mean square values of hecke l series formed with r th order characters let l be a number field containing the r th roots of unity starting with the rankin selberg convolution of a metaplectic eisenstein series on the r fold cover of gl 2 with itself we construct a dirichlet series defined over l whose coefficients involve the r th order twists of a fixed hecke l function we then observe that a group of functional equations can be naturally associated with this construction combining this with the convexity theorem for holomorphic functions of several complex variables we show that this object as a function of two complex variables admits meromorphic continuation to 2 as an application we obtain asymptotic formulae for mean square values of the r th order twists of an arbitrary hecke l function defined over l

combined 5 2 cv f test for comparing supervised classification learning algorithms dietterich 1998 reviews five statistical tests and proposes the 5 2 cv t test for determining whether there is a significant difference between the error rates of two classifiers in our experiments we noticed that the 5 2 cv t test result may vary depending on factors that should not affect the test and we propose a variant the combined 5 2 cv f test that combines multiple statistics to get a more robust test simulation results show that this combined version of the test has lower type i error and higher power than 5 2 cv proper

constructing weyl group multiple dirichlet series let be a reduced root system of rank r a weyl group multiple dirichlet series for is a dirichlet series in r complex variables s1 sr initially converging for re si sufficiently large that has meromorphic continuation to c and satisfies functional equations under the transformations of c corresponding to the weyl group of a heuristic definition of such series was given in bbc06 and they have been investigated in certain special cases in bbc06 bbf06 bbfh07 bbf08 bbfa cg07 cfg08 cg chi05 in this paper we generalize results in cg to construct weyl group multiple dirichlet series by a uniform method and show in all cases that they have the expected properties

cubic twists of gl 2 automorphic l functions let k mathbb q sqrt 3 and let be a cuspidal automorphic representation of gl 2 mathbb a k consider the family of twisted l functions l s where ranges over the cubic hecke characters of k in this paper the mean value of this family of l functions is computed the result is consistent with the generalized lindelof hypothesis from this mean value result a nonvanishing theorem is established for given s there are infinitely many cubic twists such that the l value at s is nonzero at the center of the critical strip the number of such characters of norm less than x is gg x 1 2 epsilon these results are obtained by introducing and studying three different families of weighted double dirichlet series these series are related by functional equations some of which are obtained through the study of higher metaplectic eisenstein series and the hasse davenport relation the authors establish the continuation of such series and then obtain their main result by tauberian methods

studies on error propagation for certain nonlinear approximations to hyperbolic equations discontinuities in derivatives the accuracy of numerical approximations to piecewise smooth solutions of hyperbolic partial differential equations is greatly influenced by the presence of singularities in the solution in the presence of coupling through lower order terms or variable coefficients high order numerical approximations can lose accuracy in large regions where the analytical solution is known to be smooth due to the spreading of the errors that the singularities introduce in the computation this phenomenon which has been analyzed in the past fifteen years for a number of classical linear methods is studied here for numerical approximations obtained with nonlinear essentially non oscillatory eno schemes the study of the local rate of convergence allows for the identification of the necessary techniques to reduce the spread of errors and to avoid the accuracy loss of the computed approximations the techniques developed can be applied to nonlinear hyperbolic partial differential equations and systems to sharpen the r

adaptive eulerian lagrangian finite element method for advection dispersion a new adaptive finite element method is proposed for the advection dispersion equation using an eulerian lagrangian formulation the method is based on a decomposition of the concentration field into two parts one advective and one dispersive in a rigorous manner that does not leave room for ambiguity the advective component of steep concentration fronts is tracked forward with the aid of moving particles clustered around each front away from such fronts the advection problem is handled by an efficient modified method of characteristics called single step reverse particle tracking when a front dissipates with time its forward tracking stops automatically and the corresponding cloud of particles is eliminated the dispersion problem is solved by an unconventional lagrangian finite element formulation on a fixed grid which involves only symmetric and diagonal matrices preliminary tests against analytical solutions of one and two dimensional dispersion in a uniform steady state velocity field suggest that the proposed adaptive method can handle the entire range of peclet numbers from 0 to with courant numbers well in excess of 1

