Nontrivial Galois module structure of cyclotomic fields

We say a tame Galois field extension with Galois group has trivial Galois module structure if the rings of integers have the property that is a free -module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes so that for each there is a tame Galois field extension of degree so that has nontrivial Galois module structure. However, the proof does not directly yield specific primes for a given algebraic number field For any cyclotomic field we find an explicit so that there is a tame degree extension with nontrivial Galois module structure.

     

Lower bounds for the total stopping time of iterates

The total stopping time of a positive integer is the minimal number of iterates of the function needed to reach the value , and is if no iterate of reaches . It is shown that there are infinitely many positive integers having a finite total stopping time such that 6.14316 \log n.$"> The proof involves a search of trees to depth 60, A heuristic argument suggests that for any constant , a search of all trees to sufficient depth could produce a proof that there are infinitely many such that \gamma\log n.$">It would require a very large computation to search trees to a sufficient depth to produce a proof that the expected behavior of a ``random' iterate, which is occurs infinitely often.

     

Newman polynomials with prescribed vanishing and integer sets with distinct subset sums

We study the problem of determining the minimal degree of a polynomial that has all coefficients in and a zero of multiplicity at . We show that a greedy solution is optimal precisely when , mirroring a result of Boyd on polynomials with coefficients. We then examine polynomials of the form , where is a set of positive odd integers with distinct subset sums, and we develop algorithms to determine the minimal degree of such a polynomial. We determine that satisfies inequalities of the form . Last, we consider the related problem of finding a set of positive integers with distinct subset sums and minimal largest element and show that the Conway-Guy sequence yields the optimal solution for , extending some computations of Lunnon.

     

Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of only. For example, when polynomials of degree are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order in the -norm, whereas the post-processed approximation is of order ; if the exact solution is in only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order in , where is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

     

Finite element approximation of -surfaces

In this paper a piecewise linear finite element approximation of -surfaces, or surfaces with constant mean curvature, spanned by a given Jordan curve in is considered. It is proved that the finite element -surfaces converge to the exact -surfaces under the condition that the Jordan curve is rectifiable. Several numerical examples are given.

     

The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension

Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation.

Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor is bounded by for some exponent and some positive constant . The -exponent of tractability is defined as the smallest power of in these bounds. It is shown by using Monte Carlo quadrature that the -exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the -exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.

     

About the sharpness of the stability estimates in the Kreiss matrix theorem

One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices under consideration. This so-called resolvent condition is known to imply, for all , the upper bounds and . Here is the spectral norm, is the constant occurring in the resolvent condition, and the order of is equal to .

It is a long-standing problem whether these upper bounds can be sharpened, for all fixed 1$">, to bounds in which the right-hand members grow much slower than linearly with and with , respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved: for each 0$">, there are fixed values 0, K>1$"> and a sequence of matrices , satisfying the resolvent condition, such that for .

The result proved in this paper is also relevant to matrices whose -pseudospectra lie at a distance not exceeding from the unit disk for all 0$">.

     

Tractability of quasilinear problems II: Second-order elliptic problems

In a previous paper, we developed a general framework for establishing tractability and strong tractability for quasilinear multivariate problems in the worst case setting. One important example of such a problem is the solution of the Helmholtz equation in the -dimensional unit cube, in which depends linearly on , but nonlinearly on . Here, both and  are -variate functions from a reproducing kernel Hilbert space with finite-order weights of order . This means that, although  can be arbitrarily large, and  can be decomposed as sums of functions of at most  variables, with independent of .

In this paper, we apply our previous general results to the Helmholtz equation, subject to either Dirichlet or Neumann homogeneous boundary conditions. We study both the absolute and normalized error criteria. For all four possible combinations of boundary conditions and error criteria, we show that the problem is tractable. That is, the number of evaluations of and  needed to obtain an -approximation is polynomial in  and , with the degree of the polynomial depending linearly on . In addition, we want to know when the problem is strongly tractable, meaning that the dependence is polynomial only in  , independently of . We show that if the sum of the weights defining the weighted reproducing kernel Hilbert space is uniformly bounded in  and the integral of the univariate kernel is positive, then the Helmholtz equation is strongly tractable for three of the four possible combinations of boundary conditions and error criteria, the only exception being the Dirichlet boundary condition under the normalized error criterion.

     

Linear quintuple-product identities

In the first part of this paper, series and product representations of four single-variable triple products , , , and four single-variable quintuple products , , , are defined. Reduced forms and reduction formulas for these eight functions are given, along with formulas which connect them. The second part of the paper contains a systematic computer search for linear trinomial identities. The complete set of such families is found to consist of two 2-parameter families, which are proved using the formulas in the first part of the paper.

