We consider a two-dimensional (2-D) model of an autophoretic particle. Beyond a certain emission/absorption rate (characterized by a dimensionless Péclet number, $Pe$) the particle is known to undergo a bifurcation from a non-motile to a motile state, with different trajectories, going from a straight to a chaotic motion by increasing $Pe$. From the full model, we derive a reduced closed model which involves only two time-dependent complex amplitudes $C_1(t)$ and $C_2(t)$ corresponding to the first two Fourier modes of the solute concentration field. It consists of two coupled nonlinear ordinary differential equations for $C_1$ and $C_2$ and presents several advantages: (i) the straight and circular motions can be handled fully analytically; (ii) complex motions such as chaos can be analysed numerically very efficiently in comparison with the numerically expensive full model involving partial differential equations; (iii) the reduced model has a universal form dictated only by symmetries (meaning that the form of the equations does not depend on a given phoretic model); (iv) the model can be extended to higher Fourier modes. The derivation method is exemplified for a 2-D model, for simplicity, but can easily be extended to three dimensions, not only for the presently selected phoretic model, but also for any model in which chemical activity triggers locomotion. A typical example can be found, for example, in the field of cell motility involving acto-myosin kinetics. This strategy offers an interesting way to cope with swimmers on the basis of ordinary differential equations, allowing for analytical tractability and efficient numerical treatment.

We report on our recent progress in the study of single Dibenzoterrylene (DBT) molecules as single photon sources and nanoscale probes. We consider DBT molecules embedded in thin anthracene films, a system that allows stable single photon emission both at room and at cryogenic temperatures. We investigate the most important optical properties of the DBT:anthracene system as a whole. We then perform a full statistical study of the coupling between single DBT molecules by measuring the lifetimes of DBT both in the coupled and in the uncoupled case. The experimental results are framed into a simple universal scaling model, where the magnitude of coupling depends solely on universal parameters and on the distance d between the single emitter and the graphene monolayer. We apply this model to infer d and provide a proof of principle for a position ruler at the nanoscale [1].

We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings $R$ and $S$ , if $R$ and $S$ are Morita equivalent, then so are the quasi-Baer right ring hulls ${{\widehat{\text{Q}}}_{\mathfrak{q}}}\mathfrak{B}(R)$ and ${{\widehat{\text{Q}}}_{\mathfrak{q}}}\mathfrak{B}(S)$ of $R$ and $S$ , respectively. As an application, we prove that if unital ${{C}^{*}}$ -algebras $A$ and $B$ are Morita equivalent as rings, then the bounded central closure of $A$ and that of B are strongly Morita equivalent as ${{C}^{*}}$ -algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring $A[G]$ of a torsion-free Abelian group $G$ over a commutative semiprime quasi-continuous ring $A$ . Examples that illustrate and delimit the results of this paper are provided.

It is shown that every finitely generated projective module PR over a semiprime ring R has the smallest FI-extending essential module extension (called the absolute FI-extending hull of PR) in a fixed injective hull of PR. This module hull is explicitly described. It is proved that , where is the smallest right FI-extending right ring of quotients of End(PR) (in a fixed maximal right ring of quotients of End(PR). Moreover, we show that a finitely generated projective module PR over a semiprime ring R is FI-extending if and only if it is a quasi-Baer module and if and only if End(PR) is a quasi-Baer ring. An application of this result to C*-algebras is considered. Various examples which illustrate and delimit the results of this paper are provided.

The surface antigens of Trypanosoma cruzi trypomastigotes were identified by immunoprecipitation and were compared with metabolically labelled excretory—secretory products (ES) released by the parasites in vitro. A series of major immunogenic components in the ES antigens were revealed (160 kDa, 130 kDa and 80–110 kDa). The trypomastigote surface also bears the 130 kDa band and the 80–110 kDa complex. Competition experiments demonstrated the common antigenic structure of the ES and the surface antigens. Two-dimensional analysis of ES antigens immunoprecipitated by human Chagasic serum revealed several spots in the 80–110 kDa region with a wide range of isoelectric points (PI between 5·4 and 6·7). This reflects a charge heterogeneity of these polypeptides. The trypomastigote 85 kDa polypeptide was also identified in the ES antigens by using a monoclonal antibody against this antigen. Two-dimensional analysis of the 85 kDa proteins shed from the surface of trypomastigotes and immunoprecipitated by the monoclonal antibody 155D3 showed 2 major spots: a major part of the 85 kDa polypeptide was found at pH 6·5–6·6, whereas a substantial amount of the antigen was found at pH 5·7. An additional component with molecular weight of approximately 58 kDa and isoelectric points of 6·5 and 6·6, was also visualized. Detection of the 85 kDa polypeptide circulating in serum from patients with acute and chronic Chagas' disease was achieved using an enzyme-linked immunosorbent assay. In addition, the data obtained showed that a polyclonal antibody to the 85 kDa polypeptide could be used to passively induce a partial protection of Fischer rats against acute lethal infection. Thus, the antigens recognized by polyclonal antibody appear to play a role in the development of protective immunity against T. cruzi.

The notation in this paper will be standard and it may be found in [3], for example. In particular, the notation A ⊂′ B stands for the statement “A is an essential submodule of B”. As is customary, we say that a ring R is a Goldie ring when R is both left and right Goldie. Similarly, a ring is noetherian if and only if it is both right and left noetherian, etc.