On successors of Jónsson cardinals

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming does not exist. Namely, if is a Jónsson cardinal then , provided that there is no non-trivial elementary embedding . There are a number of related results in ZFC concerning in V and inner models, for a Jónsson or singular cardinal. Received: 8 December 1998     

Consistency of V = HOD with the wholeness axiom

The Wholeness Axiom (WA) is an axiom schema that can be added to the axioms of ZFC in an extended language , and that asserts the existence of a nontrivial elementary embedding . The well-known inconsistency proofs are avoided by omitting from the schema all instances of Replacement for j-formulas. We show that the theory ZFC + V = HOD + WA is consistent relative to the existence of an embedding. This answers a question about the existence of Laver sequences for regular classes of set embeddings: Assuming there is an -embedding, there is a transitive model of ZFC +WA + “there is a regular class of embeddings that admits no Laver sequence.” Received: 7 July 1998 / Revised version: 5 November 1998     

Multiplicities of Dupin hypersurfaces

All known examples of isoparametric hypersurfaces of spheres are of the following two types:?(1) The homogeneous examples, which are principal orbits of isotropy representations of symmetric spaces of rank 2.?(2) The Clifford examples of Ferus–Karcher–Münzner, which are constructed from modules over Clifford algebras.?Our main result, which extends work of Münzner, Thorbergsson and Grove–Halperin, is that any isoparametric hypersurface has the same number of principal curvatures and the same multiplicities as some hypersurface of type (1) or (2). More generally, this holds for any Dupin hypersurface. The proof is homotopy theoretic; the main tools are the Hopf invariant and the EHP–sequence. Oblatum 19-III-1997 & 9-III-1999 / Published online: 5 August 1999     

Multilevel ILU decomposition

Summary. In this paper, the multilevel ILU (MLILU) decomposition is introduced. During an incomplete Gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels. On these levels, some smoothing steps are computed. The MLILU decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices. Convergence rates independent of the number of unknowns are shown numerically for several examples. Many numerical experiments including unsymmetric and anisotropic problems, problems with jumping coefficients as well as realistic problems are presented. They indicate a very robust convergence behavior of the MLILU method. Received June 13, 1997 / Revised version received March 17, 1998     

Interpolatory product quadratures for Cauchy principal value integrals with Freud weights

Summary. We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value integrals on the real line with Freud–type weight functions. The formulas are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration. As a by–product, we obtain new bounds for the derivative of the functions of the second kind for these weight functions. Received July 15, 1997 / Revised version received August 25, 1998     

A residual based error estimator for mortar finite element discretizations

Summary. A residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite element methods is introduced and analyzed. In particular, we consider mortar finite element techniques restricting ourselves to geometrically conforming domain decomposition methods using P1 approximations in each subdomain. Additionally, a residual based error estimator for Crouzeix-Raviart elements of lowest order is presented and compared with the error estimator obtained in the more general mortar situation. It is shown that the computational effort of the error estimator can be considerably reduced if the special structure of the Lagrange multiplier is taken into account. Received July 18, 1997 / Revised version received July 27, 1998 / Published online September 7, 1999     

Sets of Periods for Expanding Maps on Flat Manifolds

 It is proven that the sets of periods for expanding maps on n-dimensional flat manifolds are uniformly cofinite, i.e. there is a positive integer m ₀, which depends only on n, such that for any integer , for any n-dimensional flat manifold ℳ and for any expanding map F on ℳ, there exists a periodic point of F whose least period is exactly m. (Received 10 April 1998; in revised form 20 January 1999)     

Squares of Primes and Powers of 2

 As an extension of the Linnik-Gallagher results on the “almost Goldbach” problem, we prove, among other things, that there exists a positive integer k ₀ such that every large even integer is a sum of four squares of primes and k ₀ powers of 2. (Received 7 September 1998; in revised form 3 May 1999)     

Towards Minimal Addition Chains Using Ant Colony Optimisation

Modular exponentiation is one of the most important operations in almost all modern cryptosystems. It is performed using a series of modular multiplications. This operation is time consuming for large operands as is always the case in cryptography. Hence fast public-key cryptography software or hardware requires optimisation of the time consumed by a single modular multiplication and/or the reduction of the total number of modular multiplications required. This paper introduces a novel idea based on the principles of ant colony optimisation for finding a minimal addition chain that allows one to reduce the number of modular multiplications so that modular exponentiation can be implemented efficiently. The best addition chain reached by the ant system is compared to the one used in the m-ary and sliding window methods as well as with the best addition chain evolved by genetic algorithms. We demonstrate that the ant system significantly outperforms all these methods for any exponent size. ^★★ Research supported by FAPERJ () and CNPq ().     

Asymptotic behaviour of reaction–diffusion equations with non-damped impulsive effects

In this paper we consider non-autonomous reaction–diffusion systems with impulsive effects at fixed moments of time from the point of view of the theory of global attractors. For a translation-compact nonlinear term which does not provide the uniqueness of the Cauchy problem, and for different classes of non-damped multivalued impulse perturbations, we construct a multivalued non-autonomous dynamical system and prove for it the existence of a compact global attractor.