     

A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377

The standard algorithm for testing reducibility of a trinomial of prime degree over requires bits of memory. We describe a new algorithm which requires only bits of memory and significantly fewer memory references and bit-operations than the standard algorithm.

If is a Mersenne prime, then an irreducible trinomial of degree is necessarily primitive. We give primitive trinomials for the Mersenne exponents , , and . The results for extend and correct some computations of Kumada et al. The two results for are primitive trinomials of the highest known degree.

     

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 .

     

New quadratic polynomials with high densities of prime values

Hardy and Littlewood's Conjecture F implies that the asymptotic density of prime values of the polynomials , is related to the discriminant of via a quantity The larger is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant . A technique of Bach allows one to estimate accurately for any , given the class number of the imaginary quadratic order with discriminant , and for any 0$"> given the class number and regulator of the real quadratic order with discriminant . The Manitoba Scalable Sieve Unit (MSSU) has shown us how to rapidly generate many discriminants for which is potentially large, and new methods for evaluating class numbers and regulators of quadratic orders allow us to compute accurate estimates of efficiently, even for values of with as many as decimal digits. Using these methods, we were able to find a number of discriminants for which, under the assumption of the Extended Riemann Hypothesis, is larger than any previously known examples.

     

Upper bounds for the prime divisors of Wendt's determinant

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.

     

On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces

We develop and justify an algorithm for the construction of quasi-Monte Carlo (QMC) rules for integration in weighted Sobolev spaces; the rules so constructed are shifted rank-1 lattice rules. The parameters characterising the shifted lattice rule are found ``component-by-component': the ()-th component of the generator vector and the shift are obtained by successive -dimensional searches, with the previous components kept unchanged. The rules constructed in this way are shown to achieve a strong tractability error bound in weighted Sobolev spaces. A search for -point rules with prime and all dimensions 1 to requires a total cost of operations. This may be reduced to operations at the expense of storage. Numerical values of parameters and worst-case errors are given for dimensions up to 40 and up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.

     

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.

     

The convergence of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz vectors

This paper concerns a harmonic projection method for computing an approximation to an eigenpair of a large matrix . Given a target point and a subspace that contains an approximation to , the harmonic projection method returns an approximation to . Three convergence results are established as the deviation of from approaches zero. First, the harmonic Ritz value converges to if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector converges to if the Rayleigh quotient matrix is uniformly nonsingular and remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of are derived when converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue --in other words, the method can miss if it is very close to . To this end, we propose to compute the Rayleigh quotient of with respect to and take it as a new approximate eigenvalue. is shown to converge to once tends to , no matter how is close to . Finally, we show that if the Rayleigh quotient matrix is uniformly nonsingular, then the refined harmonic Ritz vector, or more generally the refined eigenvector approximation introduced by the author, converges. We construct examples to illustrate our theory.

     

Modular curves of genus 2

We prove that there are exactly genus two curves defined over such that there exists a nonconstant morphism defined over and the jacobian of is -isogenous to the abelian variety attached by Shimura to a newform . We determine the corresponding newforms and present equations for all these curves.

     

The hyperdeterminant and triangulations of the 4-cube

The hyperdeterminant of format is a polynomial of degree in unknowns which has terms. We compute the Newton polytope of this polynomial and the secondary polytope of the -cube. The regular triangulations of the -cube are classified into -equivalence classes, one for each vertex of the Newton polytope. The -cube has coarsest regular subdivisions, one for each facet of the secondary polytope, but only of them come from the hyperdeterminant.

     

Constructing representations of higher degrees of finite simple groups and covers

Let be a finite group and an irreducible character of . A simple method for constructing a representation affording can be used whenever has a subgroup such that has a linear constituent with multiplicity 1. In this paper we show that (with a few exceptions) if is a simple group or a covering group of a simple group and is an irreducible character of of degree between 32 and 100, then such a subgroup exists.

     

Solving norm equations in relative number fields using -units

In this paper, we are interested in solving the so-called norm equation , where is a given arbitrary extension of number fields and a given algebraic number of . By considering -units and relative class groups, we show that if there exists at least one solution (in , but not necessarily in ), then there exists a solution for which we can describe precisely its prime ideal factorization. In fact, we prove that under some explicit conditions, the -units that are norms are norms of -units. This allows us to limit the search for rational solutions to a finite number of tests, and we give the corresponding algorithm. When is an algebraic integer, we also study the existence of an integral solution, and we can adapt the algorithm to this case.