on totally real cubic fields the authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant d 500000 thereby extending the existing table of fields with d 100000 by i 0 angell 1 serious defects in angell s table are pointed out for each field running number discriminant coefficients of a generating polynomial integral basis class number and a fundamental pair of units are listed the article contains statistics about the following subjects distribution of class numbers fields in which every norm positive unit is totally positive nonconjugate fields with the same discriminant fields with noncyclic class group the fields are tabulated by means of a method due to davenport and heilbronn 7 8 which leads to a unique normalized generating polynomial the given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced

secondary terms in counting functions for cubic fields we prove the existence of secondary terms of order x5 6 in the davenport heilbronn theorems on cubic fields and 3 torsion in class groups of quadratic fields for cubic fields this confirms a conjecture of datskovsky wright and roberts we also prove a variety of generalizations including to arithmetic progressions where we discover a curious bias in the secondary term roberts s conjecture has also been proved independently by bhargava shankar and tsimerman in contrast to their work our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms developed by shintani and datskovsky wright

levels of distribution for sieve problems in prehomogeneous vector spaces in our companion paper 28 we developed an efficient algebraic method for computing the fourier transforms of certain functions defined on prehomogeneous vector spaces over finite fields and we carried out these computations in a variety of cases here we develop a method based on fourier analysis and algebraic geometry which exploits these fourier transform formulas to yield level of distribution results in the sense of analytic number theory such results are of the shape typically required for a variety of sieve methods as an example of such an application we prove that there are gg frac x log x quartic fields whose discriminant is squarefree bounded above by x and has at most eight prime factors

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

on the existence of sequences and matrices with prescribed partial sums of elements abstract we prove necessary and sufficient conditions for the existence of sequences and matrices with elements in given intervals and with prescribed lower and upper bounds on the element sums corresponding to the sets of an orthogonal pair of partitions we use these conditions to generalize known results on the existence of nonnegative matrices with a given zero pattern and prescribed row and column sums we also generalize recently proven results on the existence of a real or nonnegative square matrix a with a given zero pattern and with prescribed row sums such that a a t is prescribed we also introduce hadamard adjustments by means of which we generalize known results on the scaling of matrices with a given pattern to achieve prescribed row and column sums

integral points on elliptic curves and 3 torsion in class groups we give new bounds for the number of integral points on elliptic curves the method may be said to interpolate between approaches via diophantine techniques bp hbr and methods based on quasi orthogonality in the mordell weil lattice sil6 gs he we apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the 3 torsion part of the class group of a quadratic field the same ideas can be used to count rational points on curves of higher genus

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

efficient multigrid methods for solving upwind biased discretizations of the convection equation this paper studies a novel multigrid approach to the solution for a second order upwind biased discretization of the convection equation in two dimensions this approach is based on semi coarsening and well balanced explicit correction terms added to coarse grid operators to maintain on coarse grids the same cross characteristic interaction as on the target fine grid multicolor relaxation schemes are used on all the levels allowing a very efficient parallel implementation the results of the smoothing analysis and numerical tests can be summarized as follows 1 the residual asymptotic convergence rate is about 3 per cycle which far surpasses the theoretical limit 4 3 predicted for standard multigrid algorithms with full coarsening 2 the full multigrid fmg algorithm with two v 0 2 cycles on the target grid and just one v 0 2 cycle on all the coarse grids always provides an approximate solution with the algebraic error less than the discretization error 3 a novel algorithm for deriving a discrete solution approximating the true continuous solution with a relative accuracy given in advance is developed the computational complexity of this algorithm is nearly optimal comparable with the complexity of the fmg algorithm applied to solve the problem on the optimally spaced target grid

multigrid line smoothers for higher order upwind discretizations of convection dominated problems in this paper we present new multigrid line smoothers for the solution of higher order discretizations of convection dominated problems directly the behavior of the smoothers is analyzed theoretically with fourier smoothing and two grid analysis a parallel tri line variant is presented and evaluated the smoothers are applied to scalar convection diffusion problems discretized with limiters and systems of incompressible navier stokes and euler equations

sparsity of orthogonal matrices with restrictions abstract the sparsity of orthogonal matrices which have k 1 columns of nonzeros is studied it is shown that the minimum number of nonzero entries in such an m by m matrix is lg m k k 2 m k2 lg m k 1 as a consequence it is shown that if a is an m by n matrix with m and the properties that its rows are pairwise orthogonal and it has less than lg n n m 2 n n m 2 lg n n m 1 nonzero entries then each vector orthogonal to the rows of a has at least one entry equal to 0 also for integers k and n with k n the minimum number of nonzero entries in an n by n connected orthogonal matrix having a column with at least k nonzero entries is determined

