The 192 solutions of the Heun equation

A machine-generated list of local solutions of the Heun equation is given. They are analogous to Kummer's  solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with  singular points as the Coxeter group  . Each of the expressions is labeled by an element of  . Of the ,  are equivalent expressions for the local Heun function  , and it is shown that the resulting order- group of transformations of  is isomorphic to the symmetric group . The isomorphism encodes each transformation as a permutation of an abstract four-element set, not identical to the set of singular points.

     

On the smallest value of the maximal modulus of an algebraic integer

The house of an algebraic integer of degree is the largest modulus of its conjugates. For , we compute the smallest house of degree , say m. As a consequence we improve Matveev's theorem on the lower bound of m We show that, in this range, the conjecture of Schinzel-Zassenhaus is satisfied. The minimal polynomial of any algebraic integer whose house is equal to m is a factor of a bi-, tri- or quadrinomial. The computations use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in They give better bounds than the classical ones for the coefficients of the minimal polynomial of an algebraic integer whose house is small.

     

Heuristics for class numbers and lambda invariants

Let be an imaginary quadratic field and let be the associated real quadratic field. Starting from the Cohen-Lenstra heuristics and Scholz's theorem, we make predictions for the behaviors of the 3-parts of the class groups of these two fields as varies. We deduce heuristic predictions for the behavior of the Iwasawa -invariant for the cyclotomic -extension of and test them computationally.

     

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.

     

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.

     

Multigrid methods for the computation of singular solutions and stress intensity factors I: Corner singularities

We consider the Poisson equation with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain with re-entrant angles. A multigrid method for the computation of singular solutions and stress intensity factors using piecewise linear functions is analyzed. When , the rate of convergence to the singular solution in the energy norm is shown to be , and the rate of convergence to the stress intensity factors is shown to be , where is the largest re-entrant angle of the domain and can be arbitrarily small. The cost of the algorithm is . When , the algorithm can be modified so that the convergence rate to the stress intensity factors is . In this case the maximum error of the multigrid solution over the vertices of the triangulation is shown to be .

     

Adaptive wavelet methods for elliptic operator equations: Convergence rates

This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution of the equation by a linear combination of wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to by an arbitrary linear combination of wavelets (so called -term approximation), which would be obtained by keeping the largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to with error in the energy norm, whenever such a rate is possible by -term approximation. The range of 0$"> for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to . The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization.

     

Littlewood polynomials with high order zeros

Let be the minimal length of a polynomial with coefficients divisible by . Byrnes noted that for each , and asked whether in fact . Boyd showed that for all , but . He further showed that , and that is one of the 5 numbers , or . Here we prove that . Similarly, let be the maximal power of dividing some polynomial of degree with coefficients. Boyd was able to find for . In this paper we determine for .

     

A new superconvergent collocation method for eigenvalue problems

Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree , we show that the proposed method exhibits an error of the order of for eigenvalue approximation and of the order of for spectral subspace approximation. In the case of a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of the order can be obtained. This improves upon the order for eigenvalue approximation in the collocation/iterated collocation method and the orders and for spectral subspace approximation in the collocation method and the iterated collocation method, respectively. We illustrate this improvement in the order of convergence by numerical examples.

     

Optimal order collocation for the mixed boundary value problem on polygons

In usual boundary elements methods, the mixed Dirichlet-Neumann problem in a plane polygonal domain leads to difficulties because of the transition of spaces in which the problem is well posed. We build collocation methods based on a mixed single and double layer potential. This indirect method is constructed in such a way that strong ellipticity is obtained in high order spaces of Sobolev type. The boundary values of this potential define a bijective boundary operator if a modified capacity adapted to the problem is not . This condition is analogous to the one met in the use of the single layer potential, and is not a problem in practical computations. The collocation methods use smoothest splines and known singular functions generated by the corners. If splines of order are used, we get quasi-optimal estimates in -norm. The order of convergence is optimal in the sense that it is fixed by the approximation properties of the first missed singular function.     

