performance advantage of complex lms for controlling narrow band adaptive arrays in narrow band adaptive array applications the mean square convergence of the discrete time real least mean square lms algorithm is slowed by image frequency noises generated in the lms loops the complex lms algorithm proposed by widrow et al is shown to eliminate these noises yielding convergence of the mean squared error mse at slightly over twice the rate this paper includes a comprehensive analysis of the mse of adaptation for lms the analysis is based upon the method developed in the 1968 dissertation by k d senne and it represents the most complete treatment of the subject published to date

class groups of number fields numerical heuristics extending previous work of h w lenstra jr and the first author we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four given the signature and the galois group of the galois closure the theoretical justifications for these conjectures will appear elsewhere but the agreement with the existing tables is quite good 1 introduction and notations in 3 h w lenstra jr and the first author developed a method for conjecturing quantitative results on class groups of quadratic fields and cyclic extensions of prime degree in a forthcoming paper 4 we shall show that this technique can be extended to a much wider class of number fields and also to relative extensions the aim of the present paper is to rapidly make available the numerical conjec tures obtained for people not really interested in our heuristic reasoning or not wanting to wait for 4 to appear hence apart from a total lack of justifications for the conjectures that we present this paper is essentially self contained the plan is as follows in the rest of this section we present the notations used in the sequel some of them being nonstandard and in general differing from the notations of 3 we urge the reader to read the notations carefully before applying the conjectures in the next section we present templates for the subsequent conjectures and then the conjectures themselves illustrated by numerical examples first for their own sake and second as a double check for the reader to understand the templates these conjectures are given for all types of fields of degree less than or equal to four in the final section we comment on the consistency of the conjectures with existing tables which is quite good combinatorial notations if x is a set xi denotes its cardinality

on class groups of random number fields the main aim of the present paper is to disprove the cohen lenstra martinet heuristics in two different ways and to offer possible corrections we also recast the heuristics in terms of arakelov class groups giving an explanation for the probability weights appearing in the general form of the heuristics we conclude by proposing a rigorously formulated cohen lenstra martinet conjecture

error estimates for the davenport heilbronn theorems we obtain the first known power saving remainder terms for the theorems of davenport and heilbronn on the density of discriminants of cubic fields and the mean number of 3 torsion elements in the class groups of quadratic fields in addition we prove analogous error terms for the density of discriminants of quartic fields and the mean number of 2 torsion elements in the class groups of cubic fields these results prove analytic continuation of the related dirichlet series to the left of the line r s 1

an optimally accurate discrete regularization for second order timestepping methods for navier stokes equations abstract we propose a new optimally accurate numerical regularization stabilization for a family of second order timestepping methods for the navier stokes equations nse the method combines a linear treatment of the advection term together with stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure we rigorously prove that the entire new family of methods are unconditionally stable and o t 2 accurate the idea of curvature stabilization is new to cfd and is intended as an improvement over the commonly used speed stabilization which is only first order accurate in time and can have an adverse effect on important flow quantities such as drag coefficients numerical examples verify the predicted convergence rate and show the stabilization term clearly improves the stability and accuracy of the tested flows

the long time l2 and h1 stability of linearly extrapolated second order time stepping schemes for the 2d incompressible navier stokes equations abstract herein we present a study on the long time stability of finite element discretizations of a generalized class of semi implicit second order time stepping schemes for the 2d incompressible navier stokes equations these remarkably efficient schemes require only a single linear solve per time step through the use of a linearly extrapolated advective term our result develops a class of sufficient conditions such that if external forcing is uniformly bounded in time velocity solutions are uniformly bounded in time in both the l2 and h1 norms we provide numerical verification of these results we also demonstrate that divergence free finite elements are critical for long time h1 stability

on the computation of a table of complex cubic fields with discriminant d 10 6 a method for finding all the nonisomorphic complex cubic fields with discriminant d 106 is described three different methods were used to find the class number of each of these fields the speed of these techniques is discussed and several tables illustrating the computational results are presented these include tables of the distribution of the fields and the class numbers and the class group structures of these fields

