A Three Squares Theorem with almost Primes

As an application of the vector sieve and uniform estimateson the Fourier coefficients of cusp forms of half-integral weight,it is shown that any sufficiently large number n 3 (mod 24)with 5 n is expressible as a sum of three squares of integershaving at most 521 prime factors. 2000 Mathematics Subject Classification11P05, 11N36, 11N75, 11E25.     

On the Hecke Equivariance of the Borcherds Isomorphism

The Borcherds isomorphism is proved to be Hecke equivariantif one considers multiplicative Hecke operators acting on theintegral weight meromorphic modular forms. This answers a partof a question of Borcherds (see ‘Automorphic forms onO_{s+2, 2}(R) and infinite products’, Invent. Math. 120 (1995)161–213, 17.10), using his suggestion to define the multiplicativeHecke operators. 2000 Mathematics Subject Classification 11F37.     

On Primes and Powers of a Fixed Integer

According to a 1904 conjecture of Dickson [2] unless preventedby congruence conditions, any finite collection of linear formsin Z[x] with positive leading coefficients infinitely oftensimultaneously represent primes. For the forms x, x+2 this includesthe conjectured infinitude of prime-pairs.     

Zagier Duality for the Exponents of Borcherds Products for Hilbert Modular Forms

A certain sequence of weight modular forms arises in the theoryof Borcherds products for modular forms for SL₂(Z). Zagier proveda family of identities between the coefficients of these weight forms and a similar sequence of weight 3/2 modular forms, whichinterpolate traces of singular moduli. We obtain the analogousresults for modular forms arising from Borcherds products forHilbert modular forms.     

Modularity of Galois traces of class invariants

Zagier showed that the Galois traces of the values of j-invariant at CM points are Fourier coefficients of a weakly holomorphic modular form of weight 3/2 and Bruinier–Funke expanded his result to the sums of the values of arbitrary modular functions at Heegner points. In this paper, we identify the Galois traces of real-valued class invariants with modular traces of the values of certain modular functions at Heegner points so that they are Fourier coefficients of weight 3/2 weakly holomorphic modular forms.     

Oscillation theory and numerical solution of fourth-order Sturm--Liouville problems

A shooting method is developed to approximate the eigenvaluesand eigenfunc-tions of a fourth-order Sturm-Liouville problem.The main tool is a miss-distance function M(), which countsthe number of eigenvalues less than A. The method approximatesthe coefficients of the differential equation by piecewise-constantfunctions, which enables an exact solution to be found on eachmesh interval. In order to calculate N() for the approximateproblem, certain oscillation numbers N_L and N_R must be computed.These consist of sums of nullities (or rank deficiencies) of2 x 2 matrices obtained from the solutions of the approximatedifferential equation. Although these solutions can be foundexplicitly, the calculation of N_L and N_R is non-trivial, andis obtained by using certain properties of M().     

Traces of Singular Values and Borcherds Products

Let p be a prime for which the congruence group ₀(p)* is ofgenus zero, and let be the corresponding Hauptmodul. Let f be a nearly holomorphic modularform of weight 1/2 on ₀(4p) which satisfies some congruencecondition on its Fourier coefficients. We interpret f as a vectorvalued modular form. Applying Borcherds lifting of vector valuedmodular forms we construct infinite products associated to and relate them to Zagier's traceformula for the singular values of . 2000 Mathematics Subject Classification 11F03, 11F30 (primary);11F22, 11F37, 11F50 (secondary).     

Spatial decay in a cross-diffusion problem

In this paper, the authors investigate the decay of end effectsfor a cross-diffusion problem defined on a semi-infinite cylindricalregion. With homogeneous Dirichlet or Neumann conditions prescribedon the lateral surface of the cylinder, it is shown that forfixed finite time and under certain restrictions on the coefficients,solutions decay point-wise as the distance d from the finiteend of the cylinder tends to infinity at least of order e^–kd2.Under less restrictive conditions, it is shown that solutionsdecay in L₂ at least as fast as e^–kd. In both cases, kis a computable function of time.     

