The Regularity of Optimal Irrigation Patterns

A branched structure is observable in draining and irrigation systems, in electric power supply systems, and in natural objects like blood vessels, the river basins or the trees. Recent approaches of these networks derive their branched structure from an energy functional whose essential feature is to favor wide routes. Given a flow s in a river, a road, a tube or a wire, the transportation cost per unit length is supposed in these models to be proportional to s ^α with 0 < α < 1. The aim of this paper is to prove the regularity of paths (rivers, branches,...) when the irrigated measure is the Lebesgue density on a smooth open set and the irrigating measure is a single source. In that case we prove that all branches of optimal irrigation trees satisfy an elliptic equation and that their curvature is a bounded measure. In consequence all branching points in the network have a tangent cone made of a finite number of segments, and all other points have a tangent. An explicit counterexample disproves these regularity properties for non-Lebesgue irrigated measures.     

MR detection of mechanical vibrations using a radiofrequency field gradient

A new method for NMR characterization of mechanical waves, based upon radiofrequency field gradient for motion encoding, is proposed. A binomial B1 gradient excitation scheme was used to visualize the mobile spins undergoing a periodic transverse mechanical excitation. A simple model was designed to simulate the NMR signal as a function of the wave frequency excitation and the periodicity of the NMR pulse sequence. The preliminary results were obtained on a gel phantom at low vibration frequencies (0-200 Hz) by using a ladder-shaped coil generating a nearly constant RF field gradient along a specific known direction. For very small displacements and/or B1 gradients, the NMR signal measured on a gel phantom was proportional to the vibration amplitude and the pulse sequence was shown to be selective with respect to the vibration frequency. A good estimation of the direction of vibrations was obtained by varying the angle between the motion direction and the B1 gradient. The method and its use in parallel to more conventional MR elastography techniques are discussed. The presented approach might be of interest for noninvasive investigation of elastic properties of soft tissues and other materials.     

Resin dynamics contributes to the NMR line broadening of organic molecules grafted onto a polystyrene resin

Despite the use of high resolution magic angle spinning NMR, the NMR linewidth of anchored molecules on the commonly used Merrifield solid phase resins remains larger than that of the corresponding molecules in solution. We investigate the different mechanisms that might be at the origin of this line broadening. Experimentally, we use the CPMG method to determine the (15)N relaxation times of a tethered tripeptide and show that the slow resin dynamics significantly contributes to the transverse relaxation.     

The Atiyah-Singer theorems: A probabilistic approach. II. The Lefschetz fixed point formulas

In this paper the Lefschetz fixed point formulas of M. F. Atiyah and I. M. Singer (Ann. of Math.87 (1968), 484–530, 546–604) for classical elliptic complexes are proved by using probabilistic methods.     

Practical aspects of shimming a high resolution magic angle spinning probe

High resolution magic angle spinning (HRMAS) has become an extremely versatile tool to study heterogeneous systems. HRMAS relies on magic angle spinning of the sample to average out to zero magnetic susceptibility differences in the sample and to obtain resonance linewidths approaching those of liquid state NMR. Shimming such samples therefore becomes an important issue. By analyzing the different sources of magnetic field perturbations present in a sample under MAS conditions, we propose a simple protocol to obtain optimum shim settings in HRMAS. In the case of aqueous samples, we show that the lock level cannot be used as a reliable indicator of the quality of the shims at high spinning speeds. This effect is explained by the presence of temperature gradients imparted by the sample rotation.     

Stepwise and concerted electron-transfer/bond breaking reactions. solvent control of the existence of unstable pi ion radicals and of the activation barriers of their heterolytic cleavage

Available data from various sources seem to indicate an important role of solvation in the cleavage rates of intermediate pi ion radicals, in the passage from concerted to stepwise electron-transfer/bond breaking reaction pathways and even in the very existence of pi ion radicals. After preliminary computations treating the solvent as dielectric continuum, these expectations are examined with the help of a simple model system involving the anion radical of ONCH(2)Cl and two molecules of water, which allows the application of advanced computational techniques and a treatment of these solvent effects that emphasizes the role of solvent molecules that sit close to the charge centers of the molecule. A pi ion radical minimum indeed appears upon introduction of the two water molecules, and cleavage is accompanied by their displacement toward the leaving anion, thus offering a qualitative mimicry of the experimental observations.     

Photoinduction of surface-relief gratings during all-optical poling of polymer films

Using the all-optical poling method, we observed the formation of a surface-relief grating (SRG) in an amorphous polymer film containing azo dye molecules in side chain positions. The experiment consists of a seeding-type process. We used a recently described experimental setup that permits a periodic nonlinear pattern to be produced by the index dispersion of glass. The particular configuration permits photoinduced translation diffusion of the azo chromophores to be observed as the origin of the SRG formation. Analyses of the gratings recorded by use of s (TE) and p (TM) polarization of the writing beams are conducted by atomic-force microscopy. The effect is attributed to mass transport from regions of high isomerization activity to regions of lower activity.     

Activation barriers in the homolytic cleavage of radicals and ion radicals

As revealed by several experimental examples, radicals and ion radicals may, in contrast with closed-shell molecules, undergo exothermic homolytic cleavages (.A..B --> A: +.B) with substantial activation barriers. A two-state semiclassical model is proposed for explaining the existence of the barrier and estimating its magnitude. It is based on the intersection of the potential energy surfaces characterizing the dissociation of a bonding state, .A..B -->.A. +.B, on one hand, and the approach to bonding distance of a repulsive state, A: +.B --> A therefore B, on the other. After inclusion of the bond cleavage and formation as Morse curves in the normal-mode analysis, a simple activation driving force relationship is obtained, the two main ingredients of the intrinsic barrier being the triplet excitation energy of the A moiety and the pi*--> sigma* excitation energy in .A-B. The model is then tested by quantum chemical calculations, first on a simplified system to evaluate the calculation techniques and then on a real system. A comparison of the model predictions with experiment is finally performed using the rate data recently gathered for the cleavage of 4-cyanophenyl alkyl ether anion radicals, which cover a respectable range of driving forces, showing satisfactory agreement between theoretical predictions and experimental data.