Abstract: The second part of Hilbert’s 16th problem deals with polynomial differential equations in the plane. It remains unsolved even for quadratic polynomials. There were several attempts to solve it that failed. Yet the problem inspired significant progress in the geometric theory of planar differential equations, as well as bifurcation theory, normal forms, foliations and some topics in algebraic geometry. The dramatic history of the problem, as well as related developments, are presented below. §1. The problem and its counterparts What may be said about the number and location of limit cycles of a planar polynomial vector field of degree n? (The limit cycle is an isolated closed orbit of a vector field.) This second part of Hilbert’s 16th problem appears to be one of the most persistent in the famous Hilbert list [H], second only to the Riemann ζ-function conjecture. Traditionally, Hilbert’s question is split into three, each one requiring a stronger answer. Problem 1. Is it true that a planar polynomial vector field has but a finite number of limit cycles? Problem 2. Is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only? The bound on the number of limit cycles in Problem 2 is denoted by H(n) and known as the Hilbert number. Linear vector fields have no limit cycles; hence H(1) = 0. It is still unknown whether or not H(2) exists. Problem 3. Give an upper bound for H(n). A solution to any of these problems implies a solution for the previous ones. Only the first problem is solved now. The positive answer was established in [E92], [I91]. There are analytic counterparts of Problems 1 and 2. Received by the editors December 2001. 2000 Mathematics Subject Classification. Primary 34Cxx, 34Mxx, 37F75.

Abstract: We report here on our numerical study of the two-dimensional Riemann problem for the com- pressible Euler equations. Compared with the relatively simple 1-D congurations, the 2-D case consists of a plethora of geometric wave patterns which pose a computational challenge for high- resolution methods. The main feature in the present computations of these 2-D waves is the use of the Riemann-solvers-free central schemes presented in (11). This family of central schemes avoids the intricate and time-consuming computation of the eigensystem of the problem, and hence oers a considerably simpler alternative to upwind methods. The numerical results illustrate that despite their simplicity, the central schemes are able to recover with comparable high-resolution, the various features observed in the earlier, more expensive computations. AMS subject classication: Primary 65M10; Secondary 65M05

Abstract: The main result is an upper bound for the Riemann zeta function in the critical strip: $2000 Mathematical Subject Classification: primary 11M06, 11N05, 11L15; secondary 11D72, 11M35.

Abstract: We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the Riemann-Hilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via Deift-Zhou steepest-descent/stationary phase method for Riemann-Hilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of $\beta=2$ symmetry class.

Abstract: This chapter introduces the basic concepts and results of the elementary theory of numbers. Its purpose is twofold: Provide a solid foundation of elementary number theory for Computational, Algorithmic, and Applied Number Theory of the next two chapters of the book. Provide independently a self-contained text of Elementary Number Theory for Computing, or in part a text of Mathematics for Computing.

Abstract: In this paper we introduce and investigate the concepts of Riemann's delta and nabla integrals on time scales. Main theorems of the integral calculus on time scales are proved.

Abstract: The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-) periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula is used to construct a uniform approximation formula, again with arbitrary precision.

Abstract: A small part of the Generalized Riemann Hypothesis asserts that L-functions do not have zeros on the line segment ( 2 , 1]. The question of vanishing at s = 2 often has deep arithmetical significance, and has been investigated extensively. A persuasive view is that L-functions vanish at 2 either for trivial reasons (the sign of the functional equation being negative), or for deep arithmetical reasons (such as the L-function of an elliptic curve of positive rank) and that the latter case happens very rarely. N. Katz and P. Sarnak [7] have formulated precise conjectures on the low lying zeros in families of L-functions which support this view. In the case of Dirichlet L-functions it is expected that L( 2 , χ) is never zero, and so L(σ, χ) = 0 for all 2 ≤ σ ≤ 1. This conjecture appears to have been first enunciated by S.D. Chowla [2] in the special case of quadratic characters χ. Progress towards these non-vanishing questions has been in two directions: zero-density type results which establish that very few L-functions have a zero in ( 2 + , 1] (see for example A. Selberg [10], M. Jutila [6] and D.R. Heath-Brown [4]), and a growing body of work on non-vanishing at 2 (see for example R. Balasubramanian and V.K. Murty [1], H. Iwaniec and Sarnak [5], and K. Soundararajan [11]). Further much numerical evidence for the GRH has been accumulated, and these calculations support Chowla’s conjecture (see [8] and [9]). However the state of knowledge could not exclude the possibility that every Dirichlet L-function of sufficiently large conductor has a non-trivial real zero. In this

Abstract: We deduce four new integral representations for � (2n +1 ) ;n ∈ N , where � (s) is the Riemann zeta function. c

Abstract: Let $\rho(x)=x-[x]$, $\chi=\chi_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{\rho_a | a \geq 1\}$ where $\rho_a(x):=\rho(\frac{1}{ax})$. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement $\chi\in\bar{\B}$. For some time it has been conjectured, and proved in this paper, that the Riemann hypothesis is equivalent to the stronger statement that $\chi\in\bar{\Bnat}$ where $\Bnat$ is the much smaller subspace generated by $\{\rho_a | a\in\Nat\}$.

Abstract: Abstract In this paper we compare the accuracy and efficiency of a finite volume scheme solving the ideal MHD equations in one and two space dimensions for interface fluxes obtained by six different MHD Riemann solvers. We suggest a new solver of HLLEM-type (MHD–HLLEM) and show that it is the cheapest one in our comparison. Moreover, we study if the extra effort required for higher order schemes is justified by the gain in accuracy. The two-dimensional version of our code is based on unstructured triangular meshes.