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

cycle integrals of the j function and mock modular forms in this paper we construct certain mock modular forms of weight 1 2 whose fourier coecients are given in terms of cycle integrals of the modular j function their shadows are weakly holomorphic forms of weight 3 2 these new mock modular forms occur as holomorphic parts of weakly harmonic maass forms we also construct a generalized mock modular form of weight 1 2 having a real quadratic class number times a regulator as a fourier coecient as an application of these forms we study holomorphic modular integrals of weight 2 whose rational period functions have poles at certain real quadratic integers the fourier coecients of these modular integrals are given in terms of cycle integrals of modular functions such a modular integral can be interpreted in terms of a shimura type lift of a mock modular form of weight 1 2 and yields a real quadratic analogue of a borcherds product

formulation and validation of boundary conditions at a branched junction for nonlinear waves abstract a numerical model is presented for the calculation of nonlinear waves through the branched junctions in a gas transmission pipe system in the one dimensional time domain analysis of the wave dynamics the behaviour of the nonlinear waves is determined by quasi steady boundary conditions at junctions in a pipe system the quasi steady treatment through the junction yields a system of nonlinear balance equations the iterative solution procedure of the system of nonlinear equations causes the problems of divergence and multiple solutions in order to overcome the difficulties a new set of quasi linear balance equations is formulated without a sacrifice of accuracy the present model therefore requires a non iterative solution procedure that results in a unique solution and a smaller computational effort thompson s boundary condition is used at each pipe a new time derivative form of balance equations based thompson boundary condition gives a set of linear algebraic equations for the balance equations when implementing the new time derivative form of the balance equations corrected terms are added numerically the present numerical model is implemented and tested for the calculation of behaviour of nonlinear waves in a branched pipe and the calculation of sound attenuation of a helmholtz resonator and the prediction of radiated orifice noise of an internal combustion engine calculation results show good agreements with previous iterative calculations and measurement data

a finite volume method with a modified eno scheme using a hermite interpolation to solve advection diffusion equations in this paper we have developed a finite volume eno scheme third order accurate based on cell averages and a tvd runge kutta time discretization to solve advection diffusion equations in a two dimensional spatial domain we have designed a special interpolating polynomial based on a modified eno scheme and a hermite procedure which avoids the excessive smearing in regions with sharpconcentration fronts and the overcompression effects produced by the modified eno technique thesemodifications do not affect the non oscillatory philosophy since we compare divided differences inthe modified eno scheme and in the evaluation of the hermite polynomial derivatives numericalresults compare favourably with their respective analytical solutions copyright 2001 john wiley sons ltd

model selection in omnivariate decision trees using structural risk minimization as opposed to trees that use a single type of decision node an omnivariate decision tree contains nodes of different types we propose to use structural risk minimization srm to choose between node types in omnivariate decision tree construction to match the complexity of a node to the complexity of the data reaching that node in order to apply srm for model selection one needs the vc dimension of the candidate models in this paper we first derive the vc dimension of the univariate model and estimate the vc dimension of all three models univariate linear multivariate or quadratic multivariate experimentally second we compare srm with other model selection techniques including akaike s information criterion aic bayesian information criterion bic and cross validation cv on standard datasets from the uci and delve repositories we see that srm induces omnivariate trees that have a small percentage of multivariate nodes close to the root and they generalize more or at least as accurately as those constructed using other model selection techniques

comparative study of high resolution shock capturing schemes for a real gas recently developed second order explicit shock capturing methods in conjunction with generalized flux vector splittings and a generalized approximate riemann solver for a real gas are studied the comparisons are made on different one dimensional riemann shock tube problems for equilibrium air with various ranges of mach numbers densities and pressures six different riemann problems are considered these tests provide a check on the validity of the generalized formulas since theoretical prediction of their properties appears to be difficult because of the non analytical form of the state equation the numerical results in the supersonic and low hypersonic regimes indicate that these produce good shock capturing capability and that the shock resolution is only slightly affected by the state equation of equilibrium air the difference in shock resolution between the various methods varies slightly from one riemann problem to the other but the overall accuracy is very similar for the one dimensional case the relative efficiency in terms of operation count for the different methods is within 30 percent the main difference between the methods lies in their versatility in being extended to multidimensional problems with efficient implicit solution procedures

