Non-explosion en temps grand et stabilité de solutions globales des équations de Navier–StokesNon-blowup at large times and stability for global solutions to the Navier–Stokes equations

Suppose there exists a global solution u to the incompressible Navier–Stokes equations, such that $\text{u∈}\text{C}{}_{\mathrm{t}}\text{(}\text{H}\text{?}{}^{\mathrm{1/2}}\text{)}$. We prove that its $\text{H}\text{?}{}^{\mathrm{1/2}}$ norm goes to 0 at infinity. We next use this fact to control the $\text{L}{}^2{}_{\mathrm{t}}\text{(}\text{H}\text{?}{}^{\mathrm{3/2}}\text{)}$ norm of u, and finally we prove that such a solution is stable. To cite this article: I. Gallagher et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 289–292.     

The geometry of Hamilton spaces - An introduction

A short introduction of Hamilton, Cartan and Generalized Hamilton spaces is provided. The Hamilton--Jacobi equations are deduced by variational calculus.     

Measure functions for frames

This paper addresses the natural question: “How should frames be compared?” We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a function which maps each frame to a continuous function. The comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set. We define the ultrafilter measure function, an explicit frame measure function that we show is contained both algebraically and topologically inside all frame measure functions. We explore additional properties of frame measure functions, showing that they are additive on a large class of supersets—those that come from so called non-expansive frames. We apply our results to the Gabor setting, computing the frame measure function of Gabor frames and establishing a new result about supersets of Gabor frames.     

The geometry of Ingarden spaces

The Randers spaces RFⁿ were introduced by R. S. Ingarden. They are considered as Finsler spaces Fⁿ = (M, α + β) equipped with the Cartan nonlinear connection. In the present paper we define and study what we call the Ingarden spaces, I Fⁿ, as Finsler spaces I Fⁿ = (M, α + β) equipped with the Lorentz nonlinear connection. The spaces R Fⁿ and I Fⁿ are completely different. For I Fⁿ we discuss: the variational problem, Lorentz nonlinear connection, canonical N-metrical connection and its structure equations, the Cartan 1-form ω, the electromagnetic 2-form tF and the almost symplectic 2-form 0. The formula dω = F+θ is established. It has as a consequence the generalized Maxwell equations. Finally, the almost Hermitian model of I Fⁿ is constructed.     

Hydrolysis-condensation of tetraethoxysilane in nonparental solvents studied by GC-MS

Gas-chromatography coupled with mass spectrometry was used to investigate the hydrolysis and condensation of tetra-ethoxysilane (TEOS) in nonparental solvents (MeOH and nPrOH). In the presence of nonparental solvents, the reaction that advanced at highest rate is alkoxy group scrambling, that changes the precursor at the molecular level. Subsequent stages of hydrolysis-polycondensation process are essentially determined by the reactivity of the new molecular species formed.     

Steric control over hydrogen bonding in crystalline organic solids: a structural study of N,N'-dialkylthioureas

Hydrogen bonding in crystalline N,N'-dialkylthioureas was examined with the help of single-crystal X-ray diffraction, DFT calculations, and Cambridge Structural Database (CSD) analysis. A CSD survey indicated that unlike the related urea derivatives, which persistently self-assemble into one-dimensional hydrogen-bonded chains, the analogous thioureas can form two different hydrogen-bonding motifs in the solid state: chains, structurally similar with those found in ureas, and dimers, that further associate into hydrogen-bonded layers. The formation of one motif or another can be manipulated by the bulkiness of the organic substituents on the thiourea group, which provides a clear example of steric control over the hydrogen bonding arrangement in crystalline organic solids.     

