Minimisers of the Allen–Cahn equation and the asymptotic Plateau problem on hyperbolic groups

We investigate the existence of non-constant uniformly-bounded minimal solutions of the Allen–Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace term in the Allen–Cahn equation is small enough, there exist minimal solutions satisfying a large class of prescribed asymptotic behaviours. For a phase field model on a hyperbolic group, such solutions describe phase transitions that asymptotically converge towards prescribed phases, given by asymptotic directions. In the spirit of de Giorgi's conjecture, we then fix an asymptotic behaviour and let the Laplace term go to zero. In the limit we obtain a solution to a corresponding asymptotic Plateau problem by Γ-convergence.     

Improved liquid chromatographic method with pulsed electrochemical detection for the analysis of gentamicin

The official method for the determination of the composition and related substances of gentamicin prescribed by the European Pharmacopoeia (Ph. Eur.) is liquid chromatography combined with pulsed electrochemical detection (LC-PED). However, this method utilizes a polymer stationary phase which shows rather low efficiency towards the separation of the main gentamicin components. Moreover, the mobile phase contains a lot of non volatile salts: sodium sulphate and sodium octanesulphonate. Following a comparative study, the most performant LC-PED method has been evaluated and validated using a reversed phase C18 column (C18, 250 x 4.6mm ID, 110 A, 5 microm) kept at 35 degrees C with a mobile phase containing volatile ion pairing agents: trifluoroacetic acetic acid (TFA) and pentafluoropropionic acid (PFPA). In addition to the selectivity of the main gentamicin components and its related substances, the method is repeatable, linear and proves to be robust. It is also applicable to a wider number of C18 columns.     

The degree of maps of free G-manifolds

Suppose that M and N are orientable, closed, connected manifolds with free actions of compact Lie groups G and H of the same dimension, and suppose that u : G → H is a homomorphism. We study the degree of maps f : M → N that are “equivariant up to u”. For abelian actions and for a power map such maps satisfy the condition f(λx) = λ ^r x. To Albrecht Dold and Edward Fadell     

A miniaturized and integrated galvanic cell for the potentiometric measurement of ions in biological liquids

A procedure for an all-plastic electrochemical cell comprising miniaturized planar indicator and reference electrodes is described. All electrodes are based on conducting polymers, are fully integrated, and contain no internal electrolyte. The reference microsensors were deposited via electrochemical polymerization from a water solution containing the monomer 3,4-ethylenedioxythiophene (EDOT) or 1-methylpyrrole (MPy) and a biochemical buffer 3-(N-morpholino) propanesulfonic acid [MOPS], 2-(N-morpholino) ethanesulfonic acid [MES], or 2-hydroxy-5-sulfobenzoic acid [SSA]). Ion-sensitive microelectrodes were prepared by the deposition of the ion-sensitive membrane solution (Ca²⁺, K⁺, and Cl⁻) directly onto the mediating poly-EDOT (PEDOT), PEDOT–SSA, PEDOT–MES, PEDOT–MOPS, or poly-MPy–MOPS layers.     

Ghost circles in lattice Aubry–Mather theory

Monotone lattice recurrence relations such as the Frenkel–Kontorova lattice, arise in Hamiltonian lattice mechanics, as models for ferromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multi-dimensional counterpart of monotone twist maps.Such recurrence relations often admit a variational structure, so that the solutions $x:{\mathbb{Z}}^d\to \mathbb{R}$ are the stationary points of a formal action function $W(x)$. Given any rotation vector $\omega \in {\mathbb{R}}^d$, classical Aubry–Mather theory establishes the existence of a large collection of solutions of $\mathrm{?}W(x)=0$ of rotation vector ω. For irrational ω, this is the well-known Aubry–Mather set. It consists of global minimizers and it may have gaps.In this paper, we study the parabolic gradient flow $\frac{dx}{dt}=?\mathrm{?}W(x)$ and we will prove that every Aubry–Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called ‘ghost circle’. The existence of these ghost circles is known in dimension $d=1$, for rational rotation vectors and Morse action functions. The main technical result of this paper is therefore a compactness theorem for lattice ghost circles, based on a parabolic Harnack inequality for the gradient flow. This implies the existence of lattice ghost circles of arbitrary rotation vectors and for arbitrary actions.As a consequence, we can give a simple proof of the fact that when an Aubry–Mather set has a gap, then this gap must be filled with minimizers, or contain a non-minimizing solution.     

