optimal order preconditioners for the morse ingard equations abstract the morse ingard equations of thermoacoustics morse and ingard 1986 are a system of coupled time harmonic equations for the temperature and pressure of an excited gas they form a critical aspect of modeling trace gas sensors in this paper we analyze a reformulation of the system that has a weaker coupling between the equations than the original form we give a garding type inequality for the system that leads to optimal order asymptotic finite element error estimates we also develop preconditioners for the coupled system these are derived by writing the system as a 2 2 block system with pressure and temperature unknowns segregated into separate blocks and then using either the block diagonal or block lower triangular part of this matrix as a preconditioner consequently the preconditioner requires inverting smaller helmholtz like systems individually for the pressure and temperature rigorous eigenvalue bounds are given for the preconditioned system and these are supported by numerical experiments

a novel multigrid based preconditioner for heterogeneous helmholtz problems an iterative solution method in the form of a preconditioner for a krylov subspace method is presented for the helmholtz equation the preconditioner is based on a helmholtz type differential operator with a complex term a multigrid iteration is used for approximately inverting the preconditioner the choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with fourier analysis multigrid analysis results are verified by numerical experiments high wavenumber helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner

applying gmres to the helmholtz equation with shifted laplacian preconditioning what is the largest shift for which wavenumber independent convergence is guaranteed there has been much recent research on preconditioning discretisations of the helmholtz operator delta k 2 k2 subject to suitable boundary conditions using a discrete version of the so called shifted laplacian delta k 2 mathrm i varepsilon k2 i for some varepsilon 0 0 this is motivated by the fact that as varepsilon increases the shifted problem becomes easier to solve iteratively despite many numerical investigations there has been no rigorous analysis of how to chose the shift in this paper we focus on the question of how large varepsilon can be so that the shifted problem provides a preconditioner that leads to k k independent convergence of gmres and our main result is a sufficient condition on varepsilon for this property to hold this result holds for finite element discretisations of both the interior impedance problem and the sound soft scattering problem with the radiation condition in the latter problem imposed as a far field impedance boundary condition note that we do not address the important question of how large varepsilon should be so that the preconditioner can easily be inverted by standard iterative methods

the tate shafarevich group for elliptic curves with complex multiplication let e be an elliptic curve over q with complex multiplication we give an explicit upper bound for the number of copies of qp zp which can occur in the tate shafarevich group of e for all sufficiently large good ordinary primes p

finite element approximation and preconditioners for a coupled thermal acoustic model we consider a time harmonic mathematical model for a coupled thermal acoustic model we demonstrate coercivity of the pde system and hence well posedness and optimal error estimates for standard galerkin methods the large scale and ill conditioning of the discrete system motivates the study of preconditioners three block preconditioners are studied the first two applying block jacobi and block gauss seidel approaches to our problem give provably mesh independent spectral bounds the third preconditioner based on problem specific physical reasoning does not have this property however on moderate sized meshes it considerably outperforms the existing mesh techniques numerical experiments in two and three dimensions confirm the theoretical results

a preconditioned finite element solution of the coupled pressure temperature equations used to model trace gas sensors quartz enhanced photoacoustic spectroscopy qepas is a technique for detecting trace gases which relies on a quartz tuning fork resonator to amplify and measure the weak acoustic pressure waves th

stickelberger elements and modular parametrizations of elliptic curves in the present paper we shall give evidence to support the claim conjecture i below and 1 3 that every elliptic curve a o which can be parametrized by modular functions admits a canonical modular parametrization whose properties can be related to intrinsic properties of a in particular we will see how such a parametrizat ion can be used to prove some rather pleasant integrality properties of stickelberger elements ad p adic l functions attached to a in addition if conjecture i is true then we can give an intrinsic characterization of the isomorphism class of a special elliptic curve in the q isogeny class of a distinguished by modular considerations for most of the paper we have opted for the concrete approach and defined modular parametrizations in terms of x i n definition 1 1 however to justify our view of these parametrizations as being canonical we begin here with a more intrinsic definition recall that shimura 19 chap 6 see w 1 of this paper has defined a compatible system of canonical models of modular curves xs ses where 5 p is a certain collection of open subgroups of the group gl 2 az over the finite adeles a i of q we define the adelic upper half plane to be the pro variety ll m xs and give the q structure induced by the s field of modular functions whose q expansions at the 0 cusp have coefficients in q a modular parametrization of a is a q morphism a which sends