directional agglomeration multigrid techniques for high reynolds number viscous flows a preconditioned directional implicit agglomeration algorithm is developed for solving two and three dimensional viscous flows on highly anisotropic unstructured meshes of mixed element types the multigrid smoother consists of a pre conditioned point or line implicit solver which operates on lines constructed in the unstructured mesh using a weighted graph algorithm directional coarsening or agglomeration is achieved using a similar weighted graph algorithm a tight coupling of the line construction and directional agglomeration algorithms enables the use of aggressive coarsening ratios in the multigrid algorithm which in turn reduces the cost of a multigrid cycle convergence rates which are independent of the degree of grid stretching are demonstrated in both two and three dimensions further improvement of the three dimensional convergence rates through a gmres technique is also demonstrated

sign patterns of orthogonal matrices and the strong inner product property abstract a new condition the strong inner product property is introduced and used to construct sign patterns of row orthogonal matrices using this property infinite families of sign patterns allowing row orthogonality are found these provide insight into the underlying combinatorial structure of row orthogonal matrices algorithmic techniques for verifying that a matrix has the strong inner product property are also presented these techniques lead to a generalization of the strong inner product property and can be easily implemented using various software

stabilisation of discrete steady adjoint solvers a new implicit time stepping scheme which uses runge kutta time stepping and krylov methods as a smoother inside fas cycle multigrid acceleration is proposed to stabilise the flow solver and its discrete adjoint counterpart the algorithm can fully converge the discrete adjoint solver in a wide range of cases where conventional point implicit methods fail due to either physical or numerical instability this enables the discrete adjoint to be applied to a much wider range of flow regimes in addition the new algorithm offers improved efficiency when applied to stable cases for which the conventional block jacobi solver can fully converge both stable and unstable cases are presented to demonstrate the improved robustness and performance of the new scheme eigen analysis is presented to outline the mechanism of the adjoint stabilisation effect

half space analysis of the defect correction method for fromm discretization of convection a novel comprehensive discrete half space analysis for the defect correction method has been developed this analysis plays the same role for nonelliptic problem solvers as the full space fourier mode analysis plays for elliptic problem solvers numerical simulations confirm the accuracy of the half space analysis the following important findings about the defect correction method applied to the fromm discretization of the two dimensional convection equation are reported the initial convergence rate of the defect correction method is principally a function of the relative accuracy of the operators involved in the defect correction iterations the asymptotic convergence rate is about 0 5 per defect correction iteration if the driver operator is first order accurate then the initial convergence rates may be slow the number of iterations required to get into the asymptotic convergence regime or and to converge the algebraic error below the discretization error level can be proportional to h 1 3 this h dependent delay is a multidimensional phenomenon it cannot be observed in one dimensional problems and it disappears in the case of close alignment between the grid and the convection equation characteristic if the driver operator is second order accurate the defect correction solver demonstrates the asymptotic convergence rate from the very beginning only one defect correction iteration is required to converge algebraic error substantially below the discretization error level

sign pattern matrices that allow orthogonality abstract a pair of sign pattern row vectors or column vectors allows orthogonality if the two vectors are the sign patterns for two real orthogonal row vectors respectively column vectors a square sign pattern matrix that does not have a zero row or zero column is sign potentially orthogonal spo if every pair of rows and every pair of columns allows orthogonality it has been conjectured that every spo matrix a allows orthogonality that is there is a real orthogonal matrix h whose sign pattern equals a a counterexample to this conjecture is presented and sufficient conditions are given for a spo matrix to allow orthogonality

compatible relaxation and coarsening in algebraic multigrid we introduce a coarsening algorithm for algebraic multigrid amg based on the concept of compatible relaxation cr the algorithm is significantly different from standard methods most notably because it does not rely on any notion of strength of connection we study its behavior on a number of model problems and evaluate the performance of an amg algorithm that incorporates the coarsening approach finally we introduce a variant of cr that provides a sharper metric of coarse grid quality and demonstrate its potential with two simple examples

