A chemical method for the convenient surface functionalisation of polymers

A method for the application of carbenes as reactive intermediates for surface modification of polymeric substrates has been developed; the efficacy of the process has been demonstrated by irreversible dyeing of polystyrene, polythene, nylon, silica, and controlled pore glass, and to a limited extent, polytetrafluoroethylene.     

Spontaneous formation of vesicles of diblock copolymer EO6BO11 in water: a SANS study

Small angle neutron scattering (SANS) is used to study the structures formed in water by a diblock copolymer EO6BO11 (having 6 ethylene oxide, EO, and 11 butylene oxide, BO, units). The data show that polymer solutions over a broad concentration range (0.05-20 wt %) contain vesicular structures at room temperature. Interestingly, these vesicles could be formed without any external energy input, such as extrusion, which is commonly required for the formation of other block copolymer or lipid vesicles. The EO6BO11 vesicles are predominantly unilamellar at low polymer concentrations, whereas at higher polymer concentrations or temperatures there is a coexisting population of unilamellar and multilamellar vesicles. At a critical concentration and temperature, the vesicular structures fuse into lyotropic arrays of planar lamellar sheets. The findings from this study are in broad agreement with the work of Harris et al. (Langmuir, 2002, 18, 5337), who used electron microscopy to identify the vesicle phase in the same system.     

Experimental study of the stability and optimization of multipulse passive mode locking

We present an experimental study of the stability of passively mode-locked pulses against noise in multipulse operation of an erbium-doped fiber laser. The laser properties are determined by two dimensionless combinations of the laser parameters. Measurements of the pulses' destabilization threshold as a function of those laser parameters show the optimal regions that maximize the mode-locked pulse stability. We find good agreement between the experimental observations and the theoretical predictions.     

Comparison of graphs associated to a commutative Artinian ring

Let R be a commutative ring with \(1\ne 0\) and the additive group \(R^+\). Several graphs on R have been introduced by many authors, among zero-divisor graph \(\Gamma _1(R)\), co-maximal graph \(\Gamma _2(R)\), annihilator graph AG(R), total graph \( T(\Gamma (R))\), cozero-divisors graph \(\Gamma _\mathrm{c}(R)\), equivalence classes graph \(\Gamma _\mathrm{E}(R)\) and the Cayley graph \(\mathrm{Cay}(R^+ ,Z^*(R))\). Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to \(\mathrm{Cay}(R^+ ,Z^*(R))\). Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when R is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if R has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results, we prove that for a commutative finite ring R with \(|\mathrm{Max}(R)|=n \ge 3\), \( \Gamma _1(R) \simeq \Gamma _2(R)\) if and only if \(R\simeq \mathbb {Z}^n_2\); if and only if \(\Gamma _1(R) \simeq \Gamma _\mathrm{E}(R)\). Also the annihilator graph is identical to the cozero-divisor graph if and only if R is a Frobenius ring.     

Asymptotic windings of horocycles

Let Γ be a torsion-free hyperbolic group. We study Γ-limit groups which, unlike the fundamental case in which Γ is free, may not be finitely presentable or geometrically tractable. We define model Γ-limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic). Given a strict resolution of an arbitrary Γ-limit group L, we canonically construct a strict resolution of a model Γ-limit group, which encodes all homomorphisms L → Γ that factor through the given resolution. We propose this as the correct framework in which to study Γ-limit groups algorithmically. We enumerate all Γ-limit groups in this framework.     

Deforming the Lie superalgebra of contact vector fields on <Emphasis Type="Italic">S</Emphasis><Superscript>1|2</Superscript> inside the Lie superalgebra of pseudodifferential operators on <Emphasis Type="Italic">S</Emphasis><Superscript>1|2</Superscript>

We classify deformations of the standard embedding of the Lie superalgebra $ \mathcal{K} $ \mathcal{K} (2) of contact vector fields on the (1, 2)-dimensional supercircle into the Lie superalgebra SΨD(S ^1|2 ) of pseudodifferential operators on the supercircle S ^1|2 . The proposed approach leads to the deformations of the central charge induced on $ \mathcal{K} $ \mathcal{K} (2) by the canonical central extension of SΨD(S ^1|2 ).     

Cyclicity in the Dirichlet space

Let be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We give a new sufficient condition, not far from the known necessary condition, for a function f∈ to be cyclic, i.e. for {pf: p is a polynomial} to be dense in . The proof is based on the notion of Bergman–Smirnov exceptional set introduced by Hedenmalm and Shields. Our methods yield the first known examples of such sets that are uncountable. One of the principal ingredients of the proof is a new converse to the strong-type inequality for capacity.     

The de Schipper formula and squares of Riesz spaces

In this paper we introduce and study the square mean and the geometric mean in Riesz spaces. We prove that every geometric mean closed Riesz space is square mean closed and give a counterexample to the converse. We define for positive a, b in a square mean closed Riesz space E an addition via the formulaab=sup {(cos x)a + (sin x)b: 0 x 2π},which goes back to a formula by de Schipper. In case that E is geometric mean closed this turns the mldeflying set of the positive cone of E into a lattice ordered semigroup, which in turn is the positive cone ofa Riesz space E^□. We prove, under the additional condition that E is geometric mean closed, that E^□ is Riesz isomorphic to the square of E as introduced earlier by Buskes and van Rooij.