a class of discontinuous petrov galerkin methods part iv the optimal test norm and time harmonic wave propagation in 1d the phase error or the pollution effect in the finite element solution of wave propagation problems is a well known phenomenon that must be confronted when solving problems in the high frequency range this paper presents a new method with no phase errors for one dimensional 1d time harmonic wave propagation problems using new ideas that hold promise for the multidimensional case the method is constructed within the framework of the discontinuous petrov galerkin dpg method with optimal test functions we have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm in this paper we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm this is answered in the specific case of the helmholtz equation with l 2 norm as the energy norm we obtain uniform stability with respect to the wave number we illustrate the method with a number of 1d numerical experiments using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally a 1d theoretical stability analysis is also developed

the performance of time dependent adaptive filters for interference rejection time dependent adaptive filters tdafs that allow for the cyclostationary nature of communication signals by periodically changing the filter and adaptation parameters are examined the tdaf has an advantage over the conventional time independent adaptive filter in achieving better performance i e reduced mean square error mse for signals with periodic statistics the basic theory of the tdaf is presented the tdaf is shown to be more effective than the time independent adaptive filter for interference rejection this is verified by theoretical analysis and computer simulation of specific cases of extracting a signal in noise or interference the criteria for judging the performance of the tdaf for interference rejection are mse bit error rate measurements and constellation diagrams

sommes de dedekind associees a un corps de nombres totalement reel we define dedekind sums attached to a totally real number field of class number one we prove that they satisfy some reciprocity law then we relate them to special values of hecke l functions we conclude that they are ruled by stark s conjecture

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

searching for kummer congruences in an infinite slope family we consider powers of a grossencharacter the corresponding l functions twisted with quadratic dirichlet characters and their central critical values we state several conjectures concerning kummer type congruences between these numbers for a ramified prime and describe specific numerical data in support of these conjectures

spectral correlation of modulated signals part i analog modulation the importance of the concept of cyclostationarity in design and analysis of signal detectors synchronizers and extractors in communication systems is briefly discussed and the central role of spectral correlation in the characterization of random processes that are cyclostationary in the wide sense is explained a spectral correlation function that is a generalization of the power spectral density function is described and a corresponding generalization of the wiener khinchine relation and several other fundamental spectral correlation relations also are described explicit formulas for the spectral correlation function for various types of analog modulated signals are derived this includes pulse and carrier amplitude modulation quadrature amplitude carrier modulation and phase and frequency carrier modulation to illustrate the differing spectral correlation characteristics of different modulation types the magnitudes of the spectral correlation functions are graphed or described in graphical terms as the heights of surfaces above a bifrequency plane

the elliptic apostol dedekind sums generate odd dedekind symbols with laurent polynomial reciprocity laws dedekind symbols are generalizations of the classical dedekind sums symbols there is a natural isomorphism between the space of dedekind symbols with laurent polynomial reciprocity laws and the space of modular forms we will define a new elliptic analogue of the apostol dedekind sums then we will show that the newly defined sums generate all odd dedekind symbols with laurent polynomial reciprocity laws our construction is based on machide s result j number theory 128 1060 1073 2008 on his elliptic dedekind rademacher sums as an application of our results we discover eisenstein series identities which generalize certain formulas by ramanujan collected papers of srinivasa ramanujan pp 136 162 ams chelsea publishing providence 2000 van der pol indag math 13 261 271 272 284 1951 rankin proc r soc edinburgh sect a 76 107 117 1976 and skoruppa j number theory 43 68 73 1993

from functional analysis to iterative methods we examine condition numbers preconditioners and iterative methods for finite element discretizations of coercive pdes in the context of the fundamental solvability result the lax milgram lemma working in this hilbert space context is justified because finite element operators are restrictions of infinite dimensional hilbert space operators to finite dimensional subspaces moreover useful insight is gained as to the relationship between hilbert space and matrix condition numbers and translating hilbert space fixed point iterations into matrix computations provides new ways of motivating and explaining some classic iteration schemes in this framework the simplest preconditioner for an operator from a hilbert space into its dual is the riesz isomorphism simple analysis gives spectral bounds and iteration counts bounded independent of the finite element subspaces chosen moreover the abstraction allows us to consider not only riesz map preconditioning for convection diffusion equations in h 1 but also operators on other hilbert spaces such as planar elasticity in left h 1 right 2

on l functions of elliptic curves and cyclotomic towers in this paper we shall strengthen a recent theorem of ralph greenberg 1 concerning l functions of elliptic curves and anticyclotomic towers let k be an imaginary quadratic field and m an abelian extension of k possibly of infinite degree we say that the extension m k is anticyclotomic if m is galois over q and if the nontrivial element of gal k q acts on gal m k by inversion now fix a finite set p of rational primes and consider the compositum in an algebraic closure of k of all anticyclotomic extensions of k which are unramified outside p we denote this compositum by l and refer to it as the maximal anticyclotomic extension of k unramified outside p the field l may also be described as the union of all ring class fields of k with conductor divisible only by primes in p class field theory shows that gal l k is isomorphic to the product of a finite group and the group