weyl group multiple dirichlet series constructed from quadratic characters we construct multiple dirichlet series in several complex variables whose coefficients involve quadratic residue symbols the series are shown to have an analytic continuation and satisfy a certain group of functional equations these are the first examples of an infinite collection of unstable weyl group multiple dirichlet series in greater than two variables having the properties predicted in 2

an artificial compression method for eno schemes the slope modification method abstract a simple and efficient method of artificial compression is introduced this method is based on a modification of the slopes of the eno reconstruction and with the help of suitably chosen parameters greatly improves the resolution of the contact discontinuities numerical examples are provided to test the performance of the method and to give some suggestions as to the choice of the parameters

prehomogeneous vector spaces and field extensions letk be an infinite field of characteristic not equal to 2 3 5 in this paper we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of galois extensions ofk which are splitting fields of equations of certain degrees and prove that the inverse image of this map corresponds bijectively with conjugacy classes of galois homomorphisms

a spectral proof of class number one we continue our previous work on the subject of re proving the heegner baker stark theorem giving another effective resolution of this conjecture of gauss namely there are exactly 9 imaginary quadratic fields mathbf q sqrt q with class number 1 specifically the list is q in 3 4 7 8 11 19 43 67 163 we again follow an ansatz of goldfeld considering a modular form l function of analytic rank 2 and reducing the situation to showing sufficient cancellation of the dirichlet series coefficients when restricted to the principal form previously we then chose a specific rank 2 elliptic curve with complex multiplication by mathbf q sqrt 1 and deduced the desired cancellation from an equi distribution result of hooley s for roots of a quadratic polynomial to varying moduli herein we instead use spectral techniques filling in details of work of templier and tsimerman to complete the proof relying on the duke iwaniec bound for fourier coefficients of half integral weight maass forms unlike the previous method we do not require complex multiplication and any analytic rank 2 modular form whose level has an even number of prime factors suffices

the random subspace method for constructing decision forests much of previous attention on decision trees focuses on the splitting criteria and optimization of tree sizes the dilemma between overfitting and achieving maximum accuracy is seldom resolved a method to construct a decision tree based classifier is proposed that maintains highest accuracy on training data and improves on generalization accuracy as it grows in complexity the classifier consists of multiple trees constructed systematically by pseudorandomly selecting subsets of components of the feature vector that is trees constructed in randomly chosen subspaces the subspace method is compared to single tree classifiers and other forest construction methods by experiments on publicly available datasets where the method s superiority is demonstrated we also discuss independence between trees in a forest and relate that to the combined classification accuracy

explicit and implicit multidimensional compact high resolution shock capturing methods two families of explicit and implicit compact high resolution shock capturing methods for the multidimensional compressible euler equations for fluid dynamics are constructed some of these schemes can be fourth and sixth order accurate away from discontinuities for the semi discrete case their shock capturing properties are of the total variation diminishing tvd total variation bounded tvb total variation diminishing in the mean tvdm essentially nonoscillatory eno or positive type of scheme for 1d scalar hyperbolic conservation laws and are positive schemes in more than one dimension these higher order compact schemes require the same grid stencil per spatial direction as their second order noncompact cousins the added terms over the second order noncompact cousins involve extra vector additions but no added flux evaluations due to the construction these schemes can be viewed as approximations to genuinely multidimensional schemes in the sense that they might produce less distortion in spherical type shocks and are more accurate in vortex type flows than schemes based purely on 1d extensions the extension of these families of compact schemes to coupled nonlinear systems can be accomplished using the roe approximate riemann solver the generalized steger and warming flux vector splitting or the van leer type flux vector splitting modification to existing high resolution second or third order noncompact shock capturing computer codes is minimal high resolution shock capturing properties can also be achieved via a variant of the second order lax friedrichs numerical flux without the use of riemann solvers for coupled nonlinear systems with comparable operations count to their classical shock capturing counterparts an efficient and compatible high resolution shock capturing filter for spatially fourth and sixth order classical compact and noncompact schemes is discussed the simplest extension to viscous flows can be achieved by using the standard fourth order compact or noncompact formula for the viscous terms