Polypyrrole-calcion film as a membrane and solid-contact in an indicator electrode for potentiometric titrations

This paper shows the application of conducting polymers (CPs) for constructing potentiometric indicator electrodes. Two types of polypyrrole (PPy)-based calcium sensors are presented, one sensor with PPy-calcion film as the active part and the other sensor with PPy-calcion as a solid-state contact coated with a conventional membrane selective towards calcium ions. It is shown that the PPy-calcion film, due to the complexing properties of calcion ensuring high loading of the film with calcium, is sufficiently selective to be used as the active part or as a mediating layer of the indicator electrode. The electrode, with PPy-calcion film as the active part, was used as the indicator electrode in potentiometric titrations of calcium in mixed solvents, where conventional PVC-based electrode can not be used. For the first time, the practical applicability of PPy-based electrodes in titrations is demonstrated.     

Ascending and descending regions of a discrete Morse function

We present an algorithm which produces a decomposition of a regular cellular complex with a discrete Morse function analogous to the Morse–Smale decomposition of a smooth manifold with respect to a smooth Morse function. The advantage of our algorithm compared to similar existing results is that it works, at least theoretically, in any dimension. Practically, there are dimensional restrictions due to the size of cellular complexes of higher dimensions, though. We prove that the algorithm is correct in the sense that it always produces a decomposition into descending and ascending regions of the critical cells in a finite number of steps, and that, after a finite number of subdivisions, all the regions are topological disks. The efficiency of the algorithm is discussed and its performance on several examples is demonstrated.     

Agglutination of like-charged red blood cells induced by binding of beta2-glycoprotein I to outer cell surface

Plasma protein-mediated attractive interaction between membranes of red blood cells (RBCs) and phospholipid vesicles was studied. It is shown that beta(2)-glycoprotein I (beta(2)-GPI) may induce RBC discocyte-echinocyte-spherocyte shape transformation and subsequent agglutination of RBCs. Based on the observed beta(2)-GPI-induced RBC cell shape transformation it is proposed that the hydrophobic portion of beta(2)-GPI molecule protrudes into the outer lipid layer of the RBC membrane and increases the area of this layer. It is also suggested that the observed agglutination of RBCs is at least partially driven by an attractive force which is of electrostatic origin and depends on the specific molecular shape and internal charge distribution of membrane-bound beta(2)-GPI molecules. The suggested beta(2)-GPI-induced attractive electrostatic interaction between like-charged RBC membrane surfaces is qualitatively explained by using a simple mathematical model within the functional density theory of the electric double layer, where the electrostatic attraction between the positively charged part of the first domains of bound beta(2)-GPI molecules and negatively charged glycocalyx of the adjacent RBC membrane is taken into account.     

Combinatorial solution-phase synthesis of (2S,4S)-4-acylamino-5-oxopyrrolidine-2-carboxamides

Solution-phase combinatorial synthesis of (2S,4S)-4-acylamino-5-oxopyrrolidine-2-carboxamides was studied. First, di-tert-butyl (2S,4S)-4-amino-5-oxopyrrolidine-1,2-dicarboxylate hydrochloride was prepared as the key intermediate in five steps from (S)-pyroglutamic acid. Acylation of the amino group followed by acidolytic deprotection gave (2S,4S)-4-acylamino-5-oxopyrrolidine-2-carboxylic acids, which were then coupled with amines to furnish a library of (2S,4S)-4-acylamino-5-oxopyrrolidine-2-carboxamides. Four coupling reagents, BPC, EEDQ, TBTU, and PFTU, were tested for the amidation reactions in the final step. Amidations with EEDQ and TBTU led to the desired carboxamides. On the other hand, BPC and PFTU were not suited, since diketopiperazines were sometimes obtained instead of the desired carboxamides.