the cohen lenstra heuristic methodology and results in number theory great efforts have been undertaken to study the cohen lenstra probability measure on the set of all finite abelian p groups on the other hand group theorists have studied a probability measure on the set of all partitions induced by the probability that a randomly chosen n n matrix over fp is contained in a conjugacy class associated with this partitions for n this paper shows that both probability measures are identical as a consequence a multitude of results can be transferred from each theory to the other one the paper contains a survey about the known methods to study the probability measure and about the results that have been obtained so far from both communities

unconditional long time stability of a velocity vorticity method for the 2d navier stokes equations we prove unconditional long time stability for a particular velocity vorticity discretization of the 2d navier stokes equations the scheme begins with a formulation that uses the lamb vector to couple the usual velocity pressure system to the vorticity dynamics equation and then discretizes with the finite element method in space and implicit explicit bdf2 in time with the vorticity equation decoupling at each time step we prove the method s vorticity and velocity are both long time stable in the l 2 l2 and h 1 h1 norms without any timestep restriction moreover our analysis avoids the use of gronwall type estimates which leads us to stability bounds with only polynomial instead of exponential dependence on the reynolds number numerical experiments are given that demonstrate the effectiveness of the method

a new family of stable mixed finite elements for the 3d stokes equations a natural mixed element approach for the stokes equations in the velocity pressure formulation would approximate the velocity by continuous piecewise polynomials and would approximate the pressure by discontinuous piecewise polynomials of one degree lower however many such elements are unstable in 2d and 3d this paper is devoted to proving that the mixed finite elements of this p k p k 1 type when k 3 satisfy the stability condition the babuska brezzi inequality on macro tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra this type of mesh simplifies the implementation since it has no restrictions on the initial mesh the new element also suits the multigrid method

heuristics for p class towers of imaginary quadratic fields cohen and lenstra have given a heuristic which for a fixed odd prime p leads to many interesting predictions about the distribution of p class groups of imaginary quadratic fields we extend the cohen lenstra heuristic to a non abelian setting by considering for each imaginary quadratic field k the galois group of the p class tower of k i e g k mathrm gal k infty k where k infty is the maximal unramified p extension of k by class field theory the maximal abelian quotient of g k is isomorphic to the p class group of k for integers c ge 1 we give a heuristic of cohen lenstra type for the maximal p class c quotient of g k and thereby give a conjectural formula for how frequently a given p group of p class c occurs in this manner in particular we predict that every finite schur sigma group occurs as g k for infinitely many fields k we present numerical data in support of these conjectures

on the long time stability of the implicit euler scheme for the two dimensional navier stokes equations in this paper we study the stability for all positive time of the fully implicit euler scheme for the two dimensional navier stokes equations more precisely we consider the time discretization scheme and with the aid of the discrete gronwall lemma and the discrete uniform gronwall lemma we prove that the numerical scheme is stable

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

homological stability for hurwitz spaces and the cohen lenstra conjecture over function fields we prove a homological stabilization theorem for hurwitz spaces moduli spaces of branched covers of the complex projective line this has the following arithmetic consequence let 2 be prime and a a nite abelian group then there exists q q a such that for q greater than q a positive fraction of quadratic extensions of fq t have the part of their class group isomorphic to a

comments on subband coding of images the commenter feels that an equation in the above named work by j w woods and s o neil ibid vol assp 34 pp 1278 1288 oct 1986 is incorrect as given a corrected version is provided along with another equation which together allow one to obtain another equation as originally stated the analysis following the equation remains valid

long time stability of a classical efficient scheme for two dimensional navier stokes equations this paper considers the long time stability property of a popular semi implicit scheme for the two dimensional incompressible navier stokes equations in a periodic box that treats the viscous term implicitly and the nonlinear advection term explicitly we consider both the semidiscrete discrete in time but continuous in space and fully discrete schemes with either fourier galerkin spectral or fourier pseudospectral collocation methods we prove that in all cases the scheme is long time stable provided that the timestep is sufficiently small the long time stability in the l 2 and h 1 norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the navier stokes equations at vanishing timestep

binary quartic forms having bounded invariants and the boundedness of the average rank of elliptic curves we prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants this extends to the quartic case the classical results of gauss and davenport in the quadratic and cubic cases respectively our techniques are quite general and may be applied to counting integral orbits in other representations of algebraic groups we use these counting results to prove that the average rank of elliptic curves over q when ordered by their heights is bounded in particular we show that when elliptic curves are ordered by height the mean size of the 2 selmer group is 3 this implies that the limsup of the average rank of elliptic curves is at most 1 5