optimized compact finite difference schemes with maximum resolution direct numerical simulations and computational aeroacoustics require an accurate finite difference scheme that has a high order of truncation and high resolution characteristics in the evaluation of spatial derivatives compact finite difference schemes are optimized to obtain maximum resolution characteristics in space for various spatial truncation orders an analytic method with a systematic procedure to achieve maximum resolution characteristics is devised for multidiagonal schemes based on the idea of the minimization of dispersive phase errors in the wave number domain and these are applied to the analytic optimization of multidiagonal compact schemes actual performances of the optimized compact schemes with a variety of truncation orders are compared by means of numerical simulations of simple wave convections and in this way the most effective compact schemes are found for tridiagonal and pentadiagonal cases respectively from these comparisons the usefulness of an optimized high order tridiagonal compact scheme that is more efficient than a pentadiagonal scheme is discussed for the optimized high order spatial schemes the feasibility of using classical high order runge kutta time advancing methods is investigated

the fourth moment of quadratic dirichlet l functions we study the fourth moment of quadratic dirichlet l functions at s frac 1 2 we show an asymptotic formula under the generalized riemann hypothesis and obtain a precise lower bound unconditionally the proofs of these results follow closely arguments of soundararajan and young j eur math soc 12 5 1097 1116 2010 and soundararajan ann math 2 152 2 447 488 2000

on the density of unnormalized tamagawa numbers of orthogonal groups ii this is the second part of a series of three papers in this series of papers we determine the density of unnormalized tamagawa numbers of projective special orthogonal groups defined over a fixed number field in this part we determine local constants at infinite places which appear in the density theorem and prove the density theorem when the number of variables n is odd for that purpose we make a dirichlet series whose coefficients are unnormalized tamagawa numbers of projective special orthogonal groups and prove that it is rather simple when n is odd

local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws this paper constructs a local third order accurate shock capturing method for hyperbolic scalar conservation laws based on numerical fluxes with a total variation diminishing tvd runge kutta evolution in time using the idea recently introduced by c w shu and s j osher for essentially nonoscillatory eno methods the constructed method is an upwind conservative scheme that is local in the sense that numerical fluxes are reconstructed without using extrapolation from the data of the smoothest neighboring cell to design the method a new concept of local smoothing is introduced to prevent the increasing of total variation of the solution near discontinuities and to achieve third order accuracy the method becomes third order accurate in smooth regions of the solution except at local extrema where it may degenerate to o h 3 2 thus giving better accuracy than tvd methods at local extrema the main advantage of this method lies on the property that is more local than that of eno and tvd upwind

a local riemann hypothesis i daniel bump1 kwok kwong choi2 par kurlberg3 jeffrey vaaler4 1 department of mathematics stanford university stanford ca 94305 2125 usa e mail bump math stanford edu 2 department of mathematics the university of hong kong pokfulam road hong kong hong kong e mail choi maths hku hk 3 department of mathematics university of georgia athens ga 30602 usa e mail kurlberg math uga edu 4 university of texas at austin department of mathematics rlm 8 100 austin tx 78712 1082 usa e mail vaaler math utexas edu

multiresolution based on weighted averages of the hat function ii nonlinear reconstruction techniques in this paper we describe and analyze a class of nonlinear mr schemes for the multiresolution setting which corresponds to discretization by local averages with respect to the hat function these schemes are based on the essentially non oscillatory eno interpolatory procedure described in a harten b engquist s osher and s chakravarthy j comput phys 71 1987 pp 231 302 we show that by allowing the approximation to fit the local nature of the data one can improve the compression capabilities of the multiresolution algorithms the question of stability for nonlinear data dependent reconstruction techniques is also addressed

