Abstract: The convolution is defined as the sum Open image in new window where Open image in new window for n≠0 Open image in new window and W0,W1 are arbitrary smooth functions Question: how to express these sums in the form of the combinations of the N-th Fourier coefficients of the eigenfunctions of the automorphic Laplacian? The answer is given in terms of the bilinear form of the Hecke series associated with the eigenfunctions of the automorphic Laplacian and with regular cusp forms The final identity may lead to a new possibility for the solution of the moment problem of the Riemann zeta-function

Abstract: It is shown that assuming Generalized Riemann Hypothesis, the roots of ƒ(x) = O mod p, where p is a prime and f(x) is an integral Abilene polynomial can be found in deterministic polynomial time. The method developed for solving this problem is also applied to prime decomposition in Abelian number fields, and the following result is obtained: assuming Generalized Riemann Hypotheses, for Abelian number fields K of finite extension degree over the rational number field Q, the decomposition pattern of a prime p in K, i.e. the ramification index and the residue class degree, can be computed in deterministic polynomial time, providing p does not divide the extension degree of K over Q. It is also shown, as a theorem fundamental to our algorithm, that for q, p prime and m the order of p mod q, there is a q-th nonresidue in the finite field Fpm that can be written as ao + a1w + … + am-1wm-1, where |a1| ≤ cq2 log2(pq), c is an absolute effectively computable constant, and 1, w, …, wm-1 form a basis of Fpm over Fp. More explicitly, w is a root of the q-th cyclotomic polynomial over Fp. This result partially generalizes, to finite field extensions over Fp, a classical result in number theory stating that assuming Generalized Riemann Hypothesis, the least q-th nonresidue mod p for p,q prime and q dividing p - t is bounded by c log2p, where c is an absolute, effectively computable constant.

Abstract: This paper discusses some new integer factoring methods involving cyclotomic polynomials. There are several polynomials f(X) known to have the following property: given a multiple of f(p), we can quickly split any composite number that has p as a prime divisor. For example -- taking f(X) to be X- 1 -- a multiple of p - 1 will suffice to easily factor any multiple of p, using an algorithm of Pollard. Other methods (due to Guy, Williams, and Judd) make use of X + 1, X2 + 1, and X2 ± X + 1. We show that one may take f to be Φk, the k-th cyclotomic polynomial. In constrast to the ad hoc methods used previously, we give a universal construction based on algebraic number theory that subsumes all the above results. Assuming generalized Riemann hypotheses, the expected time to factor N (given a multiple E of Φk(p)) is bounded by a polynomial in k, logE, and logN.

Abstract: A geometric representation for information theory is introduced by recourse to a covariant formulation, according to the customary procedure employed in connection with Riemann spaces. The central tool of this representation is the metric tensor that characterizes the particular dynamics of a given system and yields the corresponding quantal invariants.

Abstract: : We are interested in classifying the solutions of Riemann problems for the 2 x 2 conservation laws which have homogeneous quadratic flux functions. Such flux functions approximate an arbitrary 2 x 2 system in a neighborhood of an isolated point where strict hyperbolicity fails. This problem was motivated by Marchesin and Paes-Leme who discovered such a singularity in a system of equations arising in oil reservoir simulation. Schaeffer, Shearer, Marchesin and Paes-Leme solved the Riemann problem for this system in a neighborhood of the singular point. Isaacson and Temple outlined a program for classifying such singularities by means of locating normal forms for the equivalence classes of equations generated by linear changes in dependent variables. A 2-parameter family of such normal forms were found by Plohr. In the important work of Schaeffer and Shearer a new normal form was found which reduced the classification of integral curves to a theorem of Darboux on the classification of umbilic points for homogeneous cubic equations. The integral curves fall into four isomorphism classes, called Regions I-IV. In this paper we give the solution of the Riemann problem for the systems in Region IV which exhibit up-down symmetry. A presentation of the solutions of the corresponding systems in Regions II and III will follow.

Abstract: Several exact representations (as an integral and as an infinite series) for the partial derivative delta zeta (z,q)/ delta z mod z=-1 of the generalized Riemann zeta function zeta (z,q) are given.

Abstract: On obtient un theoreme d'existence pour le probleme de Riemann pour des systemes non lineaires de lois de conservation. On utilise l'inegalite d'entropie pour obtenir une estimation a priori des forces des chocs et des ondes de refraction formant une solution; il s'ensuit l'existence d'une telle solution par une application de la theorie du degre de dimension finie

Abstract: We produce examples of Riemann surfaces whose n-first eigenvalues of the laplacian are of large multiplicity On donne des exemples de surfaces de Riemann dont les n premieres valeurs propres du laplacien sont de multiplicite elevee

Abstract: The author proves two theorems of a metric character on the distribution of zeros of odd order of the Riemann zeta-function that lie on the critical line. Bibliography: 5 titles.

Abstract: An efficient algorithm is presented for solving the Riemann problem for a polytropic gas. It enables the user to compute the solution for all physically reasonable data. The convergence of the algorithm is proved. The accuracy of the solution is limited only by the accuracy of the computer. There is an a-priori estimation of the required number of iterations. The rate of convergence turns out to be much higher than that of the usual fixed point iteration scheme.