p ell torsion points in finite abelian groups and combinatorial identities abstract the main aim of this article is to compute all the moments of the number of p l torsion elements in some type of finite abelian groups the averages involved in these moments are those defined for the cohen lenstra heuristics for class groups and their adaptation for tate shafarevich groups in particular we prove that the heuristic model for tate shafarevich groups is compatible with the recent conjecture of poonen and rains about the moments of the orders of p selmer groups of elliptic curves for our purpose we are led to define certain polynomials indexed by integer partitions and to study them in a combinatorial way moreover from our probabilistic model we derive combinatorial identities some of which appearing to be new the others being related to the theory of symmetric functions in some sense our method therefore gives for these identities a somehow natural algebraic context

convergence analysis of lms filters with uncorrelated gaussian data statistical analysis of the least mean squares lms adaptive algorithm with uncorrelated gaussian data is presented exact analytical expressions for the steady state mean square error mse and the performance degradation due to weight vector misadjustment are derived necessary and sufficient conditions for the convergence of the algorithm to the optimal wiener solution within a finite variance are derived it is found that the adaptive coefficient which controls the rate of convergence of the algorithm must be restricted to an interval significantly smaller than the domain commonly stated in the literature the outcome of this paper therefore places fundamental limitations on the mse performance and rate of convergence of the lms adaptive scheme

an efficient second order in time scheme for approximating long time statistical properties of the two dimensional navier stokes equations we investigate the long time behavior of the following efficient second order in time scheme for the 2d navier stokes equations in a periodic box begin array ll frac 3 omega n 1 4 omega n omega n 1 2k nabla perp 2 psi n psi n 1 cdot nabla 2 omega n omega n 1 nu delta omega n 1 f n 1 quad delta psi n omega n end array the scheme is a combination of a 2nd order in time backward differentiation and a particular explicit adams bashforth treatment of the advection term therefore only a linear constant coefficient poisson solver is needed at each time step we prove uniform in time bounds on this scheme in dot l 2 dot h 1 per and dot h 2 per provided that the time step is sufficiently small these time uniform estimates further lead to the convergence of long time statistics stationary statistical properties of the scheme to that of the nse itself at vanishing time step

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

divergence free finite elements on tetrahedral grids for k 6 it was shown two decades ago that the p k p k 1 mixed element on triangular grids approximating the velocity by the continuous p k piecewise polynomials and the pressure by the discontinuous p k 1 piecewise polynomials is stable for all k 4 provided the grids are free of a nearly singular vertex the problem with the method in 3d was posted then and remains open the problem is solved partially in this work it is shown that the p k p k 1 element is stable and of optimal order in approximation on a family of uniform tetrahedral grids for all k 6 the analysis is to be generalized to non uniform grids when we can deal with the complicity of 3d geometry for the divergence free elements the finite element spaces for the pressure can be avoided in computation if a classic iterated penalty method is applied the finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity numerical tests are provided

long term stability estimates and existence of a global attractor in a finite element approximation of the navier stokes equations with numerical subgrid scale modeling variational multiscale methods lead to stable finite element approximations of the navier stokes equations dealing with both the indefinite nature of the system pressure stability and the velocity stability loss for high reynolds numbers these methods enrich the galerkin formulation with a subgrid component that is modeled in fact the effect of the subgrid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum thus they also act as effective large eddy simulation turbulence models and allow one to compute flows without the need to capture all the scales in the system in this article we consider a dynamic subgrid model that enforces the subgrid component to be orthogonal to the finite element space in the l 2 sense we analyze the long term behavior of the algorithm proving the existence of appropriate absorbing sets and a compact global attractor the improvements with respect to a finite element galerkin approximation are the long term estimates for the subgrid component which are translated to effective pressure and velocity stability thus the stabilization introduced by the subgrid model into the finite element problem does not deteriorate for infinite time intervals of computation

finite element approximation of the nonstationary navier stokes problem part ii stability of solutions and error estimates uniform in time in this paper assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary navier stokes problem for the purpose of extending local a priori error estimates and local a posteriori error estimates globally in time