artin schreier extensions of the rational function field in this paper we study the arithmetic of artin schreier extensions of mathbb f q t we determine the integral closure of mathbb f q t in artin schreier extension of mathbb f q t we also investigate the average values of the l functions of orders of artin schreier extensions and study the average values of ideal class numbers when p 3 in detail

decision trees using model ensemble based nodes decision trees recursively partition the instance space by generating nodes that implement a decision function belonging to an a priori specified model class each decision may be univariate linear or nonlinear alternatively in omnivariate decision trees one of the model types is dynamically selected by taking into account the complexity of the problem defined by the samples reaching that node the selection is based on statistical tests where the most appropriate model type is selected as the one providing significantly better accuracy than others in this study we propose the use of model ensemble based nodes where a multitude of models are considered for making decisions at each node the ensemble members are generated by perturbing the model parameters and input attributes experiments conducted on several datasets and three model types indicate that the proposed approach achieves better classification accuracies compared to individual nodes even in cases when only one model class is used in generating ensemble members

double dirichlet series and the n th order twists of hecke l series let n 3 be a fixed integer and let f be a global field containing the n th roots of unity in this paper we study the collective behavior of the n th order twists of a fixed hecke l series for f to do so we introduce a double dirichlet series in two complex variables s w which is a weighted sum of the twists and obtain its meromorphic continuation we also study related sums of n th order gauss sums these objects together satisfy a nonabelian group of functional equations in s w of order 32

a converse theorem for double dirichlet series we prove that a certain vector valued double dirichlet series satisfying appropriate functional equations is a mellin transform of a vector valued metaplectic eisenstein series we establish an analogous result for scalar double dirichlet series of the type studied by siegel

asymptotic behaviors of class number sums associated with pell type equations it is well known that the pell equation t 2 du 2 4 has infinitely many integer solutions t u for a given d 0 with d equiv 0 1 bmod 4 and d ne l 2 for any integer l however for a square free integer n ne 1 the equation t 2 du 2 4n does not always have integer solutions and verifying its solubility insolubility is not easy in the present paper we propose the asymptotic formulas of the sum of the class numbers h d of the primitive indefinite binary quadratic forms over the discriminants d 0 for which the pell type equation t 2 du 2 4n has an integer solution to study the distributions of such d

a lagrangian eulerian method with zoomable hidden fine mesh approach to solving advection dispersion equations a lagrangian eulerian method with zoomable hidden fine mesh approach lezoom that can be adapted with either finite element or finite difference methods is used to solve the advection dispersion equation the approach is based on automatic adaptation of zooming a hidden fine mesh in regions where the sharp front is located application of lezoom to four bench mark problems indicates that it can handle the advection dispersion diffusion problems with mesh peclet numbers ranged from 0 to infinity and with mesh courant numbers well in excess of 1 difficulties that can be resolved with lezoom include numerical dispersion oscillations the clipping of peaks and the effect of grid orientation nonuniform grid as well as spatial temporally variable flow pose no problems with lezoom both initial and boundary value problems can be solved accurately with lezoom it is shown that although the mixed lagrangian eulerian le approach lezoom without zooming also produces excessive numerical dispersion as the upstream finite element ufe method the le approach is superior to the ufe method

first moment of hecke l functions with quartic characters at the central point in this paper we study the first moment of central values of hecke l functions associated with quartic characters

mean values of biquadratic zeta functions let h d be the number of primitive inequivalent binary quadratic forms of discriminant d in disquisitiones 302 and 304 8 gauss gives conjectures for the average value of h d if d 0 where in the latter case d is derived from the fundamental solution to pell s equation gauss s conjectures were first proved by lipschitz 12 for negative discriminants and siegel for positive discriminants 14 when d is a fundamental discriminant and d the primitive real dirichlet character associated to the quadratic extension q d dirichlet s class number formula relates the value of the l series of d at s 1 to the class number h d thus gauss s conjectured asymptotics are equivalent to conjectures for sums of the type