Recognizing ℙn in Classical and Modern Setting

In classical projective algebraic geometry, ?ⁿ was seen mainly as a linear subspace. The modern setting has produced in the last 40 years several remarkable abstract characterizations of projective space. We survey some interaction between these two points of view.     

On a Theorem of Van de Ven

A result by Van de Ven characterizes linear subspaces as the only closed submanifolds X ? ? ^N for which the normal bundle exact sequence splits. We show that X is linear assuming only the splitting of the same exact sequence when restricted to some curve contained in X .     

Optical manipulation of complex molecular systems by high density green photons: experimental and theoretical evidence

The recent revolution in modern optical techniques revealed that light interaction with matter generates a force, known as optical force, which produces material properties known in physics as optical matter. The basic technique of the domain uses forces exerted by a strongly focused beam of light to trap small objects and subsequently to manipulate their local structures. The purpose of this paper is to develop an alternative approach, using irradiations with high-density-green-photons, which induce electric dipoles by polarization effects. The materials used for the experiments were long carbon chains which represent the framework of biological macromolecules. The physical techniques used to reveal the locally induced molecular arrangements were: dynamic viscosity, zeta potential, chemiluminescence, liquid chromatography; mass spectrometry, and Raman and infrared spectroscopy. The principal result of our experiments was the detection of different molecular arrangements within the mixture of alkane chains, generated by our optical manipulations. This induced “optical matter” displayed two material properties: antioxidant effects and large molecular aggregation effects. In order to bring the experimental results in relation with theory, we developed a physical model and the interacting force between polarizable bodies was computed. By numerical calculations stable structures for N = 6 and N = 8 particles were obtained.     

Functional expansions for finding traveling wave solutions

The paper proposes a generalized analytic approach which allows to find traveling wave solutions for some nonlinear PDEs. The solutions are expressed as functional expansions of the known solutions of an auxiliary equation. The proposed formalism integrates classical approaches as tanh method or $G^{\prime }/G$ method, but it open the possibility of generating more complex solutions. A general class of second order PDEs is analyzed from the perspective of this formalism, and clear rules related to the balancing procedure are formulated. The KdV equation is used as a toy model to prove how the results obtained before through the $G^{\prime }/G$ approach can be recovered and extended, in an unified and very natural way.     

Mauldin-Williams graphs, Morita equivalence and isomorphisms

We describe a method for associating some non-self-adjoint algebras to Mauldin-Williams graphs and we study the Morita equivalence and isomorphism of these algebras.

We also investigate the relationship between the Morita equivalence and isomorphism class of the -correspondences associated with Mauldin-Williams graphs and the dynamical properties of the Mauldin-Williams graphs.

     

Hawking’s Local Rigidity Theorem Without Analyticity

We prove the existence of a Hawking Killing vector-field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth vacuum Einstein manifold. The result extends a previous result of Friedrich, Rácz and Wald, see [FRW, Prop.B.1], which was limited to the domain of dependence of the bifurcate horizon. So far, the existence of a Killing vector-field in a full neighborhood has been proved only under the restrictive assumption of analyticity of the space-time. Using this result we provide the first unconditional proof that a stationary black-hole solution must possess an additional, rotational Killing field in an open neighborhood of the event horizon. This work is accompanied by a second paper, where we prove a uniqueness result for smooth stationary black-hole solutions which are close (in a very precise, geometric sense) to the Kerr family of solutions, for arbitrary 0 < a < m.     

Electropolymerizable IrIII Complexes with β‐Ketoiminate Ancillary Ligands

A series of electropolymerizable cyclometallated Ir^III complexes were synthesized and their electrochemical and photophysical properties studied. The triphenylamine electropolymerizable fragment was introduced by using triphenylamine‐2‐phenylpyridine and, respectively, triphenylamine‐benzothiazole as cyclometalated ligands. The coordination sphere was completed by two differently substituted β‐ketoiminate ligands deriving from the condensation of acetylacetone or hexafluoroacetylacetone with para‐bromoaniline. The influence of the ‐CH₃/‐CF₃ substitution to the electrochemical and photophysical properties was investigated. Both complexes with CH₃ substituted β‐ketoiminate were emissive in solution and in solid state. Highly stable films were electrodeposited onto ITO coated glass substrates. Their emission was quenched by electron trapping within the polymeric network as proven by electrochemical studies. The ‐CF₃ substitution of the β‐ketoiminate leads instead to the quenching of the emission and inhibits electropolymerization.     

Schwarz method for earthquake source dynamics

Dynamic faulting under slip-dependent friction in a linear elastic domain (in-plane and 3D configurations) is considered. The use of an implicit time-stepping scheme (Newmark method) allows much larger values of the time step than the critical CFL time step, and higher accuracy to handle the non-smoothness of the interface constitutive law (slip weakening friction).The finite element form of the quasi-variational inequality is solved by a Schwarz domain decomposition method, by separating the inner nodes of the domain from the nodes on the fault. In this way, the quasi-variational inequality splits into two subproblems. The first one is a large linear system of equations, and its unknowns are related to the mesh nodes of the first subdomain (i.e. lying inside the domain). The unknowns of the second subproblem are the degrees of freedom of the mesh nodes of the second subdomain (i.e. lying on the domain boundary where the conditions of contact and friction are imposed). This nonlinear subproblem is solved by the same Schwarz algorithm, leading to some local nonlinear subproblems of a very small size.Numerical experiments are performed to illustrate convergence in time and space, instability capturing, energy dissipation and the influence of normal stress variations. We have used the proposed numerical method to compute source dynamics phenomena on complex and realistic 2D fault models (branched fault systems).