Polyphosphate chains by addition of oxiranes to phosphoric acid

Addition of diepoxides to the acids of phosphorus provides a new, attractive route to polyphosphates. In order to understand better the mechanism of this reaction and the structure of the products, the model reactions, namely additions of simple oxiranes to the acids of phosphorus, have been studied. It was shown that this reaction is catalysed by the P-OH groups of the acids. Activation of the oxirane molecule occurs predominantly by formation of the hydrogen bonded species while ionized form of acid plays only a minor role at the studied conditions (1,4-dioxane solution, 25°C). Reactivity of subsequent P-OH groups in phosphoric acid increases with increasing degree of substitution by -OCH₂CH₂OH groups. The ratio of the rate constants of formation of mono-, di-, and triesters of phosphoric acid is equal to 1/2/4. This has been attributed to the formation of internal hydrogen bonds, involving substituents and the P-OH groups. The observed order of the rate constants shows that reaction can not be stopped at the diester stage (linear units) and triester species (branched units) are formed fast. However, due to the large difference in the rate of hydrolysis, especially pronounced at basic conditions, triesters of phosphoric acid can easily be converted into the corresponding diesters by simple hydrolysis. Thus, the addition of diepoxides to phosphoric acid, followed by hydrolysis, leads to essentially linear polyphosphate chains.     

Modal parameters of two incomplete and complete guitars differing in the bracing pattern of the soundboard

Similarities and differences in vibrational behavior of two guitars having a symmetric Torres bracing pattern and an asymmetric pattern forming a lattice on a soundboard are investigated by means of the modal analysis technique and laser Doppler vibrometry (LDV) measurements. Instruments are investigated before and after a bridge and strings assembling (i.e., they are incomplete or complete). The bracing pattern and the absence/presence of the bridge and strings have some effect on modal frequencies and mode shapes. The bracing pattern does not affect the sequence of at least first three low frequency mode shapes of incomplete/complete instruments but affects their modal frequencies. Depending on frequency, the bridge behaves either as a rigid or a flexible structure.     

Properties of quantum Markovian master equations

In this paper we give an essentially self-contained account of some general structural properties of the dynamics of quantum open Markovian systems. We review some recent results regarding the problem of the classification of quantum Markovian master equations and the limiting conditions under which the dynamical evolution of a quantum open system obeys an exact semigroup law (weak coupling limit and singular coupling limit). We discuss a general form of quantum detailed balance and its relation to thermal relaxation and to microreversibility.     

Early evolution of transversally thermalized partons

The idea that the parton system created in relativistic heavy-ion collisions (i) emerges in a state with transverse momenta close to thermodynamic equilibrium and (ii) its evolution at early times is dominated by the 2-dimensional (transverse) hydrodynamics of the ideal fluid is investigated. It is argued that this mechanism may help to solve the problem of early equilibration.     

Envelopes of splines in the projective plane

In this paper a family of curves—Riemannian cubics—inthe unit sphere and the real projective plane is investigated.Riemannian cubics naturally arise as solutions to variationalproblems in Riemannian spaces. It is remarkable to find thatan envelope of lines generated by a Riemannian cubic in onespace is (nearly) a Riemannian cubic in another space.     

Turán numbers for odd wheels

The Turán number $\text{ex}(n,G)$ is the maximum number of edges in any $n$-vertex graph that does not contain a subgraph isomorphic to $G$. A wheel$W_n$ is a graph on $n$ vertices obtained from a $C_{n?1}$ by adding one vertex $w$ and making $w$ adjacent to all vertices of the $C_{n?1}$. We obtain two exact values for small wheels:

$\text{ex}(n,W_5)=?\frac{n^2}{4}+\frac{n}{2}?,$

$\text{ex}(n,W_7)=?\frac{n^2}{4}+\frac{n}{2}+1?.$

Given that $\text{ex}(n,W_6)$ is already known, this paper completes the spectrum for all wheels up to 7 vertices. In addition, we present the construction which gives us the lower bound $\text{ex}(n,W_{2k+1})>?\frac{n^2}{4}?+?\frac{n}{2}?$ in general case.     

Effect of pH on deagglomeration and rheology/morphology of aqueous suspensions of goethite nanopowder

The kinetics of deagglomeration in diluted suspensions of goethite nanopowder, as well as the rheology and morphology of the resulting suspensions, strongly depends on pH. At pH 3, nanopowder can be dispersed as separate nanoparticles, and the resulting suspension is Newtonian, with the viscosity only marginally higher than the viscosity of water. At pH between 5 and 12, nanoparticles tend to reaggregate and form weak aggregates/flocs. Morphology changes from a Newtonian suspension of primary nanoparticles to a non-Newtonian, shear-thinning suspension of large, porous, interconnected flocs with the yield stress reaching a maximum at an isoelectric point. The effect of pH on morphology and rheology is reversible, and as pH is reduced to 3, the suspension becomes Newtonian, with viscosity marginally higher than the viscosity of water. The rheological models based on DLVO theory do not allow prediction of the effect of pH on viscosity and yield stress, but the flow curves of goethite suspensions can be described by a fractal model with five adjustable parameters.