The generalized triangular decomposition

Given a complex matrix , we consider the decomposition , where is upper triangular and and have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where is an upper triangular matrix with the eigenvalues of on the diagonal. We show that any diagonal for can be achieved that satisfies Weyl's multiplicative majorization conditions:

where is the rank of , is the -th largest singular value of , and is the -th largest (in magnitude) diagonal element of . Given a vector which satisfies Weyl's conditions, we call the decomposition , where is upper triangular with prescribed diagonal , the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.

     

Computation of relative class numbers of CM-fields by using Hecke -functions

We develop an efficient technique for computing values at of Hecke -functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields which are abelian extensions of some totally real subfield . We note that the smaller the degree of the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing (the maximal totally real subfield of ) we can choose real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.

     

Fast algorithms for discrete polynomial transforms

Consider the Vandermonde-like matrix , where the polynomials satisfy a three-term recurrence relation. If are the Chebyshev polynomials , then coincides with . This paper presents a new fast algorithm for the computation of the matrix-vector product in arithmetical operations. The algorithm divides into a fast transform which replaces with and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes with almost the same precision as the Clenshaw algorithm, but is much faster for .

     

On strong tractability of weighted multivariate integration

We prove that for every dimension and every number of points, there exists a point-set whose -weighted unanchored discrepancy is bounded from above by independently of provided that the sequence has for some (even arbitrarily large) . Here is a positive number that could be chosen arbitrarily close to zero and depends on but not on or . This result yields strong tractability of the corresponding integration problems including approximation of weighted integrals over unbounded domains such as . It also supplements the results that provide an upper bound of the form when .

     

The irreducibility of some level 1 Hecke polynomials

Let be the characteristic polynomial of the Hecke operator acting on the space of level 1 cusp forms . We show that is irreducible and has full Galois group over  for and ,  prime.

     

On the modular curves

Let denote an elliptic curve over and the modular curve classifying the elliptic curves over such that the representations of in the 7-torsion points of and of are symplectically isomorphic. In case is given by a Weierstraß equation such that the invariant is a square, we exhibit here nontrivial points of . From this we deduce an infinite family of curves for which has at least four nontrivial points.

     

Jacobi sums and new families of irreducible polynomials of Gaussian periods

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.

     

Efficient CM-constructions of elliptic curves over finite fields

We present an algorithm that, on input of an integer together with its prime factorization, constructs a finite field and an elliptic curve over for which has order . Although it is unproved that this can be done for all , a heuristic analysis shows that the algorithm has an expected run time that is polynomial in , where is the number of distinct prime factors of . In the cryptographically relevant case where is prime, an expected run time can be achieved. We illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order and nextprime.

     

Computing the multiplicative group of residue class rings

Let be a global field with maximal order and let be an ideal of . We present algorithms for the computation of the multiplicative group of the residue class ring and the discrete logarithm therein based on the explicit representation of the group of principal units. We show how these algorithms can be combined with other methods in order to obtain more efficient algorithms. They are applied to the computation of the ray class group modulo , where denotes a formal product of real infinite places, and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields.

     

Wavelet bases in

Some years ago, compactly supported divergence-free wavelets were constructed which also gave rise to a stable (biorthogonal) wavelet splitting of . These bases have successfully been used both in the analysis and numerical treatment of the Stokes and Navier-Stokes equations. In this paper, we construct stable wavelet bases for the stream function spaces . Moreover, -free vector wavelets are constructed and analysed. The relationship between and are expressed in terms of these wavelets. We obtain discrete (orthogonal) Hodge decompositions.

Our construction works independently of the space dimension, but in terms of general assumptions on the underlying wavelet systems in that are used as building blocks. We give concrete examples of such bases for tensor product and certain more general domains . As an application, we obtain wavelet multilevel preconditioners in and .

     

Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants

In the Laurent expansion

of the Riemann-Hurwitz zeta function, the coefficients are known as Stieltjes, or generalized Euler, constants. [When , (the Riemann zeta function), and .] We present a new approach to high-precision approximation of . Plots of our results reveal much structure in the growth of the generalized Euler constants. Our results when for , and when for (for such as 53/100, 1/2, etc.) suggest that published bounds on the growth of the Stieltjes constants can be much improved, and lead to several conjectures. Defining , we conjecture that is attained: for any given , for some (and similarly that, given and , is within of for infinitely many ). In addition we conjecture that satisfies for 1$">. We also conjecture that , a special case of a more general conjecture relating the values of and for . Finally, it is known that for . Using this to define for all real 0$">, we conjecture that for nonintegral , is precisely times the -th (Weyl) fractional derivative at of the entire function . We also conjecture that , now defined for all real arguments 0$">, is smooth. Our numerical method uses Newton-Cotes integration formulae for very high-degree interpolating polynomials; it differs in implementation from, but compares in error bounding to, Euler-Maclaurin summation based methods.