On the Existence of a Solution in Weighted Sobolev Space to the Riemann-Hilbert Problem for an Elliptic System with Piecewise Continuous Boundary Data

Given a first order elliptic partial differential equation weconstruct a solution which solves a given Riemann–Hilbertboundary value problem whose coefficients have singularitiesof the first kind at a finite number of some prescribed isolatedpoints and are Holder–continuous outside those pointswhile the free term has a finite number of integrable powersingularities at some prescribed points. It is shown that thesolution belongs to some weighted Sobolev space W_1,p(D;), wherethe weight function = (z;D) is the distance of the variablepoint z from the boundary D raised to a certain power. The problemis solved by first reducing it to an analogous problem for holomorphicfunctions. The latter is then solved. 2000 Mathematics SubjectClassification 35F15, 35F30, 35J55, 35J65, 36J67.     

Global convexity in a single-source 3-D inverse scattering problem

The authors consider nonoverdetermined 3-D inverse scatteringproblems based on the telegraph equation The goal is to recover the media properties represented by thecoefficients a(x) or b(x) from, for example, backscatteringdata. Such a problem models imaging in certain biological tissues,in murky water, and in some geophysical and atmospheric phenomena.The main restriction is in the consideration of only finitelymany Fourier harmonics of the solution u(x, t) (in the timevariable t only). This seems to be acceptable for practicalcomputations. The main mathematical tool is the constructionof uniformly convex cost functionals on compact convex subsetsof the solutions. This assures a global convergence of the minimizationalgorithm, which can be applied in the case of large media inhomogeneities.Since the technique is based on the so-called Carleman's weightfunctions, the approach is called Carleman's weight method.     

A simple proof of Zagier duality for Hilbert modular forms

In this paper, we give a simple proof of an identity between the Fourier coefficients of the weakly holomorphic modular forms of weight 0 arising from Borcherds products of Hilbert modular forms and those of the weakly holomorphic modular forms of weight satisfying a certain property.

     

Domain decomposition preconditioners of Neumann-Neumann type for hp-approximations on boundary layer meshes in three dimensions

We develop and analyse Neumann–Neumann methods for hpfinite-element approximations of scalar elliptic problems ongeometrically refined boundary layer meshes in three dimensions.These are meshes that are highly anisotropic where the aspectratio typically grows exponentially with the polynomial degree.The condition number of our preconditioners is shown to be independentof the aspect ratio of the mesh and of potentially large jumpsof the coefficients. In addition, it only grows polylogarithmicallywith the polynomial degree, as in the case of p approximationson shape-regular meshes. This work generalizes our previousone on two-dimensional problems in Toselli & Vasseur (2003a,submitted to Numerische Mathematik, 2003c to appear in Comput.Methods Appl. Mech. Engng.) and the estimates derived here canbe employed to prove condition number bounds for certain typesof FETI methods.     

Primes Represented by Binary Cubic Forms

Let f(x, y) be a binary cubic form with integral rational coefficients,and suppose that the polynomial f(x, y) is irreducible in Q[x,y] and no prime divides all the coefficients of f. We provethat the set f Z⁽²⁾ contains infinitely many primes unless f(a,b) is even for each (a,b) in Z², in which case the set contains infinitely many primes. 2000Mathematical Subject Classification: primary 11N32; secondary11N36, 11R44.     

Omega Limit Sets in Positive Cones

The omega limit sets of a nonlinear operator T which is definedon a positive cone and satisfies certain ray-contractive typeconditions are discussed. Under the assumption that the restrictionof T to a compact subset is surjective, the following alternativesare proved: the omega limit set of a point in the cone eitherconsists of a fixed point or forms a 2-cycle. In addition, newproofs and extensions to relevant results are given. 2000 MathematicsSubject Classification 47H07, 47H09.     

On the Determination of Quadrature Formulae of Highest Degree of Precision for Approximating Fourier Coefficients

Let Q₀(x), Q₁(x),..., Q_n(x),... be a sequence of polynomialswhich are orthogonal with respect to the inner product . In estimating the Fourier coefficients a₁ = $$\langlef,{Q}_{i}\rangle $$, it is natural to use a quadrature formulaof highest possible degree of precision to approximate $${\int}_{a}^{b}W\left(x\right)f\left(x\right)dx$$ where W(x) = w(x)Q_i(x).Since the weight function W(x) changes sign i times in (a, b),the usual results of quadrature theory do not apply. This paperdevelops a procedure which is an initial attempt to determinewhat degree of precision is attainable.     

