Abstract: Abstract : The authors demonstrate a wider zero-free region for the Riemann zeta function than has been given before. They give improved methods for using this and a recent determination that 3,500,000 zeros lie on the critical line to develop better bounds for functions of primes.

Abstract: The convergence rate for difference approximations to mixed initial boundary value problems

Abstract: In this paper a general spectral approximation theory is developed for compact operators on a Banach space. Results are obtained on the approximation of eigenvalues and generalized eigenvectors. These results are applied in a variety of situations.

Abstract: The continued fraction method for factoring integers, which was introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation. The power of the method is demonstrated by the factorization of the seventh Fermat number F7 and other large numbers of interest. "Quand on a a' etudier un grand nombre, il faut commencer par en determiner quelques residus quadratiques." M. Kraitchik

Abstract: : The authors demonstrate a wider zero-free region for the Riemann zeta function than has been given before They give improved methods for using this and a recent determination that 3,500,000 zeros lie on the critical line to develop better bounds for functions of primes

Abstract: The hyperbolic nature of the unsteady, inviscid, gas-dynamic equations implies the existence of a similarity transformation for diagonalizing an arbitrary linear combination of coefficient matrices. It is shown that the individual matrices are simultaneously symmetrized by the similarity transformation. The transformations and their norms can be applied to the well-posedness of the Cauchy problem, linear stability theory for finite-difference approximations, and simplification of block-tridiagonal systems that arise in implicit time-split algorithms.

Abstract: A priori error estimates in the maximum norm are derived for Galerkin approximations to solutions of two-point boundary valud problems. The class of Galerkin spaces considered includes almost all (quasiuniform) piecewise-polynomial spaces that are used in practice. The estimates are optimal in the sense that no better rate of approximation is possible in general in the spaces employed.

Abstract: Let AA A A be a triangle with vertices at A1, A2 and A3. The process of "bisecting AA A A is defined as follows. We first locate the longest edge, A'Ai+i of AA 1A2A3 where A'+3 = Al, set D = (A' + Ai+l)/2, and then define two new tri- angles, AA'DAi+2 and ADAi+1Ai+2.

Abstract: This paper, in the first place, calls attention to an extraordinarily compact solution of the problem in the title, given (a trifle hidden) in the work of the late Jacques Touchard. Its main weight, however, is on properties of the several kinds of number sequences appearing.

Abstract: A Galerkin method due to Nitsche for treating the Dirichlet problem for a linear second-order elliptic equation is extended to cover the nonlinear equation V * (a(x, u)Vu) = f. The asymptotic error estimates are of the same form as in the linear case. Newton's method can be used to solve the nonlinear algebraic equations.

Abstract: : The results of this report are part of an effort to develop high accuracy efficient methods for solving hyperbolic equations. Here an efficient fourth order method for the multi-dimensional wave equation is presented. Similar techniques are being tried to solve first order hyperbolic systems for application to inviscid flow calculations.

Abstract: New asymptotic expansions are derived for the incomplete gamma functions and the incomplete beta function. In each case the expansion contains the complementary error function and an asymptotic series. The expansions are uniformly valid with respect to certain domains of the parameters.

Abstract: Methods are given for computing the LDV factorization of a matrix B and modifying the factorization when columns of B are added or deleted. The methods may be viewed as a means for updating the orthogonal (LQ) factorization of B without the use of square roots. It is also shown how these techniques lead to two numerically stable methods for updating the Cholesky factorization of a matrix following the addition or subtraction,respectively, of a matrix of rank one. The first method turns out to be one given recently by Fletcher and Powell; the second method has not appeared before.

