A finiteness result for Heegaard splittings

In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of 3g−3 disjoint essential disks. These are characterized by a combinatorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to fact that in general there can exist infinitely many such systems of disks which satisfy just the Casson-Gordon rectangle condition.     

Universal WCG banach spaces and universal Eberlein Compacts

We prove that for cardinalsτ satisfying τ^ω=τ and forτ=ω ₁, there do not exist universal Eberlein Compacts of weightτ, or universal WCG spaces of density characterτ. Ifτ is a strong limit cardinal of countable cofinality such universal spaces do exist. Thus under GCH universal spaces exist forτ iff cof(τ)=ω. The research of the second author was supported by a grant from the United States-Israel Binational Science Foundation and the Fund for the Promotion of Research at the Technion.     

Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable

We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let be a finite dimensional division algebra having center , and let be a normal subgroup of finite index. Suppose is not solvable. Then we may assume that is a minimal nonsolvable group (MNS group for short), i.e. a nonsolvable group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS group satisfies a certain property which we denote Property . This property includes the requirement that the diameter of the commuting graph should be , but is, in fact, stronger. Another ingredient is to show that if the commuting graph of has Property , then is open with respect to a nontrivial height one valuation of (assuming without loss of generality, as we may, that is finitely generated). After establishing the openness of (when is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of over its prime subfield to eliminate as a possible quotient of , thereby obtaining a contradiction and proving our main result.

     

Time course of loudness recalibration: implications for loudness enhancement

Loudness recalibration, the effect of a relatively loud 2500-Hz recalibrating tone on the loudness of a relatively soft 2500-Hz target tone, was measured as a function of the interstimulus interval (ISI) between them. The loudness of the target tone, assessed by a 500-Hz comparison tone, declined when the ISI equaled or exceeded about 200 ms and leveled off at an ISI of about 700 ms. Notably, the target tone's loudness did not change significantly at very short ISIs (< 150 ms). The latter result is incompatible with the literature reporting loudness enhancement in this time window, but is compatible with the suggestion made by Scharf, Buus, and Nieder [J. Acoust. Soc. Am. 112, 807-810 (2002)] that early measurements of enhancement were contaminated by the influence of the recalibrating tone on the comparison tone when the two shared the same frequency. In a second experiment the frequency of the comparison tone was changed to 2500 Hz and the results of a loudness enhancement paradigm was successfully predicted from the time course of recalibration obtained in experiment 1.     

Almost isometric embeddings into configuration-compacta

We show that a Polish metric space X which has ‘compact configuration spaces’ is free of almost isometric embeddings, i.e. given such an embedding into X one gets an isometric embedding. By assuming a continuous version of ℵ₀-homogeneity we get the converse statement, and also prove almost isometry uniqueness.     

Nonlinear wave interactions in patterned Silica and AlGaAs waveguides

We study nonlinear interactions between discrete optical solitons that propagate in different regimes of diffraction, and the nonlinear scattering of dispersive waves by local optical potentials. It is well known in optics that when linear coherent waves meet, they interfere without interactions. Linear waves also scatter through local optical structures not exchanging any power with the guided modes of these structures. As a focusing Kerr nonlinearity is present, such linearly-inhibited phenomena can exist. Our studies are performed in silica and AlGaAs nonlinear waveguides, excited by ultra-short pulses in the near infrared. Presented at 9-th International Workshop on Nonlinear Optics Applications, NOA 2007, May 17–20, 2007, Świnoujście, Poland     

Special Moufang sets, their root groups and their {micro}-maps

We prove Timmesfeld's conjecture that special abstract rankone groups are quasisimple. We give two characterizations ofthe root groups in special Moufang sets: a normal subgroup ofthe point stabilizer is a root group if it is either regular,or nilpotent and transitive. We prove that if a root group ofa special Moufang set contains an involution, then it is ofexponent 2. We also show that the root groups are abelian ifand only if the so-called µ-maps are involutions.     

Swelling of dextran gel and osmotic pressure of soluble dextran in the presence of salts

The swelling of dextran gels (Sephadex) in salt solutions with a water activity of 0.937, compared with the swelling in pure water, exhibited anion specificity as evidenced by an increased swelling ratio in the following order: Na₂SO₄ < H₂O < NaCl < NaSCN. The swelling ratio showed a good linear correlation with the osmotic pressure of dextran (500 kD) in these solutions. The salt‐concentration difference (imbalance) between the polymer‐solution side of the membrane and the polymer‐free permeate side during the osmotic‐pressure measurements positively correlated with the effect of the salt on the polymer osmotic pressure. These phenomena conform to Hofmeister‐type (or lyotropic) behavior. The diminishing augmentation of dextran osmotic pressure and the change in the salt‐concentration imbalance with rising NaSCN concentration imply a positive preferential interaction and adsorption of the salt onto the polymer. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2740–2750, 2001