On the Pair Correlation of Zeros of the Riemann Zeta-Function

To study the distribution of pairs of zeros of the Riemann zeta-function,Montgomery introduced the function where is real and T 2, and ' denote the imaginary parts ofzeros of the Riemann zeta-function, and w(u) = 4/(4 + u²). Assumingthe Riemann Hypothesis, Montgomery proved an asymptotic formulafor F() when || 1, and made the conjecture that F() = 1 + o(1)as T for any bounded with || 1. In this paper we use anapproximation for the prime indicator function together witha new mean value theorem for long Dirichlet polynomials andtails of Dirichlet series to prove that, assuming the GeneralizedRiemann Hypothesis for all Dirichlet L-functions, then for any > 0 we have uniformlyfor and all T T₀(). 1991Mathematics Subject Classification: primary 11M26; secondary11P32.     

Metrical Theorems on Prime Values of the Integer Parts of Real Sequences

Let a_n be an increasing sequence of positive reals with a_n as n . Necessary and sufficient conditions are obtained foreach of the sequences to take on infinitely many prime values for almost all > rß.For example, the sequence a_n is infinitely often prime for almostall > 0 if and only if there is a subsequence of the a_n,say b_n, with b_{n + 1} > b_n + 1 and with the series divergent. Asymptotic formulae areobtained when the sequences considered are lacunary. An earlierresult of the author reduces the problem to estimating the measureof overlaps of certain sets, and sieve methods are used to obtainthe correct order upper bounds. 1991 Mathematics Subject Classification:primary 11N05; secondary 11K99, 11N36.     

Higher correlations of divisor sums related to primes II: variations of the error term in the prime number theorem

We calculate the triple correlations for the truncated divisorsum _R(n). The _R(n) behave over certain averages just as theprime counting von Mangoldt function (n) does or is conjecturedto do. We also calculate the mixed (with a factor of (n)) correlations.The results for the moments up to the third degree, and thereforethe implications for the distribution of primes in short intervals,are the same as those we obtained (in the first paper with thistitle) by using the simpler approximation _R(n). However, when_R(n) is used, the error in the singular series approximationis often much smaller than what _R(n) allows. Assuming the GeneralizedRiemann Hypothesis (GRH) for Dirichlet L-functions, we obtainan _±-result for the variation of the error term in theprime number theorem. Formerly, our knowledge under GRH wasrestricted to -results for the absolute value of this variation.An important ingredient in the last part of this work is a recentresult due to Montgomery and Soundararajan which makes it possiblefor us to dispense with a large error term in the evaluationof a certain singular series average. We believe that our resultson the sums _R(n) and _R(n) can be employed in diverse problemsconcerning primes.     

A bifurcation problem governed by the boundary condition II

In this work we consider the problem u = a(x)u^p in on , where is a smooth bounded domain, isthe outward unit normal to , is regarded as a parameter and0 < p < 1. We consider both cases where a(x) > 0 in or a(x) is allowed to vanish in a whole subdomain ₀ of . Ourmain results include existence of non-negative non-trivial solutionsin the range 0 < < ₁, where ₁ is characterized by meansof an eigenvalue problem, uniqueness and bifurcation from infinityof such solutions for small , and the appearance of dead coresfor large enough .     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).     

Moduli of McKay quiver representations I: the coherent component

For a finite abelian group G GL (n, ), we describe the coherent component Y of the moduli space of-stable McKay quiver representations. This is a not-necessarily-normaltoric variety that admits a projective birational morphism obtained by variation of GeometricInvariant Theory quotient. As a special case, this gives a newconstruction of Nakamura's G-Hilbert scheme Hilb^G that avoidsthe (typically highly singular) Hilbert scheme of |G|-pointsin . To conclude, we describe the toric fan of Y and hence calculate the quiver representationcorresponding to any point of Y.     

Grothendieck's Inequalities for Real and Complex JBW*-Triples

We prove that, if and >0, if V and W are complex JBW*-triples (with preduals V_* andW_*, respectively), and if U is a separately weak*-continuousbilinear form on V x W, then there exist norm-one functionals₁, ₂ V_* and ₁, ₂ W_* satisfying for all (x, y) V x W. Here, for a norm-one functional on acomplex JB*-triple V, |·| stands for the prehilbertianseminorm on V associated to given by for all x W, where z V^** satisfies z = |z| =1. We arrive at this form of ‘Grothendieck's inequality’through results of C.-H. Chu, B. Iochum, and G. Loupias, andan amended version of the ‘little Grothendieck's inequality’for complex JB*-triples due to T. Barton and Y. Friedman. Wealso obtain extensions of these results to the setting of realJB*-triples. 2000 Mathematical Subject Classification: 17C65,46K70, 46L05, 46L10, 46L70.     

