A localized Jarník–Besicovitch theorem

Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form {x∈R:δx=δ}, where δ?1 and δx is the Diophantine approximation exponent of an irrational number x. We go beyond the classical results by computing the Hausdorff dimension of the sets {x∈R:δx=f(x)}, where f is a continuous function. Our theorem applies to the study of the approximation exponents by various approximation families. It also applies to functions f which are continuous outside a set of prescribed Hausdorff dimension.     

The Kauffman skein algebra of a surface at $${\sqrt{-1}}$$

We study the structure of the Kauffman algebra of a surface with parameter equal to \({\sqrt{-1}}\). We obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat SU(2)-connections over the surface. We analyse the asymptotics of traces of curve-operators in TQFT in non standard regimes where the root of unity parametrizing the TQFT accumulates to a root of unity. We interpret the case of \({\sqrt{-1}}\) in terms of parallel transport operators.     

New insights in quantum chemical topology studies using numerical grid-based analyses

New insights in Quantum Chemical Topology of one-electron density functions have been proposed here by using a recent grid-based algorithm (Tang et al., J Phys Condens Matter 2009, 21, 084204), initially designed for the decomposition of the electron density. Beyond the charge analysis, we show that this algorithm is suitable for different scalar functions showing a more complex topology, that is, the Laplacian of the electron density, the electron localization function (ELF), and the molecular electrostatic potential (MEP). This algorithm makes use of a robust methodology enabling to numerically assign the data points of three-dimensional grids to basin volumes, and it has the advantage of requiring only the values of the scalar function without details on the wave function used to build the grid. Our implementation is briefly outlined (program named TopChem), its capabilities are examined, and technical aspects in terms of CPU requirement and accuracy of the results are discussed. Illustrative examples for individual molecules and crystalline solids obtained with gaussian and plane-wave-based density functional theory calculations are presented. Special attention was given to the MEP because its topological analysis is complex and scarce.     

Double-hybrid density-functional theory made rigorous

We provide a rigorous derivation of a class of double-hybrid approximations, combining Hartree-Fock exchange and second-order M?ller-Plesset correlation with a semilocal exchange-correlation density functional. These double-hybrid approximations contain only one empirical parameter and use a density-scaled correlation energy functional. Neglecting density scaling leads to a one-parameter version of the standard double-hybrid approximations. We assess the performance of these double-hybrid schemes on representative test sets of atomization energies and reaction barrier heights, and we compare to other hybrid approximations, including range-separated hybrids. Our best one-parameter double-hybrid approximation, called 1DH-BLYP, roughly reproduces the two parameters of the standard B2-PLYP or B2GP-PLYP double-hybrid approximations, which shows that these methods are not only empirically close to an optimum for general chemical applications but are also theoretically supported.     

Synthesis of peptide alkylthioesters using the intramolecular N,S-acyl shift properties of bis(2-sulfanylethyl)amido peptides

The design of novel methods giving access to peptide alkylthioesters, the key building blocks for protein synthesis using Native Chemical Ligation, is an important area of research. Bis(2-sulfanylethyl)amido peptides (SEA peptides) 1 equilibrate in aqueous solution with S-2-(2-mercaptoethylamino)ethyl thioester peptides 2 through an N,S-acyl shift mechanism. HPLC was used to study the rate of equilibration for different C-terminal amino acids and the position of equilibrium as a function of pH. We show also that thioester form 2 can participate efficiently in a thiol-thioester exchange reaction with 5% aqueous 3-mercaptopropionic acid. The highest reaction rate was obtained at pH 4. These experimental conditions are significantly less acidic than those reported in the past for related systems. The method was validated with the synthesis of a 24-mer peptide thioester. Consequently, SEA peptides 1 constitute a powerful platform for access to native chemical ligation methodologies.     

Towards an accurate description of anharmonic infrared spectra in solution within the polarizable continuum model: reaction field, cavity field and nonequilibrium effects

We present a newly developed and implemented methodology to perturbatively evaluate anharmonic vibrational frequencies and infrared (IR) intensities of solvated systems described by means of the polarizable continuum model (PCM). The essential aspects of the theoretical model and of the implementation are described and some numerical tests are shown, with special emphasis towards the evaluation of IR intensities, for which the quality of the present method is compared to other methodologies widely used in the literature. Proper account of an incomplete solvation regime in the treatment of the molecular vibration is also considered, as well as inclusion of the coupling between the solvent and the probing field (cavity field effects). In order to assess the quality of our approach, comparison with experimental findings is reported for selected cases.     

Photoinduced proton-coupled electron transfers in biorelevant phenolic systems

Proton-coupled electron transfer (PCET) reactions have received much attention over the past 10 years, from an experimental as well as from a theoretical point of view. At the heart of many chemical and biological processes, such reactions are of particular interest in energy conversion and enzymatic processes. Among the numerous examples of PCET reactions, photosynthesis and particularly reactions inside the Photosystem II (PSII) subunit, involving a global four electrons and four protons process to perform water oxidation and respiration, is the most emblematic one. This review focuses on the photochemical approaches of PCET reactions involving phenolic molecules. Indeed, a significant part of photochemical PCET studies were conducted on tyrosine or phenol relevant to PSII and charge transport in enzymes. The mechanisms of these reactions, sequential or concerted, with particular emphasis on the influence of pH, temperature, solvent nature and H-bonding pattern are presented based on photochemical techniques and related theoretical analysis.     

Mathematical analysis of a scalar multidimensional conservation law with discontinuous flux

We deal in this paper with a scalar conservation law, set in a bounded multidimensional domain, and such that the convective term is discontinuous with respect to the space variable. First, we introduce a weak entropy formulation for the homogeneous Dirichlet problem associated with the first-order reaction-convection equation that we consider. Then, we establish an existence and uniqueness property for the weak entropy solution. The method of doubling variables and a pointwise reasoning along the curve of discontinuity are used to state uniqueness. Finally, the vanishing viscosity method allows us to prove the existence result. Another method to obtain the existence of a solution, which relies on the regularization of the flux, is also detailled, at least for a particular case.     

Solving coloring, minimum clique cover and kernel problems on arc intersection graphs of directed paths on a tree

Let T = (V, A) be a directed tree. Given a collection P{\mathcal{P}} of dipaths on T, we can look at the arc-intersection graph I(P,T){I(\mathcal{P},T)} whose vertex set is P{\mathcal{P}} and where two vertices are connected by an edge if the corresponding dipaths share a common arc. Monma and Wei, who started their study in a seminal paper on intersection graphs of paths on a tree, called them DE graphs (for directed edge path graphs) and proved that they are perfect. DE graphs find one of their applications in the context of optical networks. For instance, assigning wavelengths to set of dipaths in a directed tree network consists in finding a proper coloring of the arc-intersection graph. In the present paper, we give