agglomeration multigrid methods with implicit runge kutta smoothers applied to aerodynamic simulations on unstructured grids for unstructured finite volume methods an agglomeration multigrid with an implicit multistage runge kutta method as a smoother is developed for solving the compressible reynolds averaged navier stokes rans equations the implicit runge kutta method is interpreted as a preconditioned explicit runge kutta method the construction of the preconditioner is based on an approximate derivative the linear systems are solved approximately with a symmetric gauss seidel method to significantly improve this solution method grid anisotropy is treated within the gauss seidel iteration in such a way that the strong couplings in the linear system are resolved by tridiagonal systems constructed along these directions of strong coupling the agglomeration strategy is adapted to this procedure by taking into account exactly these anisotropies in such a way that a directional coarsening is applied along these directions of strong coupling turbulence effects are included by a spalart allmaras model and the additional transport type equation is approximately solved in a loosely coupled manner with the same method for two dimensional and three dimensional numerical examples and a variety of differently generated meshes we show the wide range of applicability of the solution method finally we exploit the gmres method to determine approximate spectral information of the linearized rans equations this approximate spectral information is used to discuss and compare characteristics of multistage runge kutta methods

on the average number of octahedral newforms of prime level we show that on average the number of octahedral newforms of prime level is bounded by a constant

a critical study of agglomerated multigrid methods for diffusion agglomerated multigrid techniques used in unstructured grid methods are studied critically for a model problem representative of laminar diffusion in the incompressible limit the studied target grid discretizations and discretizations used on agglomerated grids are typical of current node centered formulations agglomerated multigrid convergence rates are presented using a range of two and three dimensional randomly perturbed unstructured grids for simple geometries with isotropic and stretched grids two agglomeration techniques are used within an overall topology preserving agglomeration framework the results show that multigrid with an inconsistent coarse grid scheme using only the edge terms also referred to in the literature as a thin layer formulation provides considerable speedup over single grid methods but its convergence deteriorates on finer grids multigrid with a galerkin coarse grid discretization using piecewise constant prolongation and a heuristic correction factor is slower and also grid dependent in contrast grid independent convergence rates are demonstrated for multigrid with consistent coarse grid discretizations convergence rates of multigrid cycles are verified with quantitative analysis methods in which parts of the two grid cycle are replaced by their idealized counterparts

a fast algorithm to compute cubic fields we present a very fast algorithm to build up tables of cubic fields real cubic fields with discriminant up to 10 11 and complex cubic fields down to 10 11 have been computed

multigrid solvers for nonaligned sonic flows we investigate an approach to the solution of nonelliptic equations on a rectangular grid the multigrid algorithms presented here demonstrate the textbook multigrid efficiency even in the case that the equation characteristics do not align with the grid to serve as a model problem the two dimensional 2d and three dimensional 3d linearized sonic flow equations have been chosen efficient full multigrid fmg solvers for the problems are demonstrated

low lying zeros of artin l functions we study the one level density of artin l functions twisted by a cuspidal automorphic representation under the strong artin conjecture and certain conjectures on counting number fields our result is unconditional for s 3 fields for a non self dual pi it agrees with the unitary type text u for a self dual pi whose symmetric square l function l s pi text sym 2 has a pole at s 1 it agrees with the symplectic type text sp for a self dual pi whose exterior square l function l s pi wedge 2 has a pole at s 1 the possible symmetry types are text o text so even or text so odd when pi 1 for s 3 cubic fields and s 4 quartic fields we rediscover yang s one level density result in his thesis yang 2009 in the last section we compute the one level density of several families of artin l functions arising from parametric polynomials

an effective chebotarev density theorem for families of number fields with an application to formula see text torsion in class groups we prove a new effective chebotarev density theorem for galois extensions l mathbb q that allows one to count small primes even as small as an arbitrarily small power of the discriminant of l this theorem holds for the galois closures of almost all number fields that lie in an appropriate family of field extensions previously applying chebotarev in such small ranges required assuming the generalized riemann hypothesis the error term in this new chebotarev density theorem also avoids the effect of an exceptional zero of the dedekind zeta function of l without assuming grh we give many different appropriate families including families of arbitrarily large degree to do this we first prove a new effective chebotarev density theorem that requires a zero free region of the dedekind zeta function then we prove that almost all number fields in our families yield such a zero free region the innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non cuspidall functions this builds on and greatly generalizes the applicability of work of kowalski and michel on the average density of zeroes of a family of cuspidall functions a surprising feature of this new method which we expect will have independent interest is that we control the number of zeroes in the family of l functions by bounding the number of certain associated fields with fixed discriminant as an application of the new chebotarev density theorem we prove the first nontrivial upper bounds for ell torsion in class groups for all integers ell ge 1 applicable to infinite families of fields of arbitrarily large degree