On the laplace equation with non-linear dynamical boundary conditions

The main part of the paper deals with local existence and globalexistence versus blow-up for solutions of the Laplace equationin bounded domains with a non-linear dynamical boundary condition.More precisely, we study the problem consisting in: (1) theLaplace equation in (0, ) x ; (2) a homogeneous Dirichlet condition(0, ) x ₀; (3) the dynamical boundary condition ; (4) the initial condition u(0, x) = u₀ (x) on . Here is a regular and bounded domain in Rⁿ, with n 1, and₀ and ₁ endow a measurable partition of . Moreover, m>1,2 p < r, where r = 2 (n – 1) / (n – 2) whenn 3, r = when n = 1,2, and u₀ H^1/2 , u₀ = 0 on ₀. The final part of the paper deals with a refinement of a globalnon-existence result by Levine, Park and Serrin, which is appliedto the previous problem. 2000 Mathematics Subject Classification35K55 (primary), 35K90, 35K77 (secondary).     

On the Fredholm Alternative for the p-Laplacian in One Dimension

We investigate the existence of a weak solution u to the quasilineartwo-point boundary value problem We assume that 1 < p < p ¬ = 2, 0 < a < , andthat f L¹(0,a) is a given function. The number _k stands forthe k-th eigenvalue of the one-dimensional p-Laplacian. Let_{p p} x/a) denote the eigenfunction associated with ₁; then _p(k_p x/a) is the eigenfunction associated with _k. We show the existenceof solutions to (P) in the following cases. (i) When k=1 and f satisfies the orthogonality condition the set of solutions is bounded. (ii) If k=1 and f_t L¹(0,a) is a continuous family parametrizedby t [0,1], with then there exists some t^* [0,1] such that (P) has a solutionfor f = f_t^*. Moreover, an appropriate choice of t_* yields asolution u with an arbitrarily large L¹(0,a)-norm which meansthat such f cannot be orthogonal to _pp x/a. (iii) When k 2 and f satisfies a set of orthogonality conditionsto _p(k p x/a) on the subintervals , again, the set of solutions is bounded. is a continuous family satisfying either or another related condition, then there exists some t^* [0,1]such that (P) has a solution for f = f_t*. Prüfer's transformation plays the key role in our proofs.2000 Mathematical Subject Classification: primary 34B16, 47J10;secondary 34L40, 47H30.     

Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball

Weighted Triebel–Lizorkin and Besov spaces on the unitball B^d in ^d with weights w_µ(x)=(1–|x|²)^µ–1/2,µ0, are introduced and explored. A decomposition schemeis developed in terms of almost exponentially localized polynomialelements (needlets) {}, {} and it is shown that the membershipof a distribution to the weighted Triebel–Lizorkin orBesov spaces can be determined by the size of the needlet coefficients{f, } in appropriate sequence spaces.     

Weyl-Titchmarsh M-Function Asymptotics for Matrix-valued Schrodinger Operators

We explicitly determine the high-energy asymptotics for Weyl–Titchmarshmatrices corresponding to matrix-valued Schrödinger operatorsassociated with general self-adjoint m x m matrix potentials, where m N. More precisely,assume that for some N N and x₀R, for all c>x₀, and that x x₀ is a right Lebesgue point ofQ^(N–1). In addition, denote by I_m the mxm identity matrixand by C the open sector in thecomplex plane with vertex atzero, symmetry axis along the positive imaginary axis, and openingangle , with 0 < < . Then we prove the following asymptoticexpansion for any point M_+(z,x) of the unique limit point ora point of the limit disk associated with the differential expression in and a Dirichlet boundary condition at x=x₀: The expansion is uniform with respect to arg(z)for |z| in C and uniform in x as long as x varies in compactsubsets of R intersected with the right Lebesgue set of Q^(N–1).Moreover, the m x m expansion coefficients m_+,k(x) can be computedrecursively. Analogous results hold for matrix-valued Schrödinger operatorson the real line. 2000 Mathematics Subject Classification: 34E05,34B20, 34L40, 34A55.     