     

Comparison theorems of Kolmogorov type and exact values of -widths on Hardy-Sobolev classes

Let be a strip in complex plane. denotes those -periodic, real-valued functions on which are analytic in the strip and satisfy the condition , . Osipenko and Wilderotter obtained the exact values of the Kolmogorov, linear, Gel'fand, and information -widths of in , , and 2-widths of in , , .

In this paper we continue their work. Firstly, we establish a comparison theorem of Kolmogorov type on , from which we get an inequality of Landau-Kolmogorov type. Secondly, we apply these results to determine the exact values of the Gel'fand -width of in , . Finally, we calculate the exact values of Kolmogorov -width, linear -width, and information -width of in , , .

     

Zeta functions of a class of elliptic curves over a rational function field of characteristic two

We show how to calculate the zeta functions and the orders of Tate-Shafarevich groups of the elliptic curves with equation over the rational function field , where is a power of 2. In the range , , odd of degree , the largest values obtained for are (one case), (one case) and (three cases). We observe and discuss a remarkable pattern for the distributions of signs in the functional equation and of fudge factors at places of bad reduction. These imply strong restrictions on the precise form of the Langlands correspondence for GL over local or global fields of characteristic two.

     

Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups

We address the problem of approximating numerically the solutions of stochastic evolution equations on Hilbert spaces , with respect to Brownian motions, arising in the unraveling of backward quantum master equations. In particular, we study the computation of mean values of , where is a linear operator. First, we introduce estimates on the behavior of . Then we characterize the error induced by the substitution of with the solution of a convenient stochastic ordinary differential equation. It allows us to establish the rate of convergence of to , where denotes the explicit Euler method. Finally, we consider an extrapolation method based on the Euler scheme. An application to the quantum harmonic oscillator system is included.

     

Class numbers of real cyclotomic fields of prime conductor

The class numbers of the real cyclotomic fields are notoriously hard to compute. Indeed, the number is not known for a single prime . In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields for the primes . It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue --on the basis of the Cohen-Lenstra heuristics-- that the probability that our table is actually a table of class numbers , is at least .

     

On the total number of prime factors of an odd perfect number

We say is perfect if , where denotes the sum of the positive divisors of . No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form , where , , ..., are distinct primes and . We prove that if or for all , , then . We also prove as our main result that , where . This improves a result of Sayers given in 1986.

     

An infinite family of bounds on zeros of analytic functions and relationship to Smale's bound

Smale's analysis of Newton's iteration function induce a lower bound on the gap between two distinct zeros of a given complex-valued analytic function . In this paper we make use of a fundamental family of iteration functions , , to derive an infinite family of lower bounds on the above gap. However, even for , where coincides with Newton's, our lower bound is more than twice as good as Smale's bound or its improved version given by Blum, Cucker, Shub, and Smale. When is a complex polynomial of degree , for small the corresponding bound is computable in arithmetic operations. For quadratic polynomials, as increases the lower bounds converge to the actual gap. We show how to use these bounds to compute lower bounds on the distance between an arbitrary point and the nearest root of . In particular, using the latter result, we show that, given a complex polynomial , , for each we can compute upper and lower bounds and such that the roots of lie in the annulus . In particular, , ; and , , where . An application of the latter bounds is within Weyl's classical quad-tree algorithm for computing all roots of a given complex polynomial.

     

On the absolute Mahler measure of polynomials having all zeros in a sector. II

Let be an algebraic integer of degree , not or a root of unity, all of whose conjugates are confined to a sector . In the paper On the absolute Mahler measure of polynomials having all zeros in a sector, G. Rhin and C. Smyth compute the greatest lower bound of the absolute Mahler measure ( of , for belonging to nine subintervals of . In this paper, we improve the result to thirteen subintervals of and extend some existing subintervals.

     

Korovkin tests, approximation, and ergodic theory

We consider sequences of matrices with a block structure spectrally distributed as an -variate matrix-valued function , and, for any , we suppose that is a linear and positive operator. For every fixed we approximate the matrix in a suitable linear space of matrices by minimizing the Frobenius norm of when ranges over . The minimizer is denoted by . We show that only a simple Korovkin test over a finite number of polynomial test functions has to be performed in order to prove the following general facts:

1.

the sequence is distributed as ,

2.

the sequence is distributed as the constant function (i.e. is spectrally clustered at zero).

The first result is an ergodic one which can be used for solving numerical approximation theory problems. The second has a natural interpretation in the theory of the preconditioning associated to cg-like algorithms.