Differences Between Consecutive Primes

Let p_n be the nth prime. Then this paper is concerned with provingthe following result on the distribution of consecutive primes. The exponent of x in this theorem improves on the work of Heath-Brownwho proved (1) with exponent . Under the Riemann hypothesisone can prove (1) with exponent .The proof of the theorem startswith the Heath-Brown–Linnik identity which leads to aformula giving the number of primes in an interval in termsof coefficients of certain Dirichlet series. I then estimatethe coefficients by using, among other things, the informationwhich can be gained from Montgomery's mean value theorem andHuxley's version of the Hal' asz lemma. Furthermore, by usingfamiliar sieve arguments I am able to discard some of the coefficientsallowing us to gain an improvement over the previous resultof Heath-Brown. 1991 Mathematics Subject Classification: 11N05.     

Brauer Group Invariants Associated to Orthogonal Epsilon-Constants

In this paper, the theory of -constants associated to tame finitegroup actions on arithmetic surfaces is used to define a Brauergroup invariant µ(X, G, V) associated to certain symplecticmotives of weight one. The relationship between this invariantand w₂() (the Galois-theoretic invariant associated to tamecovers of surfaces defined by Cassou-Noguès, Erez andTaylor) is also discussed. 2000 Mathematics Subject Classification11G35 (primary), 11G40, 14G40 (secondary).     

A Class of Pencils of Matrices

A linear machine is one in which the time dependent input yis related to the output z by P(D). z = S(D). y where P andS are polynomials in D = d/dt with constant coefficients. Fornumerical computation it is necessary to replace this relationby a set of simultaneous first order differential equationsand this paper shows how to construct such equations by methodswhich extend the results of Gilder (1961). Attention is restrictedto those sets of equations that are of a special form (see (1))which is characterized by the matrix operating on the dependentvariables. This matrix forms a pencil, being linear in D, andthree theorems are given to show how such matrix pencils maybe constructed from the polynomials. The theorems also statethat any matrix pencil with the required properties can be transformedinto the canonical forms given in the theorems by pre- and post-multiplicationby suitable constant non-singular matrices. Thus the variablesof any set of equations having the required properties are linearcombinations of the variables of the equations given by thetheorems. In the paper it is assumed that the degree of P(D)is greater than that of S(D), as otherwise z would be replacedby z₁+Q(D) . y, where Q is the quotient of S(D)/P(D). Also,as the algebriac manipulations are independent of the natureof the polynomials, D is replaced by an indeterminate x andthe coefficients considered to be from an arbitrary field. Fortechnical reasons we rename y and z, y_o and y_n–_m respectively.     

Integrals and Areas Expressed as Bilinear Forms of Discrete Defining Data

An ordered set of discrete data-pairs (x_k, y_k) is supposed givenfor k = 1(1)n, derived perhaps as a consequence of experiment,and indicating some form of relationship between x and y. Itis required that these data be interpolated in some systematicway so as to establish this relationship in the form of a rectifiablecurve in the (x, y) plane, and that y is then to be integratedwith respect to x (or vice versa), thereby defining an areain this plane. We shall here consider interpolation schemeswhere by this integral can be expressed as a bilinear form _jKC_jkY_jx_k,the "weighting" coefficients C_jk being numbers independent ofthe data values (though not necessarily of their number, n). The expressions obtained sepcialize to Gregory formulae, andti Simpson's Rule and other forms of Cotes formulae in suitablecontexts, but the information is primarily of use in dealingwith unequally-spaced data values, or in estumating the areawithin a closed curve. The simplicity of the numerical algorithmis clearly unaffected by the geometric complexity of the interpolatedcurve. Values for the weighting coefficients are proposed, and theresults of a numerical experiment are described which teststhe applicability of the formulae described.     

Rational decomposition of modular forms

We prove explicit formulas decomposing cusp forms of even weight for the modular group, in terms of generators having rational periods, and in terms of generators having rational Fourier coefficients. Using the Shimura correspondence, we also give a decomposition of Hecke cusp forms of half integral weight k+1/2 with k even in terms of forms with rational Fourier coefficients, given by Rankin–Cohen brackets of theta series with Eisenstein series.