Abstract: In this work we provide a characterization of C 1,1 functions on R n (that is, dierentiable with locally Lipschitz partial derivatives) by means of second directional divided dierences. In particular, we prove that the class of C 1,1 functions is equivalent to the class of functions with bounded second directional divided dierences. From this result we deduce a Taylor’s formula for this class of functions and some optimality conditions. The characterizations and the optimality conditions proved by Riemann derivatives can be useful to write minimization algorithms; in fact, only the values of the function are required to compute second order conditions.

Abstract: For simplicity, we follow the rules: a, b are real numbers, F , G, H are finite sequences of elements of R, i, j, k are natural numbers, X is a non empty set, and x1 is a set. Let I1 be a subset of R. We say that I1 is closed-interval if and only if: (Def. 1) There exist real numbers a, b such that a ≤ b and I1 = [a, b]. Let us mention that there exists a subset of R which is closed-interval. In the sequel A is a closed-interval subset of R. One can prove the following propositions:

Abstract: We prove a relation between the asymptotic behavior of the conformal factor and the accessory parameters of the SU(1,1) Riemann- Hilbert problem. Such a relation shows the hamiltonian nature of the dynamics of N particles coupled to 2+1 dimensional gravity. A generalization of such a result is used to prove a connection between the regularized Liouville action and the accessory parameters in presence of general elliptic singularities. This relation had been conjectured by Polyakov in connection with 2-dimensional quantum gravity. An alternative proof, which works also in presence of parabolic singularities, is given by rewriting the regularized Liouville action in term of a background field.

Abstract: In this work we provide a characterization of C^(k,1) functions on R^n (that is k times differentiable with locally Lipschitz k-th derivatives) by means of (k+1)-th divided differences and Riemann derivatives. In particular we prove that the class of C^(k,1) functions is equivalent to the class of functions with bounded (k+1)-th divided difference. From this result we deduce a Taylor's formula for this class of functions and a characterization through Riemann derivatives.

Abstract: In this article, the method of differential constraintsis applied for systems written in Riemann variables. Westudied generalized simple waves. This class of solutions can beobtained by integrating a system of ordinary differentialequations. Two models from continuum mechanics are studied:traffic flow and rate-type models.

Abstract: We present an extension of the Riemann circle fit to a helix fit in space The method is studied both in barrel- and disk-type detectors We show results from two simulation experiments, including a comparison to linear regression and to the Kalman filter An implementation in C++ is described

Abstract: . We study the zero distributions of the Lerch zeta-functions L(λ, α, s) = ∑∞ n=0 e(λn) (n+α)s for the parameters 0 < λ, α ≤ 1 . Our observations show some analogies to the Riemann zeta-function (existence and number of trivial and nontrivial zeros) and some differences (asymmetrical distribution of the nontrivial zeros for almost all L(λ, α, s) ). Further, we investigate the distribution of zeros of the derivatives.

Abstract: The alternating zeta function zeta*(s) = 1 - 2^{-s} + 3^{-s} - ... is related to the Riemann zeta function by the identity (1-2^{1-s})zeta(s) = zeta*(s). We deduce the vanishing of zeta*(s) at each nonreal zero of the factor 1-2^{1-s} without using the identity. Instead, we use a formula connecting the partial sums of the series for zeta*(s) to Riemann sums for the integral of x^{-s} from x=1 to x=2. We relate the proof to our earlier paper "The Riemann Hypothesis, simple zeros, and the asymptotic convergence degree of improper Riemann sums," Proc. Amer. Math. Soc. 126 (1998) 1311-1314.

Abstract: While conducting a von Neumann stability analysis of discontinuous Galerkin methods we discovered that the classic Lax-Friedrichs Riemann solver is unstable for all time-step sizes. We describe a simple modification of the Riemann solver's dissipation returns the method to stability. Furthermore, the method has a smaller truncation error than the corresponding method with an upwind flux for the RK2-DG(1) method

Abstract: The L 2 -zeta function of an infinite graph Y (defined previously in a ball around zero) has an analytic extension. For a tower of finite graphs covered by Y, the normalized zeta functions of the finite graphs converge to the L 2 -zeta function of Y.

Abstract: Soit a fixe dans un corps quadratrique K. On note S l'ensemble des nombres premiers p pour lesquels a admet un ordre maximal modulo p. Sous G.R.H., on montre que S a une densite. Nous donnons aussi des conditions necessaires et suffisantes pour que cette densite soit strictement positive.

Abstract: This paper presents the technical details necessary to implement an exact solver for the Riemann problem of magnetohydrodynamics (MHD) and investigates the uniqueness of MHD Riemann solutions. The formulation of the solver results in a nonlinear algebraic 5 × 5 system of equations which has to be solved numerically. The equations of MHD form a non-strict hyperbolic system with non-convex fluxfunction. Thus special care is needed for possible non-regular waves, like compound waves or overcompressive shocks. The structure of the Hugoniot loci will be demonstrated and the non-regularity discussed. Several non-regular intermediate waves could be taken into account inside the solver. The non-strictness of the MHD system causes the Riemann problem also to be not unique. By virtue of the structure of the Hugoniot loci it follows, however, that the degree of freedom is reduced in the case of a non-regular solution. ?From this, uniqueness conditions for the Riemann problem of MHD are deduced.

Abstract: In this paper, by using Fourier series theory, several summing formulae for Riemann Zeta function ζ(s) and Dirichlet series are deduced.