Wide Oscillation Finite Time Blow Up for Solutions to Nonlinear Fourth Order Differential Equations

We give sufficient conditions for local solutions to some fourth order semilinear ordinary differential equations to blow up in finite time with wide oscillations, a phenomenon not visible for lower order equations. The result is then applied to several classes of semilinear partial differential equations in order to characterize the blow up of solutions including, in particular, its applications to a suspension bridge model. We also give numerical results which describe this oscillating blow up and allow us to suggest several open problems and to formulate some related conjectures.     

Effective medium representation and complex modes in 3D periodic metamaterials made of cubic resonators with large permittivity at mid-infrared frequencies

We review some of the techniques that lead to the effective medium representation of a three-dimensional (3D) periodic metamaterial. We consider a 3D lattice of lead telluride cubic resonators at mid-infrared (MIR) frequencies. Each cubic resonator is modeled with both an electric and a magnetic dipole, through a method called the dual dipole approximation. The electric and magnetic polarizabilities of a cubic resonator are computed via full-wave simulations by mapping the resonator's scattered field under electric/magnetic excitation only to the field radiated by an equivalent electric/magnetic dipole. We then analyze the allowed modes in the lattice, with transverse polarization and complex wavenumber, highlighting the attenuation that each mode experiences after one free space wavelength. We observe the presence of two modes with low attenuation constant, dominant in different frequency ranges, able to propagate inside the lattice: this allows the treatment of the metamaterial as a homogeneous material with effective parameters, evaluated by using various techniques. We then show that the metamaterial under analysis allows for the generation of artificial magnetism (i.e., relative effective permeability different than unity, including negative permeability with low losses) at MIR frequencies.     

On the integrability of Degasperis–Procesi equation: Control of the Sobolev norms and Birkhoff resonances

We consider the dispersive Degasperis–Procesi equation $u_t?u_{xxt}?\mathtt{c}{u}_{xxx}+4\mathtt{c}{u}_x?u{u}_{xxx}?3u_x{u}_{xx}+4u{u}_x=0$ with $\mathtt{c}\in \mathbb{R}?\{0\}$. In [15] the authors proved that this equation possesses infinitely many conserved quantities. We prove that there are infinitely many of such constants of motion which control the Sobolev norms and which are analytic in a neighborhood of the origin of the Sobolev space $H^s$ with $s\ge 2$, both on $\mathbb{R}$ and $\mathbb{T}$. By the analysis of these conserved quantities we deduce a result of global well-posedness for solutions with small initial data and we show that, on the circle, the formal Birkhoff normal form of the Degasperis–Procesi at any order is action-preserving.     

Boosting Conjugate Addition to Nitroolefins Using Lithium Tetraorganozincates: Synthetic Strategies and Structural Insights

We report the first transition metal catalyst- and ligand-free conjugate addition of lithium tetraorganozincates (R₄ZnLi₂) to nitroolefins. Displaying enhanced nucleophilicity combined with unique chemoselectivity and functional group tolerance, homoleptic aliphatic and aromatic R₄ZnLi₂ provide access to valuable nitroalkanes in up to 98 % yield under mild conditions (0 °C) and short reaction time (30 min). This is particularly remarkable when employing β-nitroacrylates and β-nitroenones, where despite the presence of other electrophilic groups, selective 1,4 addition to the C=C is preferred. Structural and spectroscopic studies confirmed the formation of tetraorganozincate species in solution, the nature of which has been a long debated issue, and allowed to unveil the key role played by donor additives on the aggregation and structure of these reagents. Thus, while chelating N,N,N’,N’-tetramethylethylenediamine (TMEDA) and (R,R)-N,N,N’,N’-tetramethyl-1,2-diaminocyclohexane (TMCDA) favour the formation of contacted-ion pair zincates, macrocyclic Lewis donor 12-crown-4 triggers an immediate disproportionation process of Et₄ZnLi₂ into equimolar amounts of solvent-separated Et₃ZnLi and EtLi.     

Evaluation of methoxy polyethylene glycol‐polylactide diblock copolymers as additive in hypromellose film coating

This paper deals with a new application of diblock methoxy polyethylene glycol‐polylactide block copolymers, a class of synthetic biomaterials largely studied in the pharmaceutical and biomedical fields owing to their favorable properties such as biocompatibility, biodegradability, low immunogenicity, and good mechanical properties. In this work, these materials were evaluated as additives for gastro‐soluble pharmaceutical coating aimed to reduce film stiffness and water permeability. Two copolymers with different polylactide chain lengths were synthesized and characterized in term of molecular weight and solid‐state properties. A series of free films with different hypromellose/copolymers ratio were prepared and characterized in terms of appearance, components miscibility, plasticity, and water vapor permeability. The obtained results demonstrate that copolymers effectively influence hypromellose film properties according to their concentration and molecular weight. Specifically, the addition of the copolymer with a molecular weight of 6.5 kDa in a ratio hypromellose:polymer 5:1, allowed to obtain films with good appearance, improved plasticization, and water permeability properties. For higher molecular weight, copolymer or different ratios was not possible to observe the improvement of all the properties at the same time. The results also make possible to define the critical features to improve in order to use block copolymers as additive in hypromellose film coating. The availability of new water‐soluble additives able to work as plasticizer and moisture sealer in polymeric films represents an important progress not only in the field of pharmaceutical coating but also in that of food coatings, as for example in the formulation of edible films. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonnegative Matrix Factorization based on the Geometry of Positive Numbers

We consider the problem of nonnegative matrix factorization where the typical objective function is altered based on geometrical arguments. A noneuclidean geometry on positive real numbers is used to describe the nonnegative entries of a nonnegative matrix, influencing the factorization model. We design an optimization procedure from a differential geometric point of view for the newly proposed model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)