Asymptotic Convergence of Conjugate Gradient Methods for the Partial Symmetric Eigenproblem

Recently an efficient method (DACG) for the partial solution of the symmetric generalized eigenproblem A x = δB x has been developed, based on the conjugate gradient (CG) minimization of the Rayleigh quotient over successive deflated subspaces of decreasing size. The present paper provides a numerical analysis of the asymptotic convergence rate ρ_j of DACG in the calculation of the eigenpair λ_j, u _j, when the scheme is preconditioned with A⁻¹. It is shown that, when the search direction are A-conjugate, ρ_j is well approximated by 4/ξ_j, where ξ_j is the Hessian condition number of a Rayleigh quotient defined in appropriate oblique complements of the space spanned by the leftmost eigenvectors u ₁, u ₂,…, u _j−1 already calculated. It is also shown that 1/ξ_j is equal to the relative separation between the eigenvalue λ_j currently sought and the next higher one λ_j+1 and the next higher one λ_{j + 1}. A modification of DACG (MDACG) is studied, which involves a new set of CG search directions which are made M-conjugate, with M-conjugate, with M-conjugate, with M a matrix approximating the Hessian. By distinction, MDACG has an asymptotic rate of convergence which appears to be inversely proportional to the square root of ξ_j, in complete agreement with the theoretical results known for the CG solution to linear systems. © 1997 by John Wiley & Sons, Ltd.     

The Prime Spectrum and the Extended Prime Spectrum of Noncommutative Rings

We investigate the prime spectrum of a noncommutative ring and its spectral closure, the extended prime spectrum. We construct a ring for which the prime spectrum is a spectral space different from the extended prime spectrum and we construct a von Neumann regular ring for which the prime spectrum is not a spectral space. The authors are members of the European Research Training Network RAAG (Contract No. HPRN-CT-2001-00271). The first author was also supported by the Ministry of Education, Science and Sport of Slovenia.     

Freshman rules in calculus

In this note we present a class of functions (f, g) that satisfy ‘freshman rules’ in calculus.     

Über Die Assoziierten Primideale des Bidualen

Let (R, 𝔪) be a commutative, noetherian, local ring, E the injective hull of the residue field R/𝔪, and M ^○○ = Hom _R (Hom _R (M, E), E) the bidual of an R-module M. We investigate the elements of Ass(M ^○○) as well as those of Coatt(M) = {𝔭 ∈ Spec(R)|𝔭 = Ann _R (Ann _M (𝔭))} and provide criteria for equality in one of the two inclusions Ass(M) ? Ass(M ^○○) ? Coatt(M). If R is a Nagata ring and M a minimax module, i.e., an extension of a finitely generated R-module by an artinian R-module, we show that Ass(M ^○○) = Ass(M) ∪ {𝔭 ∈ Coatt(M)| R/𝔭 is incomplete}.     

Bi-Invariant Domains in Semisimple Groups: Nonprincipal Orbits and Boundaries

Let G be a complex semisimple group with real form G . For the action of G × G on G by left and right translation, we define and study an orbit-type stratification of G. In particular, we compute the dimension of an arbitrary stratum; we identify certain strata of low codimension in G, the union of whose closures contains every nonprincipal stratum; and we describe the boundaries and adjacency relations of the maximal connected domains in G that contain only principal orbits.     

Complemented ideals in the Fourier algebra of a locally compact group

In this paper we provide a necessary condition for a closed ideal in the Fourier algebra of a locally compact amenable group to be completely complemented. The classification of completely complemented ideals is completed in the case of an amenable discrete group. We also investigate the ideals possessing a bounded approximate identity.

     

A continuous decomposition of the Menger curve into pseudo-arcs

It is proved that the Menger universal curve admits a continuous decomposition into pseudo-arcs with the quotient space homeomorphic to .

     

Optimal risk transfer for agents with germs

We introduce a new class of risk measures called generalized entropic risk measures (GERMS) that allow economic agents to have different attitudes towards different sources of risk. We formulate the problem of optimal risk transfer in terms of these risk measures and characterize the optimal transfer contract. The optimal contract involves what we call intertemporal source-dependent quotient sharing, where agents linearly share changes in the aggregate risk reserve that occur in response to shocks to the system over time, with scaling coefficients that depend on the attitudes of each agent towards the source of risk causing the shock. Generalized entropic risk measures are not dilations of a common base risk measure, so our results extend the class of risk measures for which explicit characterizations of the optimal transfer contract can be found.     

The multiplicity of abelian covers of splice quotient singularities

From a resolution graph with certain conditions, Neumann and Wahl constructed an equisingular family of surface singularities called splice quotients. For this class some fundamental analytic invariants have been computed from their resolution graph. In this paper we give a method to compute the multiplicity of an abelian covering of a splice quotient from its resolution graph and the Galois group.