Conformal two-boundary loop model on the annulus

We study the two-boundary extension of a loop model—corresponding to the dense phase of the O(n)$\mathrm{O}(n)$ model, or to the Q=n²$Q=n^2$ state Potts model—in the critical regime −2<n?2$-2<n?2$. This model is defined on an annulus of aspect ratio τ. Loops touching the left, right, or both rims of the annulus are distinguished by arbitrary (real) weights which moreover depend on whether they wrap the periodic direction. Any value of these weights corresponds to a conformally invariant boundary condition. We obtain the exact seven-parameter partition function in the continuum limit, as a function of τ, by a combination of algebraic and field theoretical arguments. As a specific application we derive some new crossing formulae for percolation clusters.     

FT‐Raman,IR and UV‐visible spectral investigations and ab initio computations of a nonlinear food dye amaranth

Amaranth (E123, Food Red 9, FD & C Red 2) is a sulfonated azo dye used as a color additive in foodstuffs, pharmaceuticals and cosmetics. FT‐IR and FT‐Raman spectra of amaranth were recorded and analyzed. Density functional theory (DFT) calculations were performed to derive the equilibrium geometry, vibrational wavenumbers, intensities and first hyperpolarizability. The results of the optimized molecular structure gave clear evidence for the intramolecular charge transfer (ICT) and intramolecular hydrogen bonding in the molecule. Azo stretching wavenumbers are lowered owing to conjugation and π‐electron delocalization. Time‐dependent density functional theory (TD‐DFT) calculations of the electronic spectra were performed on the optimized structure and compared with the experimental UV‐visible spectrum. Vibrational spectra, natural bonding orbitals (NBO) analysis and optimized geometry indicate C H·N hydrogen bonding in the molecule. The first hyperpolarizability of the molecule was calculated. The optical nonlinearity of the dye is due to the donation of the electron density from the hydroxyl group of the conjugated system via naphthalene ( 2 ) ring into π*‐orbital of the azo moiety. Copyright © 2008 John Wiley & Sons, Ltd.     

NIR‐FT Raman,FT‐IR and surface‐enhanced Raman scattering spectra,with theoretical simulations on chloramphenicol

Chloramphenicol (CLM), originally derived from the bacterium Streptomyces venezuelae, is an inhibitor of bacterial ribosomal peptidyl transferase activity. The near infrared Fourier transform (NIR‐FT) Raman, surface‐enhanced Raman spectroscopy (SERS) and Fourier transform infrared (FT‐IR) spectral analyses of CLM, a potential antibacterial drug for the treatment of typhoid fever, were carried out along with density functional computations. The vibrational spectral analysis reveals that the CH₂ asymmetric and symmetric stretching modes are shifted to higher wavenumbers than the computed values, owing to the electronic effects resulting from induction of methylene group with the adjacent electronegative atom. The lowering of CO stretching wavenumber is due to the presence of the strong electronegative atom, nitrogen, attached to the carbonyl carbon, causing large degree of molecular π‐electron delocalization and redistribution of electrons, which weakens the CO bond. The absence of a C H stretching vibration and the observed C H out‐of‐plane bending modes suggest that the CLM molecule may be adsorbed in a flat orientation with respect to the silver surface. Copyright © 2008 John Wiley & Sons, Ltd.     

Das Lösen realitätsorientierter Aufgaben zu periodischen Vorgängen mit Computeralgebra

There is a new chance in teaching and learning the relationships between mathematics and the real world while using computers and especially computer-algebra-systems in mathematics education. We discuss three real life problems in the area of periodic actions which can be handled with basic knowledge of trigonometry. First we analyse the motion of the twine of a sewing-machine. Geometrical software (Euklid and Cabri) allows the simulation of the motion and Derive gives us the analytical representation of this curve. The second example tries to answer the question how the stroke of an engine can be described mathematically. The last example shows, how a computer-algebra-system (Mathplus) helps to get an analytical expression for the relationship between the time and the monthly found out average values of the air temperature for Munich. On the one hand computer-algebra-systems are an essential help for modelling real life situations, on the other hand the given results allow to understand the situation.     

La surjectivité de l'application moyenne pour les espaces préhomogènes

Let ? be an homogeneous polynomial on $\text{R}$ⁿ. First an analog of the Borel theorem is proved for the distributions which appear at the poles of the distribution |?|^s (s ∈ $\text{C}$). If ? is the relative invariant of an irreducible regular prehomogeneous vector space, the preceding result is used to characterize the functions which are obtained by integration on the fibers of ?.     

Electrochemical Study of Phase-Transfer Catalysis Reactions: The Williamson ether synthesis

The transfer properties of the ionic species involved in the Williamson ether synthesis by phase-transfer catalysis were investigated using electrochemical techniques developed for the study of polarised liquid/liquid interfaces. This approach allows the measurement of the apparent partition coefficients of the transferring species. From these data, it is proposed that the role of the phase-transfer catalyst salt in the reaction mechanism is to establish a Galvani distribution potential difference between the two phases which in turn acts as the driving force for transferring the reactive aqueous ions to the organic phase.     

The Catline for Deep Regression

Motivated by the notion of regression depth (Rousseeuw and Hubert, 1996) we introduce thecatline, a new method for simple linear regression. At any bivariate data setZ_n={(x_i, y_i);i=1, …, n} its regression depth is at leastn/3. This lower bound is attained for data lying on a convex or concave curve, whereas for perfectly linear data the catline attains a depth ofn. We construct anO(n log n) algorithm for the catline, so it can be computed fast in practice. The catline is Fisher-consistent at any linear modely=βx+α+ein which the error distribution satisfies med(e | x)=0, which encompasses skewed and/or heteroscedastic errors. The breakdown value of the catline is 1/3, and its influence function is bounded. At the bivariate gaussian distribution its asymptotic relative efficiency compared to theL¹line is 79.3% for the slope, and 88.9% for the intercept. The finite-sample relative efficiencies are in close agreement with these values. This combination of properties makes the catline an attractive fitting method.