Abstract: A method for calculating vectors of smallest norm in a given lattice is outlined. The norm is defined by means of a convex, compact, and symmetric subset of the given space. The main tool is the systematic use of the dual lattice. The method generalizes an algorithm presented by Coveyou and MacPherson, and improved by Knuth, for the determination of vectors of smallest Euclidean norm. 1. Formulation of the Problem. Let G be a lattice in the n-dimensional Euclidean space Rn, generated by n linearly independent vectors e (1) G = {x = zie z1 integers}. The norm in Rn is defined by a convex, compact set B which has positive measure and is symmetric about the origin: (2) lIxIl = min{X E R I x E XB}. Examples of these norms for x = (xl, . . ., xn) are (i) The Euclidean norm llxl = (X2 + ? ? x2)1/2. (ii) The Maximum norm lxii = max{ixi I I i = 1, ... , n}. Here Boo = { . . ., Xn)I Ixi? 1 for all i}. (iii) The norm lixil = lx11 + -'_ + lXn 1Here B1 = {(xl, . . . , Xn) IXI I + ?* + Xn I 1 The problem is to find a nonzero vector of shortest length (norm) in G. The main tool of the presented method is the use of the dual lattice, (3) G* ={x*= E z4e*Iz*integers), k=l where the e* are defined by eie* = 6ik; here 6ik is equal to 1 if i = k and equal to 0 if i # k, and e e* denotes the scalar product V7n= e.1e* . The polar of B, namely (4) B* = {b* E Rn I Ibb*l < 1, V bEB}, induces a length or norm in G* by (5) 1ix*i*1 min{X* E R Ix* G X*B*}. It should be noted that the Euclidean norm corresponds to itself, whereas the Maximum norm lxii = maxi IxiI corresponds to iix*ii* = Ixv + i ? + Ix*l and vice versa. Received May 16, 1974; revised August 6, 1974. AMS (MOS) subject classifications (1970). Primary 10E0S, 10E20, 10E25; Secondary 65C10.

Abstract: A generalization of Gershgorin's theorem is developed for the eigenvalue problem Ax = XBx and is applied to obtain perturbation bounds for multiple eigenvalues. The results are interpreted in terms of the chordal metric on the Riemann sphere, which is especially convenient for treating infinite eigenvalues.

Abstract: Let to, t1, t2, be a sequence of elements of a field F. We give a continued fraction algorithm for tox + tlx2 + t2x3 + * . . If our sequence satisfies a linear recurrence, then the continued fraction algorithm is finite and produces this recurrence. More generally the algorithm produces a nontrivial solution of the system E ti+j j 0O < i < s-1, j=O for every positive integer s. 1. Let tO, tl, t2, be a sequence of elements of a field F. Set

Abstract: A generalization of Gershgorin's theorem is developed for the eigenvalue problem Ax = XBx and is applied to obtain perturbation bounds for multiple eigenvalues. The results are interpreted in terms of the chordal metric on the Riemann sphere, which is especially convenient for treating infinite eigenvalues.

Abstract: : The report is concerned with the use of collocation by splines to numerically solve two-point boundary-value problems. The problem is analyzed in terms of cubic splines first and then extended to the use of quintic and septic splines. Consideration is given both to convergences as the mesh is refined and to the bandwidth of the matrices involved. Comparisons are made to a similar approach using the Galerkin method rather than collocation. (Author)

Abstract: A finite element approximation of the minimal surface problem for a strictly convex bounded plane domain Q2 is considered. The approximating functions are continuous and piecewise linear on a triangulation of Q2. Error estimates of the form 0(h) in the H1 norm and 0(h 2) in the Lp-norm (p < 2) are proved,where h denotes the maximal side in the triangulation.

Abstract: In this paper we obtain, by simple means, interior maximum-norm estimates for a class of Ritz-Galerkin methods used for approximating solutions of second order elliptic boundary value problems in RN. The estimates are proved when the approximating subspaces are any of a large class of piecewise polynomial subspaces which we assume here to be defined on a uniform mesh on the interior domain. Optimal rates of convergence are obtained.

Abstract: Let d < 0 be the discriminant of a complex quadratic field of class-number h(d). In a previous paper the author has effectively shown how to find all d with h(d) = 2. In this paper, it is proved that, if h(d) = 2, then Id l < 427.

Abstract: The initial-boundary value problem for a linear parabolic equation in an infinite cylinder under the Dirichlet boundary condition is solved by applying the finite element discretization in the space dimension and A0-stable multistep discretizations in time. No restriction on the ratio of the time and space increments is imposed. The methods are analyzed and bounds for the discretization error in the L2-norm are given.