Slowly oscillating functions in semigroup compactifications and convolution algebras

The paper proposes a unified approach to many key theorems proved in the last twenty years in different areas of abstract harmonic analysis. This approach is based on the so-called slowly oscillating functions which were introduced in coarse geometry. In addition to this method being the most natural and simple, it also leads to the generalisation of some of the results and to the achievement of some new results. Several of these results concern the topological centres of convolution algebras and semigroup compactifications.     

Asymptotic oscillations

Given a function defined on the support of a ballean, we introduce the notion of slow oscillation in direction of a filter on X. We show that there exists a filter on X responsible for the rate of slow oscillation of f at infinity. We apply this result to the Stone-?ech compactifications of discrete groups.     

The extension of the operation in the Stone-Čech compactification of semigroups

For a large class of locally compact semitopological semigroups S, the Stone-Čech compactification β S is a semigroup compactification if and only if S is either discrete or countably compact. Furthermore, for this class of semigroups which are neither discrete nor countably compact, the quotient contains a linear isometric copy of ℓ _∞. These results improve theorems by Baker and Butcher and by Dzinotyiweyi.     

Weak factorizations of operators in the group von Neumann algebras of certain amenable groups and applications

Let G be a compact group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group and A(G × H) be the Fourier algebra of G × H. We prove that the group von Neumann algebra VN(G × H) = A(G × H)^* has the weak uniform A(G × H)^** factorization property of level b(G). As a corollary we show that A(G × H) is strongly Arens irregular, and the topological centre of UC ₂(G × H)^* is equal to the Fourier–Stieltjes algebra B(G × H).     

On the semigroup BetaS

Let S be an infinite, discrete, cancellative semigroup and let BetaS be the Stone-Cech compactification of S. Then BetaS is a semigroup with an operation which extends that of S and which is continous only in one variable. We generalize some algebraic properties known to hold for the additive semigroup of the integers.     

Elliptic dynamical reflection algebra and partition function of SOS model with reflecting end

In this paper we introduce an elliptic dynamical reflection algebra describing an SOS model with a reflecting end. Using a factorizing Drinfel’d twist, we compute the partition function of this model with domain wall boundary conditions. We show that it can be represented in the form of a single Izergin determinant.     

Synthesis of new spiro‐[isothiochromene‐3,5′‐isoxazolidin]‐4(1H)‐ones

A series of seven new 2′,3′,4′‐substituted spiro[isothiochromene‐3,5′‐isoxazolidin]‐4(1H)‐ones ( 7‐13 ) has been prepared in the reaction of benzylidene(phenyl)azane oxide ( 5 ) or benzylidene(methyl)azane oxide ( 6 ) with (3Z)‐3‐(4‐substituted‐benzylidene)‐1H‐isothio‐ chromen‐4(3H)‐one ( 1‐4 ). The reaction occurs by a 1,3‐dipolar cycloaddition mechanism that leads to the regiospecific formation of various spiroisoxazolidines ( 7‐13 ).     

Lower bounds of Tian's invariant under toric invariances

On some toric Fano manifolds with metrics in the first Chern class, we show that a large family of smooth almost pluri-subharmonic functions (i.e. subharmonic with respect to the metric) with maximum equal to 0 admits a lower envelope. In our previous papers (A. Ben Abdesselem (2006) [4] and A. Ben Abdesselem and B. Dridi (2008) [5]) we established such envelopes when the functions considered are invariant under the action of a larger automorphisms group. Here we only consider the invariances due to the of the toric structure of the manifolds.     

Tuning the hydrophilicity of gold nanoparticles templated in star block copolymers

We report on a simple procedure to tune the hydrophilicity of hybrid gold nanoparticles. The nanoparticles have been prepared in the core of a poly(ethylene glycol)-block-poly(epsilon-caprolactone) (PEG-b-PCL) five-arm star block copolymer. A hydrophilic corona was then added to these hybrid gold nanoparticles by direct chemisorption of trithiocarbonate-containing poly(acrylic acid) chains. These polymers were synthesized by RAFT polymerization with a trithiocarbonate as the chain-transfer agent. The efficiency of the grafting was evidenced by TEM, AFM, and DLS and by the successful transfer of these nanoparticles from organic solvent to water.