ0 5 cusp forms we develop an algorithm for numerical computation of the eisenstein series on a riemann surface of constant negative curvature we focus in particular on the computation of the poles of the eisenst