The uniform modulus of continuity of iterated Brownian motion

LetX be a Brownian motion defined on the line (withX(0)=0) and letY be an independent Brownian motion defined on the nonnegative real numbers. For allt0, we define theiterated Brownian motion (IBM),Z, by setting . In this paper we determine the exact uniform modulus of continuity of the process Z.Research supported by NSF grant DMS-9122242.     

On the homology spectral sequence for topological Hochschild homology

Marcel Bökstedt has computed the homotopy type of the topological Hochschild homology of using his definition of topological Hochschild homology for a functor with smash product. Here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of -algebras as introduced by Elmendorf, Kriz, Mandell and May. We give algebraic arguments based on naturality properties of the topological Hochschild homology spectral sequence. In the process we demonstrate the utility of the unstable ``lower' notation for the Dyer-Lashof algebra.

     

Infinite products of finite simple groups

We classify the sequences of finite simple nonabelian groups such that has uncountable cofinality.

     

Phenyldimethylsilyl-Substituted Ketenes and Bisketenes

The phenyldimethylsilyl-substituted monoketene PhMe(2)SiCH=C=O (1) and bisketene (PhMe(2)SiC=C=O)(2) (3) have been prepared and compared to the corresponding Me(3)Si- and t-BuMe(2)Si-substituted species. The (13)C, (17)O, and (29)Si NMR spectra fit the pattern shown by other silylketenes and provide no evidence for transmission of a substituent effect of the Ph group through the silicon to the ketenyl group, as has been proposed for PhMe(2)Si-substituted radicals. The UV spectrum of 1 does show a longer lambda and greater epsilon than for t-BuMe(2)SiCH=C=O, and this may indicate some interaction of the phenyl group with the ketene chromophore, while the greater reactivity of 1 in hydration compared to t-BuMe(2)SiCH=C=O is ascribed to the inductive effect of the phenyl. The very similar ring-opening reactivity of the bis(phenyldimethylsilyl)cyclobutenedione (6) to form 3 compared to the bis(Me(3)Si) analogues also provides no evidence of a significant interaction of the phenyl with the ketene. A new type of stabilized 1,8-bisketene based on the arylbis(dimethylsilyl) grouping, namely, 1,4-bis(ketenyldimethylsilyl)benzene (12), has been prepared for the first time.     

On noise reduction methods for chaotic data

Recently proposed noise reduction methods for nonlinear chaotic time sequences with additive noise are analyzed and generalized. All these methods have in common that they work iteratively, and that in each step of the iteration the noise is suppressed by requiring locally linear relations among the delay coordinates, i.e., by moving the delay vectors towards some smooth manifold. The different methods can be compared unambiguously in the case of strictly hyperbolic systems corrupted by measurement noise of infinitesimally low level. It was found that all proposed methods converge in this ideal case, but not equally fast. Different problems arise if the system is not hyperbolic, and at higher noise levels. A new scheme which seems to avoid most of these problems is proposed and tested, and seems to give the best noise reduction so far. Moreover, large improvements are possible within the new scheme and the previous schemes if their parameters are not kept fixed during the iteration, and if corrections are included which take into account the curvature of the attracting manifold. Finally, the fact that comparison with simple low-pass filters tends to overestimate the relative achievements of these nonlinear noise reduction schemes is stressed, and it is suggested that they should be compared to Wiener-type filters.     

Complex geometry,unification, and quantum gravity. I. The geometry of elementary particles

The Poincaré group is replaced byU(3, 2), the pseudounitary extension of the de Sitter groupSO(3, 2), as internal and space-time symmetries are combined in a geometric setting which invalidates the no-go theorems. A new model of elementary particles as vertical vectors on the principal fiber bundleU(3, 2) U(3, 2)/U(3, 1)×U(1) is introduced and their interactions via Lie bracket analyzed. The model accounts for the four known superselection rules: spin, electric charge, baryon number, and lepton number.