Higher order oblique derivatives problems on polyhedral domains

we investigate index properties of general oblique derivatives problems on polygons and three-dimensional polyhedra. We state conditions for induced operators to be Fredholm between Hilbert Sobolev spaces ; moreover, for two-dimensional domains, we give formulae for the dimension of the kernel. We focus on the semi-variational character of such problems.     

The distribution of points on the sphere and corresponding cubature formulae

In applications, for instance in optics and astrophysics, thereis a need for high-accuracy integration formulae for functionson the sphere. To construct better formulae than previouslyused, almost equidistantly spaced nodes on the sphere and weightsbelonging to these nodes are required. This problem is closelyrelated to an optimal dispersion problem on the sphere and tothe theories of spherical designs and multivariate Gauss quadratureformulae. We propose a two-stage algorithm to compute optimal point locationson the unit sphere and an appropriate algorithm to calculatethe corresponding weights of the cubature formulae. Points aswell as weights are computed to high accuracy. These algorithmscan be extended to other integration problems. Numerical examplesshow that the constructed formulae yield impressively smallintegration errors of up to 10^-12.     

On the construction of multi-dimensional embedded cubature formulae

Summary In order to compute an integralI[f], one needs at least two cubature formulaeQ _j ,j{1, 2}. |Q ₁[f]–Q ₂[f]| can be used as an error estimate for the less precise cubature formula. In order to reduce the amount of work, one can try to reuse some of the function evaluations needed forQ ₁, inQ ₂. The easiest way to construct embedded cubature formulae is: start with a high degree formulaQ ₁, drop (at least) one knot and calculate the weights such that the new formulaQ ₂ is exact for as much monomials as possible. We describe how such embedded formulae with positive weights can be found. The disadvantage of such embedded cubature formulae is that there is in general a large difference in the degree of exactness of the two formulae. In this paper we will explain how the high degree formula can be chosen to obtain an embedded pair of cubature formulae of degrees 2m+1/2m–1. The method works for all regions ⁿ ,n2. We will also show the influence of structure on the cubature formulae.     

Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices

We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter λ and on the vector of real physical parameters p. We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of p. Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD α ²-dynamo and circular string demonstrates the efficiency and applicability of the approach.     

A Class of Series Acceleration Formulae for Catalan's Constant

In this note, we develop transformation formulae and expansions for the log tangent integral, which are then used to derive series acceleration formulae for certain values of Dirichlet L-functions, such as Catalan's constant. The formulae are characterized by the presence of an infinite series whose general term consists of a linear recurrence damped by the central binomial coefficient and a certain quadratic polynomial. Typically, the series can be expressed in closed form as a rational linear combination of Catalan's constant and times the logarithm of an algebraic unit.     

Counting and coding identity trees with fixed diameter and bounded degree

Identity trees with bounded maximum degree play a fundamental role in applications-oriented problems, especially when the trees are classified by their diameters. This paper offers results related to enumeration of such tree classes obtained by extending the methods of Gordon and Kennedy [The counting and coding of trees of fixed diameter, SIAM J. Appl. Math. 28 376–398 (1975)]. We set our results into the context of other enumerative work on identity trees.We derive formulae for the numbers of identity trees of various types, with fixed diameter and maximum degree. This then leads to asymptotic formulae (for large diameter). By combining these with formulae derived by Gordon and Kennedy [loc. cit.] we obtain the asymptotic fractions of identity trees among trees in various classes. These fractions are juxtaposed with asymptotic results that have appeared elsewhere.Our final section derives algorithms for integer coding and decoding identity trees in a way that is highly convenient for computer applications.     

Integral formulae on foliated symmetric spaces

In this article, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan and the second author) for foliations of codimension 1 or unit vector fields and obtain an infinite series of such formulae involving invariants of the Weingarten operator of a unit vector field, of the Jacobi operator in its direction, and their products. We write several such formulae explicitly, on locally symmetric spaces as well as on arbitrary Riemannian manifolds where they involve also covariant derivatives of the Jacobi operator. We work also with foliations of codimension 1 (or vector fields) which admit “good” (in a sense) singularities.     

Rigidity of <Emphasis Type="Italic">τ</Emphasis>-quasi Ricci-harmonic metrics

We study τ-quasi Ricci-harmonic metrics. First, we shall derive some formulae which will give some integral formulae for such a class of compact manifolds that permit to obtain some rigidity results. Second, and particularly, if a τ-quasi Ricci-harmonic metric possesses constant generalized scalar curvature then we determine the generalized scalar curvature in explicit form. These results are generalizations of ones found in [1–3, 5, 10].     

