Light‐Controlled Regioselective Synthesis of Fullerene Bis‐Adducts

Multi‐functionalization and isomer‐purity of fullerenes are crucial tasks for the development of their chemistry in various fields. In both current main approaches—tether‐directed covalent functionalization and supramolecular masks—the control of regioselectivity requires multi‐step synthetic procedures to prepare the desired tether or mask. Herein, we describe light‐responsive tethers, containing an azobenzene photoswitch and two malonate groups, in the double cyclopropanation of [60]fullerene. The formation of the bis‐adducts and their spectroscopic and photochemical properties, as well as the effect of azobenzene photoswitching on the regiochemistry of the bis‐addition, have been studied. The behavior of the tethers depends on the geometry of the connection between the photoactive core and the malonate moieties. One tether lead to a strikingly different adduct distribution for the E and Z isomers, indicating that the covalent bis‐functionalization of C₆₀ can be controlled by light.     

Redox Properties and in Vivo Magnetic Resonance Imaging of Cyclodextrin-Polynitroxides Contrast Agents

This paper reports the synthesis, characterization and in vivo application of water-soluble supramolecular contrast agents (Mw: 5–5.6 kDa) for MRI obtained from β-cyclodextrin functionalized with different kinds of nitroxide radicals, both with piperidine structure ( CD2 and CD3 ) and with pyrrolidine structure ( CD4 and CD5 ). As to the stability of the radicals in presence of ascorbic acid, CD4 and CD5 have low second order kinetic constants (≤0.05 M⁻¹ s⁻¹) compared to CD2 (3.5 M⁻¹ s⁻¹) and CD3 (0.73 M⁻¹ s⁻¹). Relaxivity (r₁) measurements on compounds CD3 - CD5 were carried out at different magnetic field strength (0.7, 3, 7 and 9.4 T). At 0.7 T, r₁ values comprised between 1.5 mM⁻¹ s⁻¹ and 1.9 mM⁻¹ s⁻¹ were found while a significant reduction was observed at higher fields (r₁≈0.6-0.9 mM⁻¹ s⁻¹ at 9.4 T). Tests in vitro on HEK293 human embryonic kidney cells, L929 mouse fibroblasts and U87 glioblastoma cells indicated that all compounds were non-cytotoxic at concentrations below 1 μmol mL⁻¹. MRI in vivo was carried out at 9.4 T on glioma-bearing rats using the compounds CD3 - CD5 . The experiments showed a good lowering of T₁ relaxation in tumor with a retention of the contrast for at least 60 mins confirming improved stability also in vivo conditions.     

Representation Formulae for Solutions to Some Classes of Higher Order Systems and Related Liouville Theorems

Let m ≥ 1 be an integer and N > 2m. Let μ be a positive Radon measure on . We study necessary and sufficient conditions on possible distributional solutions of , that guarantee the validity of the representation formula a.e. on , where and c(2m) is a positive constant depending on m and N. Several consequences are derived. In particular we prove Liouville theorems for systems of higher order elliptic inequalities and weighted form of Hardy-Littlewood-Sobolev systems of integral equations. Received: March 2008     

Macroscopic models for fibroproliferative disorders: A review

In this paper we review some mathematical modelling of organ reparative processes (wound healing) for both the physiological and pathological case. The natural process of healing consists in a series of overlapping phases involving cells, chemicals, extracellular matrix (ECM) and the environment surrounding the wound site. Sometimes the healing process fails and the reparative mechanism produces pathological conditions which are commonly termed fibrosis or fibroproliferative disorders. Biological insight into the pathogenesis, progression and possible regression of fibrosis is lacking and many issues are still open. Mathematical modelling can surely play its part in this field and this paper is aimed at showing what has been done so far and what has still to be done to achieve a unified framework for studying these kinds of problems. Due to the high complexity of this phenomenon, multi-scale modelling is certainly the appropriate approach that should be used for studying these kinds of problems. Unfortunately most of the mathematical literature on this topic consists of macroscopic continuous models which fail to investigate processes occurring at smaller length scales (cellular, sub-cellular). We present a review of some of the mathematical literature, showing the widely used approaches, focusing on the interpretation of results and indicating possible developments in the study of these highly complex systems.     

Asymptotic behavior for the Navier-Stokes equations with nonzero external forces

We estimate the asymptotic behavior for the Stokes solutions, with external forces first. We found that if there are external forces, then the energy decays slowly even if the forces decay quickly. Then, we also obtain the asymptotic behavior in the temporal-spatial direction for weak solutions of the Navier-Stokes equations. We also provide a simple example of external forces which shows that the Stokes solution does not decay quickly.     

Optimal Equilibria in the Non-Cooperative Game Associated with Cost Spanning Tree Problems

We study the Pareto optimal equilibria payoffs of the non-cooperative game associated with the cost spanning tree problem. We give two characterisations of these payoffs: one based on the tree they induce and another based on the strategies played by agents. Moreover, an algorithm for computing all these payoffs is provided.     

Enumerating and decoding perfect linear Lee codes

Using group theory approach, we determine all numbers q for which there exists a linear 1-error correcting perfect Lee code of block length n over Z _q , and then we enumerate those codes. At the same time this approach allows us to design a linear time decoding algorithm.      

Asymptotic gauges: Generalization of Colombeau type algebras

We use the general notion of set of indices to construct algebras of nonlinear generalized functions of Colombeau type. They are formally defined in the same way as the special Colombeau algebra, but based on more general “growth condition” formalized by the notion of asymptotic gauge. This generalization includes the special, full and nonstandard analysis based Colombeau type algebras in a unique framework. We compare Colombeau algebras generated by asymptotic gauges with other analogous construction, and we study systematically their properties, with particular attention to the existence and definition of embeddings of distributions. We finally prove that, in our framework, for every linear homogeneous ODE with generalized coefficients there exists a minimal Colombeau algebra generated by asymptotic gauges in which the ODE can be uniquely solved. This marks a main difference with the Colombeau special algebra, where only linear homogeneous ODEs satisfying some restrictions on the coefficients can be solved.     

Laguerre geometry of surfaces with plane lines of curvature

We study surfaces with plane lines of curvature in the framework of Laguerre geometry and provide explicit representation formulae for these surfaces in terms of a potential function. As an application, we explicitly integrate allL- minimal surfaces with plane curvature lines. Partially supported by MURST 40.