on totally real cubic fields with discriminant 10 the authors have constructed a table of the 592923 nonconjugate totally real cubic number fields of discriminant d 107 thereby extending the existing table of fields with d 5 x 105 constructed by ennola and turunen 4 each field is given by its discriminant and the coefficients of a generating polynomial the method used is an improved version of the method developed in 8 the article contains an exposition of the modified method statistics and examples the decomposition of the rational primes is studied and the relative frequency of each type of decomposition is compared with the corresponding density given by davenport and heilbronn 2

on quadratic fields with large 3 rank davenport and heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields which has been used to tabulate the latter we give a simpler proof of their theorem then analyze and improve the table building algorithm it computes the multiplicities of the o x general cubic discriminants real or imaginary up to x in time o x and space o x or more generally in time o x x m and space o m x for a freely chosen positive m a variant computes the 3 ranks of all quadratic fields of discriminant up to x with the same time complexity but using only m o 1 units of storage as an application we obtain the first 1618 real quadratic fields with r3 4 and prove that q 5393946914743 is the smallest imaginary quadratic field with 3 rank equal to 5

sparse orthogonal matrices the minimum number of nonzero entries in an n by n orthogonal matrix which has a column of nonzeros is known to be f n 1g n 3 n 2 1g n 1 in this note the sparsity of orthogonal matrices which have both a column and a row of nonzeros is studied for each integer n 2 we construct an n by n orthogonal matrix which has both a row and column of nonzeros and has g n 2k 2 n 7 2 k 1 3 if 2 k n 2 k 2 k 1 2k 3 n 10 2 k 1 3 if 2 k 2 k 1 n 2 k 1 nonzero entries

on quantitative analysis methods for multigrid solutions limitations of the classical local mode fourier lmf analysis in application to multigrid solutions of variable coefficient problems are well known this paper introduces alternative very general quantitative analysis methods for multigrid solutions of partial differential equations the methods are applied to available nonperfect multigrid solvers that deal with practical problems the analysis methods considered in this paper focus on the main complementary parts of a multigrid cycle relaxation and coarse grid correction tasks for these parts e g error smoothing for relaxation and reduction of smooth errors for coarse grid correction are assumed to be assigned in advance idealized relaxation ir and idealized coarse grid icg iterations are introduced in these iterations one part of the cycle relaxation for ir iterations and coarse grid correction for icg iterations is replaced with an idealized imitation known to be efficient for the assigned task its complementary part is the actual part of a two grid cycle the analysis compares performances of the actual cycle and idealized iterations the ir and icg iterations are very general and can be directly applied in the most complicated simulations including highly variable or nonlinear coefficients complex geometries and unstructured grids the results of this analysis are not single number estimates they are rather convergence patterns of the iterations that may either confirm or refute expectations indicating what part of the actual solver is inefficient in carrying out the assigned task the generality of the analysis makes it a valuable tool for analyzing complicated large scale computational problems where no other analysis methods are currently available in this paper the ir and icg iterations are applied to model problems of progressive complexity the analysis proved sensitive to very delicate details of the actual multigrid cycle pointing clearly to each instance when a part of the algorithm was inefficient

density of cubic field discriminants in this paper we give a conjectural refinement of the davenport heilbronn theorem on the density of cubic field discriminants we explain how this refinement is plausible theoretically and agrees very well with computa tional data let an be the number of isomorphism classes of abelian cubic fields with discrim inant n let bn be the number of isomorphism classes of non abelian cubic fields with discriminant n the numbers an are very well understood the numbers bn have been the subject of extensive theoretical and computational study for at least sixty years but are less well understood the object of this note is to contribute to the study of these bn by bringing together the theoretical and computational literature for define g x

algebraic multigrid theory the symmetric case a rigorous two level theory is developed for general symmetric matrices and nonsymmetric ones using kaczmarz relaxation without assuming any regularity not even any grid structure of the unknowns the theory applies to algebraic multigrid amg processes as well as to the usual geometric multigrid it yields very realistic estimates and precise answers to basic algorithmic questions such as in what algebraic sense does gauss seidel or jacobi kaczmarz line gauss seidel etc relaxation smooth the error when is it appropriate to use block relaxation what algebraic relations must be satisfied by the coarse to fine interpolations what is the algorithmic role of the geometric origin of the problem the theory helps to rigorize local mode analyses and locally analyze cases where the latter is inapplicable