p adic l functions and the coniveau filtration on chow groups we give evidence for the refined version of the bloch beilinson conjecture involving coniveau filtrations by studying several infinite families of cm motives indexed by the integers r 1 that are irreducible of hodge type 2r 1 0 0 2r 1 and whose l functions vanish at the center in each case we construct a corresponding algebraic cycle that is homologically trivial but nontrivial in the top graded piece for the coniveau filtration consequently we obtain new explicit examples of cycles that are nontorsion in the griffiths group the proof of the main result relies crucially on p adic hodge theoretic methods and in particular on the relation between images of homologically trivial cycles under the p adic abel jacobi map and the values of p adic l functions

coates wiles towers in dimension two let k be an imaginary quadratic field and e an elliptic curve defined over k with complex multiplication by the ring of integers of k in 1977 coates and wiles proved that if e k has positive rank then the hasse weil zeta function l e k s of e has a zero at s 1 2 such curves exist precisely when the class number of k is one subsequent work 1 extended the result to a wider class of cm curves another proof emerged from rubin s important work on the tate shafarevich group 13 14 see also 10 12 coates and wiles used formal groups and iwasawa theoretic techniques to relate the special value of l e f s to elliptic units employing more classical techniques stark and gupta combined to give a different proof of the theorem 21 4 although the two proofs have different flavors they include the same ingredients let p be one of infinitely many suitably chosen primes of k and k the field obtained by adjoining the p torsion of e to k there is a unique prime p of k lying over p let q be a point of infinite order in e k q c pe k and 5 l q k q let v 2 be the gr6ssencharakter of e and f2 a fundamental

quadratic twists of elliptic curves in this paper we give the method of constructing non torsion points on elliptic curves which generalizes the classical birch lemma as an application we get more quadratic twist families of the elliptic curve x 0 49 which have rank one this report is a combination of the two joint works coates li tian zhai proc lond math soc 110 2 357 394 2015 4 cai li wang sci china math 59 7 1307 1326 2016 2

modulation by means of linear periodic filtering linear periodically time varying filters are considered as modulators and demodulators for the transmission of analog information over a stationary gaussian channel a complete analysis is carried out based on a certain general decomposition of such a filter into a number of branches each containing a linear time invariant filter and a multiplication with a complex sinusoid receiver optimization is carried out under the assumption that the above decomposition is finite finally a joint optimization of transmitter and receiver is derived under the further restriction that the message signal is strictly band limited

elliptic apostol sums and their reciprocity laws we introduce an elliptic analogue of the apostol sums which we call elliptic apostol sums these sums are defined by means of certain elliptic functions with a complex parameter r having positive imaginary part when t i these elliptic apostol sums represent the well known apostol generalized dedekind sums also these elliptic apostol sums are modular forms in the variable r we obtain a reciprocity law for these sums which gives rise to new relations between certain modular forms of one variable

on the conjecture of birch and swinnerton dyer in this paper we complete rubin s partial verification of the conjecture for a large class of elliptic curves with complex multiplication by q c

cyclic wiener filtering theory and method conventional time and space filtering of stationary random signals which amounts to forming linear combinations of time translates and space translates exploits the temporal and spatial coherence of the signals by including frequency translates as well the spectral coherence that is characteristic of cyclostationary signals can also be exploited some of the theoretical concepts underlying this generalized type of filtering called frequency shift fresh filtering are developed the theory of optimum fresh filtering which is a generalization of wiener filtering called cyclic wiener filtering is summarized and the theory is illustrated with specific examples of separating temporally and spectrally overlapping communications signals including am bpsk and qpsk the structures and performances of optimum fresh filters are presented and adaptive adjustment of the weights in these structures is discussed also specific results on the number of digital qam signals that can be separated as a function of excess bandwidth are obtained

elliptic curves with complex multiplication and the conjecture of birch and swinnerton dyer 1 quick review of elliptic curves 2 2 elliptic curves over c 4 3 elliptic curves over local fields 6 4 elliptic curves over number fields 12 5 elliptic curves with complex multiplication 15 6 descent 22 7 elliptic units 27 8 euler systems 37 9 bounding ideal class groups 43 10 the theorem of coates and wiles 47 11 iwasawa theory and the main conjecture 50 12 computing the selmer group 61

fast iterative solution of stabilised stokes systems part i using simple diagonal preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems the use of stabilisation methods for convenient but unstable mixed elements introduces stabilisation parameters we show how these can be chosen to obtain rapid iterative convergence we propose a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems describe the effects of stabilisation on the algebraic structure of the discrete stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends here we discuss the simple case of diagonal preconditioning our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable we demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete laplacian are approachable for the stokes problem