Almost Everywhere Convergence of Bochner-Riesz Means On The Heisenberg Group and Fractional Integration On The Dual

Let L denote the sub-Laplacian on the Heisenberg group H_n and the corresponding Bochner-Riesz operator. Let Q denote the homogeneous dimension and D the Euclideandimension of H_n. We prove convergence a.e. of the Bochner-Rieszmeans as r 0 for > 0and for all f L^p(H_n), provided that . Our proof is based on explicit formulas for the operators with a C, defined on the dual ofH_n by , which may be of independent interest. Here is given by for all (z,u) H_n. 2000 Mathematical Subject Classification: 22E30, 43A80.     

Unicity of Types for Supercuspidal Representations of GLN

Let F be a non-Archimedean local field, with the ring of integerso_F. Let G = GL_N(F), K = GL_N (o_F), and be a supercuspidal representationof G. We show that there exists a unique irreducible smoothrepresentation of K, such that the restriction to K of a smoothirreducible representation ' of G contains if and only if 'is isomorphic to ° det, where is an unramified quasicharacterof F^x. Moreover, we show that contains with the multiplicity1. As a corollary we obtain a kind of inertial local Langlandscorrespondence. 2000 Mathematics Subject Classification 22E50.     

Algebraic K-Theory In Low Degree and The Novikov Assembly Map

We prove that the Novikov assembly map for a group factorizes,in ‘low homological degree’, through the algebraicK-theory of its integral group ring. In homological degree 2,this answers a question posed by N. Higson and P. Julg. As adirect application, we prove that if is torsion-free and satisfiesthe Baum-Connes conjecture, then the homology group H₁(; Z)injects in and in , for any ring A such that . If moreover B is of dimension lessthan or equal to 4, then we show that H₂(; Z) injects in and in , where A is as before, and ₂ is generated by the Steinberg symbols{,}, for . 2000 Mathematical Subject Classification: primary 19D55, 19Kxx,58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.     

The Approximation Numbers of Hardy-Type Operators on Trees

The Hardy operator T_a on a tree is defined by Properties of T_a as a map from L^p() into itselfare established for 1 p . The main result is that, with appropriateassumptions on u and v, the approximation numbers a_n(T_a) ofT_a satisfy for a specified constant _p and 1 p < . This extends results of Naimark, Newmanand Solomyak for p = 2. Hitherto, for p 2, (*) was unknowneven when is an interval. Also, upper and lower estimates forthe l^q and weak-l^q norms of a_n(T_a) are determined. 2000 MathematicalSubject Classification: 47G10, 47B10.     

Dynamics of projective morphisms having identical canonical heights

Let , :^{N N} be morphisms of degree at least 2 whose canonicalheights and are identical. We draw various conclusions aboutthe Green functions, Julia sets, and canonical local heightsof and . We use this information to completely characterize and in the following cases: (i) and are polynomial mapsin one variable; (ii) is the dth-power map; (iii) is a Lattèsmap.     

Spectra of graph neighborhoods and scattering

Let (G)_>0 be a family of ‘-thin’ Riemannian manifoldsmodeled on a finite metric graph G, for example, the -neighborhoodof an embedding of G in some Euclidean space with straight edges.We study the asymptotic behavior of the spectrum of the Laplace–Beltramioperator on G, as 0, for various boundary conditions. We obtaincomplete asymptotic expansions for the kth eigenvalue and theeigenfunctions, uniformly for kC^–1, in terms of scatteringdata on a non-compact limit space. We then use this to determinethe quantum graph which is to be regarded as the limit object,in a spectral sense, of the family (G). Our method is a directconstruction of approximate eigenfunctions from the scatteringand graph data, and the use of a priori estimates to show thatall eigenfunctions are obtained in this way.     

Ideal Spaces of Banach Algebras

The ideal space Id(A) of a Banach algebra A is studied as abitopological space Id(A), _u, _n, where _u is the weakest topologyfor which all the norm functions I || a + I|| (with a A andI Id(A)) are upper semi-continuous, and _n is the de Groot dualof _u. When A is separable, _nu is either a compact, metrizabletopology, or it is neither Hausdorff nor first countable. TAF-algebrasare shown to exhibit the first type of behaviour. Applicationsto Banach bundles (which motivate the study), and to PI-Banachalgebras, are given. 1991 Mathematics Subject Classification:46H10, 46J20.