convergence analysis of the defect correction iteration for hyperbolic problems this paper explains some of the convergence behaviour of iterative implicit and defect correction schemes for the solution of the discrete steady euler equations such equations are also commonly solved by pseudo time integration the steady solution being achieved as the limit for t to infty of the solution of a time dependent problem implicit schemes are then often chosen for their favourable stability properties permitting large timesteps for efficiency an important class of implicit schemes involving first and second order accurate upwind discretisations is considered in the limit of an infinite timestep these implicit schemes approach defect correction algorithms thus our analysis is informative for both types of construction simple scalar linear model problems are introduced for the one dimensional and two dimensional cases these model problems are analyzed in detail by both fourier and matrix analyses the convergence behaviour appears to be strongly dependent on a parameter beta

beyond endoscopy and special forms on gl 2 we carry out with technical modifications some cases of a procedure proposed by r langlands in beyond endoscopy this gives a new proof of the classification of dihedral forms on gl 2 avoiding endoscopic methods instead it uses an infinite limiting process in the trace formula we also explain the relationship of langlands idea to the results of 10 that count automorphic forms associated to galois representations we prove results of a similar nature over an arbitrary number field the main tool is a version of the relative trace formula namely a slight variant for pgl 2 of the kuznetsov bruggeman miatello formula msc 11f72 11f70

coarse grid correction for nonelliptic and singular perturbation problems the suitability of coarse grid correction operators for multigrid solution of discretized nonelliptic partial differential equations is studied it is found that the requirements for the orders of intergrid transfers and the order of approximation of the coarse grid operators differ in an important way from the well known rules pertaining to elliptic operators both galerkin coarsening and coarsening schemes based on the differential operator are examined and new coarse grid correction strategies are suggested in light of this analysis

fourier analysis for multigrid methods on triangular grids in this paper a local fourier analysis technique for multigrid methods on triangular grids is presented the analysis is based on an expression of the fourier transform in new coordinate systems both in space variables and in frequency variables associated with reciprocal bases this tool makes it possible to study different components of the multigrid method in a very similar way to that of rectangular grids different smoothers for the discrete laplace operator obtained with linear finite elements are analyzed a new three color smoother has been studied and has proven to be the best choice for near equilateral triangles it is also shown that the block line smoothers are more appropriate for irregular triangles numerical test calculations validate the theoretical predictions

rigorous quantitative analysis of multigrid i constant coefficients two level cycle with l 2 norm exact numerical convergence factors for any multigrid cycle can be predicted by local mode fourier analysis for general linear elliptic partial differential equation pde systems with piecewise smooth coefficients in general domains discretized by uniform grids it is proved that in the limit of small meshsizes these predicted factors are indeed obtained provided the cycle is supplemented with a proper processing at and near the boundaries that processing it is proved costs negligible extra computer work apart from mode analysis a coarse grid approximation cga condition is introduced which is both necessary and sufficient for the multigrid algorithm to work properly the present part studies the l 2 convergence in one cycle for equations with constant coefficients in the sequel brandt rigorous quantitative analysis of multigrid ii extensions and practical implications manuscript extensions are discussed to many cycles asymptotic convergence to more levels with arbitrary cycle ty

computer aided analysis of the convergence to steady state of discrete approximations to the euler equations abstract the behaviour of a centered finite volume scheme for the isoenergetic euler equations in two space dimensions is studied by numerical differentiation and approximate eigensystem analysis the entire semidiscrete approximation including boundary conditions is formulated as a large system of odes which are linearized by numerically approximating the frechet derivative an approximate eigensystem procedure that only needs the frechet derivative is used to extract the least damped eigenmodes the overall method has been applied to the case of transonic flow past an airfoil and has revealed that the most persistent transient modes are highly structured and are associated with eigenvalues of small modulus furthermore they appear to be centered around the shock region the stagnation region and the trailing edge wake region of the airfoil the beneficial effect of local time step scaling and artificial dissipation is also demonstrated by the method

on multigrid solution of high reynolds incompressible entering flows an approach is presented for effectively separating the solution process of the elliptic component of high reynolds incompressible steady entering flow for which classical multigrid techniques are well suited from that of the non elliptic part for which other methods are more effective it is shown by analysis and numerical calculations that such an approach is very effective in terms of asymptotic convergence as well as reduction of errors well below discretization level in a 1 fmg algorithm

on the davenport heilbronn theorems and second order terms we give simple proofs of the davenport heilbronn theorems which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3 torsion elements in the class groups of quadratic fields having bounded discriminant we also establish second main terms for these theorems thus proving a conjecture of roberts our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings