Countably compact topological group topologies on free Abelian groups from selective ultrafilters

Tkachenko showed in 1990 the existence of a countably compact group topology on the free Abelian group of size c using CH. Koszmider, Tomita and Watson showed in 2000 the existence of a countably compact group topology on the free Abelian group of size c² using a forcing model in which CH holds.Wallace's question from 1955, asks whether every both-sided cancellative countably compact semigroup is a topological group. A counterexample to Wallace's question has been called a Wallace semigroup. In 1996, Robbie and Svetlichny constructed a Wallace semigroup under CH. In the same year, Tomita constructed a Wallace semigroup from MAcountable.In this note, we show that the examples of Tkachenko, Robbie and Svetlichny, and Koszmider, Tomita and Watson can be obtained using a family of selective ultrafilters. As a corollary, the constructions presented here are compatible with the total failure of Martin's Axiom.     

On cellularization for simplicial presheaves and motivic homotopy

We construct cellular homotopy theories for categories of simplicial presheaves on small Grothendieck sites and discuss applications to the motivic homotopy category of Morel and Voevodsky.     

QN-spaces, wQN-spaces and covering properties

The main results of the paper are as follows: covering characterizations of wQN-spaces, covering characterizations of QN-spaces and a theorem saying that Cp(X) has the Arkhangel'ski?ˇ property (α₁) provided that X is a QN-space. The latter statement solves a problem posed by M. Scheepers [M. Scheepers, Cp(X) and Arhangel'ski?ˇ's αi-spaces, Topology Appl. 89 (1998) 265-275] and for Tychonoff spaces was independently proved by M. Sakai [M. Sakai, The sequence selection properties of Cp(X), Preprint, April 25, 2006]. As the most interesting result we consider the equivalence that a normal topological space X is a wQN-space if and only if X has the property S₁(Γshr,Γ). Moreover we show that X is a QN-space if and only if Cp(X) has the property (α₀), and for perfectly normal spaces, if and only if X has the covering property (β₃).     

An adaptive scheme for the approximation of dissipative systems

We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, a regular explicit Euler scheme–with constant or decreasing step–may explode and implicit Euler schemes are CPU-time expensive. The algorithm we introduce is explicit and we prove that any weak limit of the weighted empirical measures of this scheme is a stationary distribution of the stochastic differential equation. Several examples are presented including gradient dissipative systems and Hamiltonian dissipative systems.     

A delta white noise functional

The additive renormalization% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabs7adaWgaaWcbaGaaeySdiaab6cacaqG0bqefeKCPfgBaGqb% diaa-bcaaeqaaOGaeyypa0Jaa8hiaiaacIcacaaIYaGaeqiWdaNaai% ykamaaCaaaleqabaGaeyOeI0IaaGymaiaac+cacaaIYaaaaGqadOGa% a4hiaiGacwgacaGG4bGaaiiCaiaacIcacqGHsislcaqGXoWaaWbaaS% qabeaacaqGYaaaaOGaai4laiaaikdacaGGPaGaa4hiaiaacQdaciGG% LbGaaiiEaiaacchacqGHXcqSdaWadiqaaiabgkHiTiaadkeacaGGNa% GaaiikaiaadshacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaai4laiaa% ikdacaGFGaGaey4kaSIaa4hiaiaabg7acaWGcbGaai4jaiaacIcaca% WG0bGaaiykaaGaay5waiaaw2faaiaacQdaaaa!6C5C!\[{\rm{\delta }}_{{\rm{\alpha }}{\rm{.t}} } = (2\pi )^{ - 1/2} \exp ( - {\rm{\alpha }}^{\rm{2}} /2) :\exp \pm \left[ { - B'(t)^2 /2 + {\rm{\alpha }}B'(t)} \right]:\]is shown to be a generalized Brownian functional. Some of its properties are derived. is shown to be a generalized Brownian functional. Some of its properties are derived.On leave from Universidade do Minho, Area de Matematica, Largo Carlos Amarante, P-4700 Braga, Portugal.     

Approximation of infinite delay and Volterra type equations

Summary We consider a class of infinite delay equations of neutral type which includes Volterrra type integral und integrodifferential equations. Using abstract approximation results (cf. Trotter-Kato-type) for strongly continuous semigroups we develop an approximation scheme which is based on approximation of the system state by Laguerre (and Legendre) polynomials. Numerical examples demonstrate the feasibility of the scheme and show infinite order convergence for smooth data.Supported in part by the Air Force Office of Scientific Research under Contracts AFORS-84-0398 and AFORS-85-0303 and the National Aeronautics and Space Administration under NASA Grant NAG-1-517 and by NSF under Grant UINT-8521208Supported in part by the Air Force Office of Scientific Research under Contract AFORS-84-0398 and in part by the Fonds zur Förderung der wissenschaftlichen Forschung, Austria, under project No. S3206     

Symmetric Boundary Value Methods for Second Order Initial and Boundary Value Problems

We introduce symmetric Boundary Value Methods for the solution of second order initial and boundary value problems (in particular Hamiltonian problems). We study the conditioning of the methods and link it to the boundary loci of the roots of the associated characteristic polynomial. Some numerical tests are provided to assess their reliability. Dedicated to the memory of Professor Aldo Cossu     

Integrating long-term care insurance purchase decisions with saving and investment for retirement

Risk related to long-term care (LTC) is high for the elderly. Planning for LTC is now regarded as the ‘third leg’ of retirement planning. In this paper, planning for LTC is integrated with saving and investment decisions for an integrated approach to retirement planning. Optimal LTC insurance purchase decisions are obtained by developing a trade-off between post-retirement LTC costs and LTC insurance premiums paid and coverage received. Integrating insurance purchase with wealth evolution, consisting of saving and investment decisions, allows addressing affordability issues.Two-way branching models are used for the stochastic health events and asset returns. The problem, formulated as a nonlinearly constrained mixed-integer optimization problem, is solved using a heuristic. Sensitivity analyses are performed for initial health and wealth status. Some important aspects of an individual’s behavioral preferences are also addressed in this framework to provide more robust decision support.     

On the pivot structure for the weighing matrix W(12,11)

In the present article we concentrate our study on the growth problem for the weighing matrix W(12,11) and show that the unique W(12,11) has three pivot structures. An improved algorithm for extending a k × k (0,+,-) matrix to a W(n,n-1), if possible, has been developed to simplify the proof. For the implementation of the algorithm special emphasis is given to the notions of data structures and parallel processing.