Hölder continuity of local minimizers of vectorial integral functionals

We study the regularity of vector-valued local minimizers in $ W^{1,p}, p > 1 $, of the integral functional where is an open set in $ \mathbb{R}^N $ and f is a continuous function, convex with respect to the last variable, such that $ 0 \leq f(x,u,t)\leq C(1+t^p) $.We prove that if f = f(x, t), or f = f(x, u, t) and $ p \leq N $, then local minimizers are locally Hölder continuous for any exponent less than 1. If f = f(x, u, t) and p < N then local minimizers are Höolder continuous for every exponent less than 1 in an open set $ \Omega_0 $ such that the Hausdorff dimension of $ \Omega \backslash \Omega_0 $ is less than N–p.AMS Subject Classification: 49N60.     

Hoeffding decompositions for exchangeable sequences and chaotic representation of functionals of Dirichlet processes

Consider an exchangeable sequence $\text{X}\text{={X}{}_{\mathrm{n}}\text{:}\text{1?n<N}}$, where $\text{N∈}\text{N}\text{∪{∞}}$, and note $\text{X}{}_{\mathrm{n}}\text{=(X}{}_1\text{,…,X}{}_{\mathrm{n}}\text{)}$. We say that $\text{X}$ is Hoeffding decomposable if, for each n, every square integrable, centered and symmetric functional of $\text{X}{}_{\mathrm{n}}$ is the orthogonal sum of n U-statistics with degenerated and symmetric kernels. We state a necessary and sufficient condition for an exchangeable sequence to be Hoeffding decomposable, named weak independence. We show that a class of weakly independent sequences is given by generalized urn sequences and, specifically, by generalized Pólya urns. We point out that this yields an orthogonal decomposition of the space of square integrable functionals of Dirichlet–Ferguson processes into orthogonal subspaces of multiple integrals. Explicit formulae are provided. To cite this article: G. Peccati, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Toggling enantioselective catalysis--a promising paradigm in the development of more efficient and versatile enantioselective synthetic methodologies

The increasingly needed synthesis of both enantiomers of a chiral compound usually requires the use of both enantiomers of a chiral catalyst. Several of the usually employed chiral ligands are naturally available in only one enantiomeric form, the antipode often being of labor-intensive preparation. Enantiodivergent asymmetric catalysis has accrued in importance in this regard, in that it allows expeditious access to both enantiomers of a product without any direct modification on the chemical structure of the chiral promoter. Various promising examples will be discussed throughout the review. If available or envisageable, a mechanistic rationale for the observed enantioinversion will be outlined.     

Chemometric investigation of the efficiency of different TiO<Subscript>2</Subscript>-based catalysts as principal components of TOC photochemical sensors under development

There is an increasing demand for measurement of organic carbon in solutions both in industry and in environmental research for the purpose of continuous water-quality monitoring. Practically all the methods used are based on a catalytic system in which metal oxides play a major role. The development of a new TOC/DOC measurement system, in our work, uses a mixed anatase-rutile form of TiO(2 )as catalyst of the mineralisation process, and direct measurement of the CO(2) produced, by a gaseous diffusion electrode. The entire research is based on heterogeneous catalysis using an immobilised catalyst, which can offer considerable advantages over other methods of catalysis. Four different catalytic systems were analysed. Three involved thermal immobilisation of TiO(2) on glass supports (glass spheres and glassy particulates) or on an Al metal grid. One further system consisted of direct oxidation of a Ti grid at high temperatures. The system was illuminated using a 350 nm UV source (350 mW cm(-2)) contained in a home-made measurement cell. Five molecules-malic acid, pentachlorophenol, sodium dodecylsulfate, hydroquinone, and citric acid-which were deemed to be representative and had been investigated in several previous studies, were used as photodegradation targets. However, it was not possible to obtain a direct expression of "catalytic efficiency" simply by reading the data obtained, nor any indirect molecular "recalcitrancy" scale. Chemometric analysis, by principal components analysis, allows the five used catalytic systems to be easily compared each other and a single PC component is able to perform classification.     

On the structure of singular sets of convex functions

Whenf is a convex function ofR ^h, andk is an integer with 0<k, then the set ^k (f)=x:dim(f(x)k may be covered by countably many manifolds of dimensionh–k and classC ² except an _h–k negligible subset.The author is supported by INdAM     

A study of the conformation of some 3,3′-bithienyl and biselenienyl derivatives by measurements of twisting powers in nematic liquid crystals and of dipole moments

The conformation of chiral 2,2′-dinitro-3,3′-bithienyl-4,4′-dicarboxylic acid (Ia), its dimethylester (Ib), and analogous selenium derivatives Ic and Id was investigated by means of induced cholesteric mesophases. A parallel investigation on the diester (Ib) and on other model 3,3′-bithienyls was carried out with dipole moment measurements. The conclusions reached with the two independent methods for derivative Ib are in excellent agreement and indicate, if a single conformation is assumed, a preferred cisoid conformation with a dihedral angle somewhat smaller than 90°. Alternatively, if the existence of both cisoid and transoid conformations having supplementary dihedral angles is assumed, the data indicate a higher population for the cisoid form.     

A convenient and versatile synthesis of 4-trifluoromethyl-substituted pyrazoles

Trifluoromethyl-substituted pyrazoles were synthesized via 1,3-dipolar cycloaddition from sydnones 2a-j and 1-aryl-3,3,3-trifluoro-propynes 5a-f. The regioselectivity of the reaction and the spectroscopic characteristics of the products are discussed.     

Quadrupole contributions to lattice vibration frequencies of benzene and naphthalene

Calculations have been made of the lattice vibration frequencies of benzene and naphthalene, in which quadrupole interactions have been added to the usual atom—atom terms. A significant improvement between theory and experiment is obtained: in particular the ordering of the highest A_g and B_g frequencies of naphthalene is given correctly for the first time.     

Variable strategy toward carbasugars and relatives. 5.(1) Focus on preparation of chiral nonracemic medium-sized carbocycles

The feasibility of sequential vinylogous aldol (intermolecular)/silylative aldol (intramolecular) addition reactions involving furan- and pyrrole-based dienoxysilanes, 6 and 12, in the synthesis of carbasugar frameworks is illustrated by the preparation of the scantily investigated carbaseptanose and carbaoctanose representatives of this class of compounds. The target compounds, 1, 2, 3, ent-2, ent-3, and 4, were obtained from readily available carbohydrate precursors (5 and 19) in yields of 21-30% over 8-12 steps. The irreversible silylative ring-closing aldolisation of gamma-substituted dihydro-5H-furan-2-one and pyrrolidin-2-one aldehydes (9, 16, ent-16, and 22) driven by the TBSOTf/Pr(i)(2)EtN Lewis acid-Lewis base couple was shown to be a practical, diastereoselective maneuver to forge the densely functionalized, medium-sized core carbocycles.     

Detailed analysis and group-type separation of natural fats and oils using comprehensive two-dimensional gas chromatography

Comprehensive gas chromatography (GC x GC) is an adequate methodology for the separation and identification of very complex samples. It is based on the coupling of two capillary columns that each give a different but substantial contribution to the unprecedented resolving power of this technique. The 2D space chromatograms that derive from GC x GC analysis have great potential for identification. This is due to the fact that the contour plot positions, pinpointed by two retention time coordinates, give characteristic patterns for specific families of compounds that can be mathematically translated. This investigation concerned the application of this principle to fatty acid methyl esters that were grouped on an equal double bond number basis. The ester samples were derived from various lipids and all underwent bidimensional analysis on two sets of columns. Peak attribution was supported by mass spectra, linear retention indices and information reported in the literature.