Explicit Error Terms for Asymptotic Expansions of Stieltjes Transforms

A technique is developed which gives explicit expressions forthe error terms associated with the asymptotic expansions ofthe Stieltjes transform of f, S _f(z) = ₀ f(t)/t + z dt. Here z is a complex parameter in the cut plane \arg z\ <, and f(t) is a locally integrable function on [0, ). Near infinitywe assume that f(t) is either algebraically decaying or oscillatory.From the explicit expressions, strict and realistic error boundscan be obtained. Explicit error terms are also given for asymptoticexpansions of Laplace and Fourier transforms of small argument.Our approach is based on the theory of distributions.     

A Refined Version of the Lang-Trotter Conjecture

Let E be an elliptic curve defined over the rational numbersand r a fixed integer. Using a probabilistic model consistentwith the Chebotarev density theorem for the division fieldsof E and the Sato–Tate distribution, Lang and Trotterconjectured an asymptotic formula for the number of primes upto x which have Frobenius trace equal to r, where r is a fixedinteger. However, as shown in this note, this asymptotic estimatecannot hold for all r in the interval with a uniform bound for the error term, becausean estimate of this kind would contradict the Chebotarev densitytheorem as well as the Sato–Tate conjecture. The purposeof this note is to refine the Lang–Trotter conjecture,by taking into account the "semicircular law," to an asymptoticformula that conjecturally holds for arbitrary integers r inthe interval , with auniform error term. We demonstrate consistency of our refinementwith the Chebotarev density theorem for a fixed division field,and with the Sato–Tate conjecture. We also present numericalevidence for the refined conjecture.     

Fourier-Mukai transforms for coherent systems on elliptic curves

We determine all the Fourier–Mukai transforms for coherentsystems consisting of a vector bundle over an elliptic curveand a subspace of its global sections, showing that these transformsare indexed by positive integers. We prove that the naturalstability condition for coherent systems, which depends on aparameter, is preserved by these transforms for small and largevalues of the parameter. By means of the Fourier–Mukaitransforms we prove that certain moduli spaces of coherent systemscorresponding to small and large values of the parameter areisomorphic. Using these results we draw some conclusions aboutthe possible birational type of the moduli spaces. We provethat for a given degree d of the vector bundle and a given dimensionof the subspace of its global sections there are at most d differentpossible birational types for the moduli spaces.     

Convolutions, Transforms, and Convex Bodies

The paper studies convex bodies and star bodies in Rⁿ by usingRadon transforms on Grassmann manifolds, p-cosine transformson the unit sphere, and convolutions on the rotation group ofRⁿ. It presents dual mixed volume characterizations of i-intersectionbodies and L_p-balls which are related to certain volume inequalitiesfor cross sections of convex bodies. It considers approximationsof special convex bodies by analytic bodies and various finitesums of ellipsoids which preserve special geometric properties.Convolution techniques are used to derive formulas for mixedvolumes, mixed surface measures, and p-cosine transforms. Theyare also used to prove characterizations of geometric functionals,such as surface area and dual quermassintegrals. 1991 MathematicsSubject Classification: 52A20, 52A40.     

Ratio Mercerian Theorems with Applications to Hankel and Fourier Transforms

We complete and complement our recent work on Drasin-Shea-Jordantheorems for Fourier and Hankel transforms. In improving onthe methods of our previous work, we were led to certain ratioMercerian theorems for general kernels; these yield definitiveversions of our earlier results for Fourier and Hankel transforms.1991 Mathematics Subject Classification: primary 44A15; secondary47B38.     

Wild Recurrent Critical Points

It is conjectured that a rational map whose coefficients arealgebraic over Q_p has no wandering components of the Fatou set.Benedetto has shown that any counterexample to this conjecturemust have a wild recurrent critical point. We provide the firstexamples of rational maps whose coefficients are algebraic over Q_p and that have a (wild) recurrent critical point. In fact,it is shown that there is such a rational map in every one-parameterfamily of rational maps that is defined over a finite extensionof Q_p and that has a Misiurewicz bifurcation.     

The Accurate Numerical Inversion of Laplace Transforms

Numerical inversion of almost arbitrary Laplace transforms,for any value of t, to any prescribed accuracy up to atleastthree-quarters of the computer precision, is effected by trapezoidalintegration along a special contour. The required number ofpoints depends on t, the accuracy, and the transform singularitypositions, and for moderate t is typically 11 for errors oforder 10^–6, 18 for order 10^–10, 35 for order 10^–20(withdouble precision working).     

A posteriori L2 error estimation on anisotropic tetrahedral finite element meshes

A new a posteriori L₂ norm error estimator is proposed for thePoisson equation. The error estimator can be applied to anisotropictetrahedral or triangular finite element meshes. The estimatoris rigorously analysed for Dirichlet and Neumann boundary conditions. The lower error bound relies on specifically designed anisotropicbubble functions and the corresponding inverse inequalities.The upper error bound utilizes non-standard anisotropic interpolationestimates. Its proof requires H² regularity of the Poisson problem,and its quality depends on how good the anisotropic mesh resolvesthe anisotropy of the problem. This is measured by a so-called‘matching function’. A numerical example supports the anisotropic error analysis.     