Twisted $$\Gamma \times \mathbb T^ n$$Γ×T n-equivaria nt degree with n-parameters: computatio nal formulae a nd applicatio ns

We introduce the generalized twisted \(\Gamma \times \mathbb {T}^{n}\)-equivariant degree with n free parameters, where \(\Gamma \) is a nonabelian finite group, present a computational formulae based on the reduction to \(\mathbb {T}^{n}\)-equivariant maps and reduction of parameters techniques. This twisted equivariant degree can be effectively applied to study Hopf bifurcation with symmetries from relative equilibria. We give an example of such bifurcation in a system of ODEs.     

The Hodge–Poincaré polynomial of the moduli spaces of stable vector bundles over an algebraic curve

Let X be a nonsingular complex projective variety that is acted on by a reductive group G and such that ${X^{ss} \neq X_{(0)}^{s}\neq \emptyset}$ . We give formulae for the Hodge–Poincaré series of the quotient ${X_{(0)}^{s}/G}$ . We use these computations to obtain the corresponding formulae for the Hodge–Poincaré polynomial of the moduli space of properly stable vector bundles when the rank and the degree are not coprime. We compute explicitly the case in which the rank equals 2 and the degree is even.     

A Widom like formula for some Toeplitz plus Hankel determinants

A well known Widom formula expresses the determinant of a Toeplitz matrix _Tn$T_n$ with Laurant polynomial symbol f in terms of the zeros of f. We give similar formulae for some even Toeplitz plus Hankel matrices. The formulae are based on an analytic representation of the determinant of such matrices in terms of Chebyshev polynomials.     

Geometrically Derived Difference Formulae for the Numerical Integration of Trajectory Problems

The term "trajectory problem" is taken to include problems thatcan arise, for instance, in connection with contour plotting,or in the application of continuation methods, or during phaseplane analysis. Geometrical techniques are used to constructdifference methods for these problems to produce in turn explicitand implicit circularly exact formulae. Based on these formulae,a predictor-corrector method is derived which, when comparedwith a closely related standard method, shows improved performance.It is found that this latter method produces spurious limitcycles, and this behaviour is partly analysed. Finally, a simplevariable-step algorithm is constructed and tested. Visting Scientists at the National Research Institute for MathematicalSciences, CSIR, P.O. BOX 395. pretoria, South Africa     

Simple expressions for finding recovery system inventory control parameter values

In recent years considerable effort has been devoted to the development of inventory control models for joint manufacturing and remanufacturing. Optimality of control policies is analyzed and algorithms for the determination of parameter values have been developed. However, there is still a lack of formulae or algorithms that allow for an easy computation of optimal or near optimal policy parameter values. This paper addresses the problem of computing the produce-up-to level S and the remanufacture-up-to level M in a periodic review inventory control model. We provide simple formulae for the policy parameter values, which can easily be implemented within spreadsheet applications. The approach is to derive news-vendor-type formulae that are based on underage and overage cost considerations. We propose different formulae depending on whether lead times for production and remanufacturing are identical or not. A numerical study shows that the obtained solutions provide relatively small cost deviations compared to the optimal solution within the investigated class of inventory control policies.     

High order methods for the numerical solution of two-point boundary value problems

In a recent paper, Cash and Moore have given a fourth order formula for the approximate numerical integration of two-point boundary value problems in O.D.E.s. The formula presented was in effect a one-off formula in that it was obtained using a trial and error approach. The purpose of the present paper is to describe a unified approach to the derivation of high order formulae for the numerical integration of two-point boundary value problems. It is shown that the formula derived by Cash and Moore fits naturally into this framework and some new formulae of orders 4, 6 and 8 are derived using this approach. A numerical comparison with certain existing finite difference methods is made and this comparison indicates the efficiency of the high order methods for problems having a suitably smooth solution.     

On the dimension of bivariate spline spaces of smoothnessr and degreed=3r+1

Summary We consider the well-known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally. To date, dimension formulae for such spaces have been established only whend3r+2, (except for the special case wherer=1 andd=4). In this paper we establish dimension formulae for allr1 andd=3r+1 for almost all triangulations.Dedicated to R. S. Varga on the occasion of his sixtieth birthdaySupported in part by National Science Foundation Grant DMS-8701121Supported in part by National Science Foundation Grant DMS-8602337     

High orderP-stable formulae for the numerical integration of periodic initial value problems

Summary Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Störmer-Cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that ofP-stability. However Lambert and Watson have shown that aP-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems.P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.     

Multidimensional occupancy problems with Poisson randomization

In this paper the concept of Poisson randomization is studied as given in [1, 2] and analogous formulae for the generalized process are derived. The generalization regards occupancy problems where different ball types are considered such that each type has an associated probability distribution of urn occupancy. Theorems are given for formulae to calculate probabilities of events and the distribution and moments of waiting time random variables. Finally, the theory is illustrated with examples.