Insights on the Synthesis of N-Heterocycles Containing Macrocycles and Their Complexion and Biological Properties

Macrocyclic chemistry has been extensively developed over the past several decades. In fact, the architecture of new macrocyclic models has undergone exponential growth to offer molecules with specific properties. In this context, an attempt is made in this study to provide an overview of some synthetic methods allowing the elaboration of N-heterocycles containing macrocycles (imidazole, triazole, tetrazole, and pyrazole), as well as their applications in the complexation of metal cations or as pharmacological agents.     

The pyridyl-tag strategy applied to the hydrocarbon/perfluorocarbon phase-switching of a porphyrin and a fullerene

A new hydrocarbon/perfluorocarbon phase-switching strategy based on coordination of pyridyl-tagged molecules to a highly fluorinated dicopper-carboxylate complex possessing two accessible axial coordination sites is described. When a chloroform solution of the tetrapyridyl-substituted porphyrin 3 (0.1 mM, 2 mL) is layered on a perfluorodecalin solution of 2 (3.25 mM, 1.5 mL), complete extraction of the porphyrin into the fluorous phase is observed after 30 min of stirring. Quantitative release of both the porphyrin and 2 is achieved simply by adding excess THF to the biphasic system, the THF acting as a pyridine competing ligand. The recovered perfluorocarbon solution containing 2 can be reused for another complexation with the same efficiency. The scope of this approach is emphasized by the phase-switching of a dipyridyl-substituted fullerene, another example of a molecule for which solubilization in perfluorocarbons is very challenging.     

A Dielectric Chirped Layered Mirror for Optical Telecommunication Wavelengths

We describe in this paper a design procedure for omnidirectional high reflectors with wide bandwidths for optical telecommunication bands. From the numerical results achieved by the transfer matrix method, it is found that a partial omnidirectional high-reflector covering the optical telecommunication wavelengths 1.3 and 1.55 μm is obtained for a partially deformed quarter-wave stack air/H(LH)¹⁵/glass by using the tellurium as a material of high refractive index (n _H = 4.6). The partially deformed stack is a periodic system deformed by a power law y = x ^1+k. Here y is the coordinates of the deformed object, x is the coordinates of the periodic stack and k is the coefficient defining the deformation degree. The study of this system leads to an omnidirectional high reflector band covering the optical telecommunication wavelengths 1.3 and 1.55 μm for k = 0.2.     

An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit

In this article, we propose a numerical scheme to solve the Kac model of the Boltzmann equation for multiscale rarefied gas dynamics. Formally, this scheme is shown to be uniformly stable with respect to the Knudsen number, consistent with the fluid-diffusion limit for small Knudsen numbers, and with the Kac equation in the kinetic regime. Our approach is based on the micro–macro decomposition which leads to an equivalent formulation of the Kac model that couples a kinetic equation with macroscopic ones. This method is validated with various test cases and compared to other standard methods.     

A generalizaton of Hardy’s uncertainty principle on compact extensions of \mathbb R ^n

Let \(K\) be a compact subgroup of automorphisms of \(\mathbb R ^n\) . We prove in this paper a generalization of Hardy’s uncertainty principle on the semi-direct product \(K\ltimes \mathbb R ^n\) , and we solve the sharpness problem. As a consequence, a complete analogue of classical Hardy’s theorem is obtained. The representation theory and the Plancherel formula play an important role in the proofs.     

Photophysics and kinetics of naphthopyran derivatives,part 1: General analytical solutions for the kinetics of AB(k,ϕ) and ABC(k,ϕ) systems

A new formalism allowing solution of the differential equations for kinetic systems comprising three species (ABC) linked by first‐order thermal and/or photochemical reactions is proposed. The approach is developed specifically with photochromic materials in mind, but the formalism developed is shown to be applicable to a variety of kinetic systems irradiated with monochromatic light where the photokinetic factor ( F ) is constant. © 2005 Wiley Periodicals, Inc. Int J Chem Kinet 37: 162–174, 2005     

H2 Convergence of Solutions of a Biharmonic Problem on a Truncated Convex Sector Near the Angle π

We consider a biharmonic problem Δ²u_ω = f_ω with Navier type boundary conditions u_ω = Δu_ω = 0, on a family of truncated sectors Ω_ω in ?² of radius r, 0 < r < 1 and opening angle ω, ω ∈ (2π/3, π] when ω is close to π. The family of right-hand sides (f_ω)_{ω∈(2π/3, π]} is assumed to depend smoothly on ω in L²(Ω_ω). The main result is that u_ω converges to u_π when ω → π with respect to the H²-norm. We can also show that the H²-topology is optimal for such a convergence result.

     

One-dimensional stochastic heat equation with discontinuous conductance

We introduce a new stochastic partial differential equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space–time white noise. Such equation could be used in mathematical modeling of diffusion phenomena in medium consisting of two kinds of materials and undergoing stochastic perturbations. We prove the existence of the solution and we present explicit expressions of its covariance and variance functions. Some regularity properties of the solution sample paths are also analyzed.     

General decay for a porous thermoelastic system with a memory

In this paper, we consider a linear damped porous thermoelastic system of memory type where the heat conduction is given by Cattaneo’s Law . We establish a general decay results using an appropriate Lyapunov functional.