On the Accuracy of Quadrature Laplace Transform Inversion and the Solution of State Space Equations

A quadrature formula is shown to be an approximation of thepower-series method of inverting Laplace transforms. This togetherwith the properties of the constants derived from a power-seriesexpansion of the Pad? approximation to exp (s) yield an importantupper limit on t which is quite sharp in determining the breakdownpoint up to and after which the approximation is accurate andinaccurate respectively. The solution of state space equationsusing the quadrature inversion formula is also discussed.     

Operational Analysis of Multiregional Nuclear Reactor Kinetics

A general procedure for solving the boundary-value problem ofthe unsteady multigroup coupled neutron diffusion and precursorequations is developed analytically for a multiregional nuclearreactor whose subregions are of arbitrary shapes and with arbitraryspace-time dependent neutron multiplication properties. Themethod which is based on the formulation and application ofcomposite region integral transforms is seen to accept variousorders of analytical approximations that originate from thetransformed space. The formalism does not cover circulatingfuel precursors.     

On the Poles of Maximal Order of the Topological Zeta Function

The global and local topological zeta functions are singularityinvariants associated to a polynomial f and its germ at 0, respectively.By definition, these zeta functions are rational functions inone variable, and their poles are negative rational numbers.In this paper we study their poles of maximal possible order.When f is non-degenerate with respect to its Newton polyhedron,we prove that its local topological zeta function has at mostone such pole, in which case it is also the largest pole; wegive a similar result concerning the global zeta function. Moreover,for any f we show that poles of maximal possible order are alwaysof the form –1/N with N a positive integer. 1991 MathematicsSubject Classification 14B05, 14E15, 32S50.     

Shift-Generated Random Permutations and The M/D/1 Queue

We consider a simple shift procedure which transforms any mappingf:{0, ..., n}{0, ..., n} into a permutation of 0, ..., n. Westudy this shift under the assumption that the values f(0),..., f(n) are uniformly and independently chosen random numbersin {0, ..., n}. The exact and asymptotic distributions of therequired shift sizes are derived, and a connection to queueingtheory is exhibited. 1991 Mathematics Subject Classification60C05, 60K25.     

Certain Rational Function Approximations to (1+x2) -1/2

With a view to numerical applications, particular rational functionapproximations to (1+x²) – are constructed by using continuedfraction expansions. It is found that the zeros of the denominatorpolynomials, which occur in the convergents of the continuedfractions, can be determined explicitly. This fact enables therational function approximations to be expressed convenientlyin partial fractions. The resulting expressions are then appliedto examples in which x is regarded either as a real variable,or as the Laplace operator, the latter case giving rise to someinteresting approximations to Bessel functions of integral orderand complex argument. The character of the errors of the variousapproximations is discussed, and expressions for the error boundsdeveloped. These are in good agreement with computed results.The method followed in this paper is capable of further development.In particular it can be extended to those functions which possesssuitable Taylor and asymptotic expansions.     

Equivalences of Triangulated Categories and Fourier-Mukai Transforms

We give a condition for an exact functor between triangulatedcategories to be an equivalence. Applications to Fourier–Mukaitransforms are discussed. In particular, we obtain a large numberof such transforms for K3 surfaces. 1991 Mathematics SubjectClassification 18E30, 14J28.     

A posteriori H1 error estimation for a singularly perturbed reaction diffusion problem on anisotropic meshes

The paper deals with a singularly perturbed reaction diffusionmodel problem. The focus is on reliable a posteriori error estimatorsfor the H¹ seminorm that can be applied to anisotropic finiteelement meshes. A residual error estimator and a local problemerror estimator are proposed and rigorously analysed. They arelocally equivalent, and both bound the error reliably. Threemodifications of these estimators are introduced and discussed. Much attention is given to the performance of the error estimatorin numerical experiments. This helps to identify those estimatorsthat are suitable for practical applications.     

Coupling of mixed finite elements and boundary elements

The symmetric coupling of mixed finite element and boundaryelement methods is analysed for a model interface problem withthe Laplacian. The coupling involves a further continuous ansatzfunction on the interface to link the discontinuous displacementfield to the necessarily continuous boundary ansatz function.Quasi-optimal a priori error estimates and sharp a posteriorierror estimates are established which justify adaptive mesh-refiningalgorithms. Numerical experiments prove the adaptive couplingas an efficient tool for the numerical treatment of transmissionproblems.     

Solution of Wiener-Hopf Integral Equations using the Fast Fourier Transform

The Wiener-Hopf method of solving the linear integral equation can be carried out conveniently and efficiently using the FastFourier Transform. Given the values of h(t) and g(t) at equidistantpoints along the t-axis, the solution f(t) is obtained by dintof seven Fourier transforms.     

Smoothing properties and approximation of time derivatives for parabolic equations: constant time steps

We study smoothing properties and approximation of time derivativesfor time discretization schemes with constant time steps fora homogeneous parabolic problem formulated as an abstract initial-valueproblem in a Banach space. The time stepping schemes are basedon using rational functions r(z) e^–z which are A()-stablefor suitable [0, /2] and satisfy |r()| < 1, and the approximationsof time derivatives are based on using difference quotientsin time. Both smooth and non-smooth data error estimates ofoptimal order for the approximation of time derivatives areproved. Further, we apply the results to obtain error estimatesof time derivatives in the supremum norm for fully discretemethods based on discretizing the spatial variable by a finite-elementmethod.