über eine Klasse Riemannscher Mannigfaltigkeiten mit raumartigen pkf-ZusammenhÄngen

One considers m-dimensional Riemannian manifoldsM with tangent spaces T_p(M), p M that are a direct sum of a spacelike m-2 planeR _p and a 2-planeH _p. It is supposed that onM there exists a connection whose space-like components are parallel conformal flat (pkf). These components are generated by a vector field X. Assuming that X belongs to a pair X,Y of reciprocal quasi-cocircular vector fields and that the Pfaffian of this pair is the 1-form associated with the connection, the following results are derived: 1. X and Y are of equal constant length (This is true for all Riemannian manifolds). 2. The immersion of the integral manifold ofH _p intoM is cylindrical and the normal connection is flat. 3. The immersion of any space-like submanifold intoM is cylindrical with respect to the sections inH _p and umbilical with respect to all spacelike sections. 4. If m 4, the integral manifoldP _p is flat.

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Herrn Prof. Dr. WERNER BURAU zum 70.Geburtstag Klaus Buchner und Radu Rosca     

Influences of changes in multitopic tris(pyrazolyl)methane ligand topology on silver(I) supramolecular structures

The reactions between silver tetrafluoroborate and the ligands 1,2,4,5-C(6)H(2)[CH(2)OCH(2)C(pz)(3)](4) (L1, pz = pyrazolyl ring), o-C(6)H(4)[CH(2)OCH(2)C(pz)(3)](2) (L2), and m-C(6)H(4)[CH(2)OCH(2)C(pz)(3)](2) (L3) yield coordination polymers of the formula (C(6)H(6)(-)(n)[CH(2)OCH(2)C(pz)(3)](n)(AgBF4)(m))( infinity ) (n = 4, m = 2, 1; n = 2, ortho substitution, m = 1, 2; meta substitution, m = 2, 3). In the solid state, L2 molecules dimerize by a pair of C-H.pi interactions, forming an arrangement that resembles the tetratopic ligand L1. In the solid-state structure of 1, each silver atom is kappa(2)-bonded to two tris(pyrazolyl)methane units from different ligands with the overall structure a polymer made up from 32-atom macrocyclic rings formed by bonding tris(pyrazolyl)methane groups from nonadjacent positions on the central arene rings to the same two silver atoms. In 2, each silver is bonded to two tris(pyrazolyl)methane units in the same kappa(2)-kappa(2) fashion as with 1, forming a polymer chain. The chains are organized into dimeric units by strong face-to-face pi-pi stacking between the central arene rings making bitopic L2 act as half of tetratopic L1. The chains in both structures are organized by weak C-H.F hydrogen bonds and pi-pi stacking interactions into very similar 3D supramolecular architectures. The structure of 3 contains three types of silvers with the overall 3D supramolecular sinusoidal structure comprised of 32-atom macrocycles. Infrared studies confirm the importance of the noncovalent interactions. Calculations at the DFT (B3LYP/6-31G) level of theory have been carried out on L2 and also support C-H.pi interactions. Electrospray mass spectral data collected from acetone or acetonitrile show the presence of aggregated species such as [(L)Ag(2)(BF(4))](+) and [(L)Ag(2)](2+), despite the fact that (1)H NMR spectra of all compounds show that acetonitrile completely displaces the ligand whereas acetone does not.     

Dynamical scaling of single chains on adsorbing substrates: diffusion processes

We study the dynamics of tethered chains of length N on adsorbing surfaces, considering the dilute case; for this we use the bond fluctuation model and scaling concepts. In particular, we focus on the mean-square displacement of single monomers and of the center of mass of the chains. The characteristic time tau of the fluctuations of a free chain in a good solvent grows as tau approximately N(a), where the coefficient a obeys a=2nu+1. We show that the same coefficient also holds at the critical point of adsorption. At intermediate time scales single monomers show subdiffusive behavior; this concurs with the behavior calculated from scaling arguments based on the dynamical exponent a. In the adsorbed state tau(perpendicular), the time scale for the relaxation in the direction perpendicular to the surface, becomes independent of N; tau(perpendicular) is then the relaxation time of an adsorption blob. In the direction parallel to the surface the motion is similar to that of a two-dimensional chain and is controlled by a time scale given by tau(parallel) approximately N(2nu(2)+1)L(-2Delta(nu/nu)), where nu(2) is the Flory exponent in two dimensions, nu is the Flory exponent in three dimensions, and Deltanu=nu(2)-nu. For the motion parallel to the surface we find dynamical scaling over a range of about four decades in time.