Abstract: On the inversion of a linear differential polynomial by C.-T. Chuang Distribution of the values of meromorphic functions by C.-T. Chuang and L. Yang A simple proof of a theorem of Fatou on the iteration and fix-points of transcendental entire functions by C.-T. Chuang Algebroid functions and their applications to ordinary differential equations by Y.-Z. He Summary of recent research accomplishments in value distribution theory at East China Normal University by R.-F. Lee, C.-J. Dai, and G.-D. Song Univalent functions by S.-Q. Liu and C.-T. Chuang Quasiconformal mappings by C.-Q. He and Z. Li Riemann surfaces by M.-Y. Zhang Approximation and interpolation in the complex domain by X.-C. Shen Dirichlet series by C.-Y. Yu Application of complex analysis to nonlinear elliptic systems of partial differential equations by G.-C. Wen Riemann zeta-function by N.-Y. Zhang and S.-Y. Zhang Application of the theory of functions of a complex variable to celestial mechanics by Z.-H. Yi Complex variable methods in elasticity by J.-K. Ju.

Abstract: The leading term of the asymptotics of a solution of the nonlinear hyperbolic system for large times is constructed and justified. A version of the method of the inverse problem reducing to the solution of a matrix problem of linear conjugation on the complex plane of the spectral parameter is used to solve this system. The coefficients of the asymptotics of are expressed explicitly in terms of the elements of the Riemann matrix realizing the linear conjugation. A theorem is proved on the approximation of the exact solution by the asymptotics constructed. Bibliography: 14 titles.

Abstract: This paper deals with the average number of nodes with a special property in binary trees with n nodes. Generating functions are set up and considered as analytic functions. A detailed singularity analysis allows to get asymptotic formulas for the considered numbers. The local expansions are derived by use of the Mellin transform. The asymptotic expansion involves periodic terms; the Fourier coefficients are computed in terms of Riemann’s ξ-functions etc.

Abstract: It is shown that the function\(\zeta ^{1/k} \) (s) (k large) can be used to derive the prime number theorem from the known zero-free regions for the Riemann Zeta-function. For the proof no upper bound for |ζ′/ζ(s)| is required.

Abstract: A linear time-optimal problem is considered. It is shown that the set of initial states for which a three-dimensional time-optimal problem at the origin, with constant coefficients, has no solution in the class of Riemann integrable controls can fill up a given ball to an arbitrary extent. For a large class of multidimensional systems it is shown that this set does not contain isolated points. Sufficient conditions for the existence of a Riemann integrable control are studied for problems with variable coefficients. Bibliography: 8 titles.

Abstract: The relationship between Poisson’s summation formula and Hamburger’s theorem [2] which is a characterization of Riemann’s zetafunction by the functional equation was already mentioned in Ehrenpreis-Kawai [1]. There Poisson’s summation formula was obtained by the functional equation of Riemann’s zetafunction. This procedure is another proof of Hamburger’s theorem. Being interpreted in this way, Hamburger’s theorem admits various interesting generalizations, one of which is to derive, from the functional equations of the zetafunctions with Grossencharacters of the Gaussian field, Poisson’s summation formula corresponding to its ring of integers [1], The main purpose of the present paper is to give a generalization of Hamburger’s theorem to some zetafunctions with Grossencharacters in algebraic number fields. More precisely, we first define the zetafunctions with Grossencharacters corresponding to a lattice in a vector space, and show that Poisson’s summation formula yields the functional equations of them. Next, we derive Poisson’s summation formula corresponding to the lattice from the functional equations.

Abstract: If one tries to build axiomatically the Einsteinian theory of gravitation by means of the basic concepts «event, light ray and freely falling particle», one arrives at a Weyl and not at a Riemann spacetime. However, Audretsch has recently proved the following remarkable theorem: if one assumes that the motion of the particle is governed by Dirac, or Klein-Gordon, wave equation, the above Weyl space-time degenerates into a Riemann space-time. In this paper, we give a synthetic—and more general—proof of Audretsch’s result.

Abstract: Chapter 5 contains more number theory than any of the remaining 20 chapters. Of the 94 formulas or statements of theorems in Chapter 5, the great majority pertain to Bernoulli numbers, Euler numbers, Eulerian polynomials and numbers, and the Riemann zeta-function. As is to be expected, most of these results are not new. The geneses of Ramanujan’s first published paper [4] (on Bernoulli numbers) and fourth published paper [7] (on sums connected with the Riemann zeta-function) are found in Chapter 5. Most of Ramanujan’s discoveries about Bernoulli numbers that are recorded here may be found in standard texts, such as those by Bromwich [1], Nielsen [5], Norlund [2], and Uspensky and Heaslet [1], for example.

Abstract: The conference started with why, for ten minutes. I do mathematics because I like it. We discussed briefly the distinction between pure mathematics and applied mathematics, which actually intermingle in such a way that it is impossible to define the boundary between one and the other precisely; and also the aesthetic side of mathematics. Then we did mathematics together. I started by defining prime numbers, and I recalled Euclid’s proof that there are infinitely many. Then I defined twin primes, (3,5), (5, 7), (11,13), (17,19), etc. which differ by 2. Is there an infinite number of those? No one knows, even though one conjectures that the answer is yes. I gave heuristic arguments describing the expected density of such primes. Why don’t you try to prove it? The question is one of the big unsolved problems of mathematics.