A composition law for Runge-Kutta methods applied to index-2 differential-algebraic equations

This paper presents a new composition law for Runge-Kutta methods when applied to index-2 differential-algebraic systems. Applications of this result to the study of the order of composite methods and of symmetric methods are given.     

Some remarks on the symmetric rank-one update

We consider the symmetric rank-one, quasi-Newton formula. The hereditary properties of this formula do not require quasi-Newton directions of search. Therefore, this formula is easy to use in constrained optimization algorithms; no explicit projections of either the Hessian approximations or the parameter changes are required. Moreover, the entire Hessian approximation is available at each iteration for determining the direction of search, which need not be a quasi-Newton direction. Theoretical difficulties, however, exist. Even for a positive-definite, quadratic function with no constraints, it is possible that the symmetric rank-one update may not be defined at some iteration. In this paper, we first demonstrate that such failures of definition correspond to either losses of independence in the directions of search being generated or to near-singularity of the Hessian approximation being generated. We then describe a procedure that guarantees that these updates are well-defined for any nonsingular quadratic function. This procedure has been incorporated into an algorithm for minimizing a function subject to box constraints. Box constraints arise naturally in the minimization of a function with many minima or a function that is defined only in some subregion of the space.     

Can a Circulating Light Beam Produce a Time Machine

In a recent paper, Mallett found a solution of the Einstein equations in which closed timelike curves (CTC’s) are present in the empty space outside an infinitely long cylinder of light moving in circular paths around an axis. Here we show that, for physically realistic energy densities, the CTC’s occur at distances from the axis greater than the radius of the visible universe by an immense factor. We then show that Mallett’s solution has a curvature singularity on the axis, even in the case where the intensity of the light vanishes. Thus it is not the solution one would get by starting with Minkowski space and establishing a cylinder of light.     

The Orthomodular Poset of Projections of A Symmetric Lattices

In a Hilbert space, there exists a natural correspondence between continuous projections and particular pairs of closed subspaces. In this paper, we generalize this situation and associate to a symmetric lattice L a subset P(L) of L× L, called its projection poset. If L is the lattice of closed subspaces of a topological vector space then elements of P(L) correspond to continuous projections and we prove that automorphisms of P(L) are determined by automorphisms of the lattice L when this lattice satisfies some basic properties of lattices of closed subspaces. Primary: 06C15, Secondary: 03G12 81P10.     

The Doss Trick on Symmetric Spaces

The Doss trick is employed to find solutions of Schrüdinger equations on symmetric spaces of compact type. The potentials and initial conditions are taken from an algebra of functions which admit an holomorphic extension to the complexification of the considered symmetric spaces.     

Energy Conservation with Non-Symplectic Methods: Examples and Counter-Examples

Energy conservation of numerical integrators is well understood for symplectic one-step methods. This article provides new insight into energy conservation with non-symplectic methods. Sufficient conditions and counter-examples are presented. AMS subject classification (2000) 65L06, 65P10, 37J99.Submitted June 2004. Accepted October 2004. Communicated by Syvert Nørsett.     

Plane Symmetric Cosmological Macro Models in Self-creation Theory of Gravitation

We have constructed a class of plane symmetric macro models in Barber's second self-creation theory, when the source of the gravitational field is a macro matter field representing perfect fluid and satisfying the gamma-law equation of state p=(-1), where =4/3 and 3/2. The solutions of the field equations are derived and their physical aspects are studied.     

Deficiency Indices of a Symmetric Ordinary Differential Operator with Infinitely Many Degeneration Points

Let H be the minimal symmetric operator in L ²() generated by the differential expression (-1) ⁿ (c(x)f ⁽ⁿ⁾)⁽ⁿ⁾, n 1, with a real coefficient c(x) that has countably many zeros without finite accumulation points and is infinitely smooth at all points x with c(x) 0. We study the value Def H of the deficiency indices of H. It is shown that DefH=+ if infinitely many zeros of c(x) have multiplicities p satisfying the inequality n – 1/2 < p < 2n – 1/2. Our second result pertains to the case in which the set of zeros of c(x) is bounded neither above nor below. Under this condition, Def H = 0 provided that the multiplicity of each zero is greater than or equal to 2n – 1/2. The multiplicities of zeros of c(x) are understood in the paper in a broader sense than in the standard definition.     

New classes of domains with explicit Bergman kernel

We introduce two classes of egg type domains, built on general bounded sym-metric domains, for which we obtain the Bergman kernel in explicit formulas. As an aux-iliary tool, we compute the integral of complex powers of the generic norm on a boundedsymmetric domains using the well-known integral of Selberg. This generalizes matrix in-tegrals of Hua and leads to a special polynomial with integer or half-integer coefficientsattached to each irreducible bounded symmetric domain.     

Two-Parametric Compound Binomial Approximations

We consider two-parametric compound binomial approximation of the generalized Poisson binomial distribution. We show that the accuracy of approximation essentially depends on the symmetry or shifting of distributions and construct asymptotic expansions. For the proofs, we combine the properties of norms with the results for convolutions of symmetric and shifted distributions. In the lattice case, we use the characteristic function method. In the case of almost binomial approximation, we apply Steins method.__________Published in Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 443–466, October–December, 2004.