Trivariate spline approximations of 3D Navier-Stokes equations

We present numerical approximations of the 3D steady state Navier-Stokes equations in velocity-pressure formulation using trivariate splines of arbitrary degree and arbitrary smoothness with . Using functional arguments, we derive the discrete Navier-Stokes equations in terms of -coefficients of trivariate splines over a tetrahedral partition of any given polygonal domain. Smoothness conditions, boundary conditions and the divergence-free condition are enforced through Lagrange multipliers. The pressure is computed by solving a Poisson equation with Neumann boundary conditions. We have implemented this approach in MATLAB and present numerical evidence of the convergence rate as well as experiments on the lid driven cavity flow problem.

     

Complexity of inverting the Euler function

Given an integer , how hard is it to find the set of all integers such that , where is the Euler totient function? We present a certain basic algorithm which, given the prime number factorization of , in polynomial time ``on average' (that is, ), finds the set of all such solutions . In fact, in the worst case this set of solutions is exponential in , and so cannot be constructed by a polynomial time algorithm. In the opposite direction, we show, under a widely accepted number theoretic conjecture, that the PARTITION PROBLEM, an NP-complete problem, can be reduced in polynomial (in the input size) time to the problem of deciding whether has a solution, for polynomially (in the input size of the PARTITION PROBLEM) many values of (where the prime factorizations of these are given). What this means is that the problem of deciding whether there even exists a solution to , let alone finding any or all such solutions, is very likely to be intractable. Finally, we establish close links between the problem of inverting the Euler function and the integer factorization problem.

     

New expansions of numerical eigenvalues for by nonconforming elements

The paper explores new expansions of the eigenvalues for in with Dirichlet boundary conditions by the bilinear element (denoted ) and three nonconforming elements, the rotated bilinear element (denoted ), the extension of (denoted ) and Wilson's elements. The expansions indicate that and provide upper bounds of the eigenvalues, and that and Wilson's elements provide lower bounds of the eigenvalues. By extrapolation, the convergence rate can be obtained, where is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.

     

Stability of Runge-Kutta methods for abstract time-dependent parabolic problems: The Hölder case

We consider an abstract time-dependent, linear parabolic problem

where , , is a family of sectorial operators in a Banach space with time-independent domain . This problem is discretized in time by means of an A() strongly stable Runge-Kutta method, . We prove that the resulting discretization is stable, under the assumption

where and . Our results are applicable to the analysis of parabolic problems in the , , norms.

     

The monic integer transfinite diameter

We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval . The monic integer transfinite diameter is defined as the infimum of all such supremums. We show that if has length , then .

We make three general conjectures relating to the value of for intervals of length less than . We also conjecture a value for where . We give some partial results, as well as computational evidence, to support these conjectures.

We define functions and , which measure properties of the lengths of intervals with on either side of . Upper and lower bounds are given for these functions.

We also consider the problem of determining when is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.

     

Class numbers of some abelian extensions of rational function fields

Let be a monic irreducible polynomial. In this paper we generalize the determinant formula for of Bae and Kang and the formula for of Jung and Ahn to any subfields of the cyclotomic function field By using these formulas, we calculate the class numbers of all subfields of when and are small.

     

The Postage Stamp Problem: An algorithm to determine the

Given an integral ``stamp" basis with and a positive integer , we define the -range as

. For given and , the extremal basis has the largest possible extremal -range

We give an algorithm to determine the -range. We prove some properties of the -range formula, and we conjecture its form for the extremal -range. We consider parameter bases , where the basis elements are given functions of . For we conjecture the extremal parameter bases for .

     

Limiting set of second order spectra

Let be a self-adjoint operator acting on a Hilbert space . A complex number is in the second order spectrum of relative to a finite-dimensional subspace iff the truncation to of is not invertible. This definition was first introduced in Davies, 1998, and according to the results of Levin and Shargorodsky in 2004, these sets provide a method for estimating eigenvalues free from the problems of spectral pollution. In this paper we investigate various aspects related to the issue of approximation using second order spectra. Our main result shows that under fairly mild hypothesis on the uniform limit of these sets, as increases towards , contain the isolated eigenvalues of of finite multiplicity. Therefore, unlike the majority of the standard methods, second order spectra combine nonpollution and approximation at a very high level of generality.

     

Error estimates for the three-field formulation with bubble stabilization

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 .