The algorithm is a structure-preserving algorithm for computing the spectrum of symplectic matrices. Any symplectic matrix can be reduced to symplectic butterfly form. A symplectic matrix in butterfly form is uniquely determined by parameters. Using these parameters, we show how one step of the symplectic algorithm for can be carried out in arithmetic operations compared to arithmetic operations when working on the actual symplectic matrix. Moreover, the symplectic structure, which will be destroyed in the numerical process due to roundoff errors when working with a symplectic (butterfly) matrix, will be forced by working just with the parameters.
Let 2$">, an -th primitive root of 1, mod a prime number, a primitive root modulo and . We study the Jacobi sums , , where is the least nonnegative integer such that mod . We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families , , of irreducible polynomials of Gaussian periods, , of degree , where is a suitable set of primes mod . We exhibit examples of such families for several small values of , and give a MAPLE program to construct more of them.
A sequence of integers in an interval of length is called admissible if for each prime there is a residue class modulo the prime which contains no elements of the sequence. The maximum number of elements in an admissible sequence in an interval of length is denoted by . Hensley and Richards showed that \pi (x)$"> for large enough . We increase the known bounds on the set of satisfying and find smaller values of such that \pi (x)$">. We also find values of satisfying 2\pi (x/2)$">. This shows the incompatibility of the conjecture with the prime -tuples conjecture.
This paper concerns the Rayleigh-Ritz method for computing an approximation to an eigenspace of a general matrix from a subspace that contains an approximation to . The method produces a pair that purports to approximate a pair , where is a basis for and . In this paper we consider the convergence of as the sine of the angle between and approaches zero. It is shown that under a natural hypothesis--called the uniform separation condition--the Ritz pairs converge to the eigenpair . When one is concerned with eigenvalues and eigenvectors, one can compute certain refined Ritz vectors whose convergence is guaranteed, even when the uniform separation condition is not satisfied. An attractive feature of the analysis is that it does not assume that has distinct eigenvalues or is diagonalizable.
Using a carefully optimized segmented sieve and an efficient checking algorithm, the Goldbach conjecture has been verified and is now known to be true up to . The program was distributed to various workstations. It kept track of maximal values of the smaller prime in the minimal partition of the even numbers, where a minimal partition is a representation with being composite for all . The maximal prime needed in the considered interval was found to be 5569 and is needed for the partition 389965026819938 = 5569 + 389965026814369.
We examine the problem of factoring the th cyclotomic polynomial, over , and distinct primes. Given the traces of the roots of we construct the coefficients of in time . We demonstrate a deterministic algorithm for factoring in time when has precisely two irreducible factors. Finally, we present a deterministic algorithm for computing the sum of the irreducible factors of in time .
Let be an even integer, . The resultant of the polynomials and is known as Wendt's determinant of order . We prove that among the prime divisors of only those which divide or can be larger than , where and is the th Lucas number, except when and . Using this estimate we derive criteria for the nonsolvability of Fermat's congruence.
A systematic search for optimal lattice rules of specified trigonometric degree over the hypercube has been undertaken. The search is restricted to a population of lattice rules . This includes those where the dual lattice may be generated by points for each of which . The underlying theory, which suggests that such a restriction might be helpful, is presented. The general character of the search is described, and, for , and , , a list of -optimal rules is given. It is not known whether these are also optimal rules in the general sense; this matter is discussed.
In this paper, we enumerate all number fields of degree of discriminant smaller than in absolute value containing a quintic field having one real place. For each one of the (resp. found fields of signature (resp. the field discriminant, the quintic field discriminant, a polynomial defining the relative quadratic extension, the corresponding relative discriminant, the corresponding polynomial over , and the Galois group of the Galois closure are given.
In a supplementary section, we give the first coincidence of discriminant of (resp. nonisomorphic fields of signature (resp. .
In this paper we prove convergence and error estimates for the so-called 3-field formulation using piecewise linear finite elements stabilized with boundary bubbles. Optimal error bounds are proved in and in the broken norm for the internal variable , and in suitable weighted norms for the other variables and . 相似文献
In this article, we prove the convergence of a splitting scheme of high order for a reaction-diffusion system of the form where is an matrix whose spectrum is included in 0 \}$">. This scheme is obtained by applying a Richardson extrapolation to a Strang formula.
The present paper is a continuation of an earlier work by the author. We propose some new definitions of -adic continued fractions. At the end of the paper we give numerical examples illustrating these definitions. It turns out that for every if then has a periodic continued fraction expansion. The same is not true in for some larger values of
By a prime gap of size , we mean that there are primes and such that the numbers between and are all composite. It is widely believed that infinitely many prime gaps of size exist for all even integers . However, it had not previously been known whether a prime gap of size existed. The objective of this article was to be the first to find a prime gap of size , by using a systematic method that would also apply to finding prime gaps of any size. By this method, we find prime gaps for all even integers from to , and some beyond. What we find are not necessarily the first occurrences of these gaps, but, being examples, they give an upper bound on the first such occurrences. The prime gaps of size listed in this article were first announced on the Number Theory Listing to the World Wide Web on Tuesday, April 8, 1997. Since then, others, including Sol Weintraub and A.O.L. Atkin, have found prime gaps of size with smaller integers, using more ad hoc methods. At the end of the article, related computations to find prime triples of the form , , and their application to divisibility of binomial coefficients by a square will also be discussed.
Let be a product of two distinct primes and . We show that for almost all exponents with the RSA pairs are uniformly distributed modulo when runs through
- the group of units modulo (that is, as in the classical RSA scheme);
- the set of -products , , where are selected at random (that is, as in the recently introduced RSA scheme with precomputation).
In this paper we tabulate all strong pseudoprimes (spsp's) to the first ten prime bases which have the form with odd primes and There are in total 44 such numbers, six of which are also spsp(31), and three numbers are spsp's to both bases 31 and 37. As a result the upper bounds for and are lowered from 28- and 29-decimal-digit numbers to 22-decimal-digit numbers, and a 24-decimal-digit upper bound for is obtained. The main tools used in our methods are the biquadratic residue characters and cubic residue characters. We propose necessary conditions for to be a strong pseudoprime to one or to several prime bases. Comparisons of effectiveness with both Jaeschke's and Arnault's methods are given.
Bounds for the distance between adjacent zeros of cylinder functions are given; and are such that ; stands for the th positive zero of the cylinder (Bessel) function , , .
These bounds, together with the application of modified (global) Newton methods based on the monotonic functions and , give rise to forward ( ) and backward ( ) iterative relations between consecutive zeros of cylinder functions.
The problem of finding all the positive real zeros of Bessel functions for any real and inside an interval , 0$">, is solved in a simple way.