Measuring the interactions among variables of functions over the unit hypercube

By considering a least squares approximation of a given square integrable function f:n[0,1]→R by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of f. This definition extends the concept of Banzhaf interaction index introduced in cooperative game theory. Our approach is partly inspired from multilinear regression analysis, where interactions among the independent variables are taken into consideration. We show that this interaction index has appealing properties which naturally generalize several properties of the Banzhaf interaction index. In particular, we interpret this index as an expected value of the difference quotients of f or, under certain natural conditions on f, as an expected value of the derivatives of f. Finally, we discuss a few applications of the interaction index in aggregation function theory.     

Remarques sur les m\'ethodes de projection pour l'approximation des \'equations de Navier--Stokes

Summary. L'objet de cet article est de montrer que les estimations de convergence sur la pression pour les m\'ethodes de projection d\'ecrites dans \cite{Shen1} et \cite{Shen2} ne sont pas obtenues correctement car elles sont toutes bas\'ees sur une in\'egalit\'e fausse. Il semble qu'on ait besoin d'une convergence en de la vitesse dans pour d\'emontrer les estimations de convergence sur la pression en . La question de savoir si la m\'ethode de projection a un taux de convergence pour la pression plus \'elev\'e que reste ouverte. Received June 1, 1993     

Sur le theoreme de Stone-Weierstrass en Algebre commutative

We show that the sufficient conditions given by Cahen, Grazzini and Haouat for a version of the Stone-Weierstrass theorem in commutative algebra are the widest. More precisely, letA be a Noetherian ring andI a proper ideal ofA such thatA is Hausdorff with respect to theI-adic topology. Note the completion ofA andC(Â,Â) the ring of continuous functions from to with uniform convergence topology. The subset of polynomial functions is dense inC(Â,Â) if and only if the radical ofI is a maximal idealm ofA and the local ringA _m is a one-dimensional analytically irreducible domain with finite residue field.     

The automatic control of boundary-layer transition — Experiments and computation

In recent years there has been an increasing interest in the control of boundary-layer transition through the use of wall suction. In the current work suction is provided through one or more suction panels situated close to the leading edge of a plate. Experiments show that boundary-layer pressure fluctuation measurements can be used to identify the position of transition. Transition can be maintained at a desired location with minimum power consumption by employing an automatic adaptive feedback control loop which regulates the suction flow rates of two independent suction panels. This can be expressed as a constrained optimization problem. To allow the suction flow rates to be updated, a modified least mean squares algorithm is used within the control loop. Experimental measurements show that the control algorithm allows fast and stable convergence towards the optimum suction distribution for a double suction panel configuration. Numerical simulations have also been performed. The two-dimensional boundary layer was calculated allowing the viscous boundary layer to interact with the inviscid outer flow. Following linear stability theory the spatial growth rates are calculated by solving an Orr-Sommerfeld type eigenvalue problem, with the streamwise location of transition predicted via thee ^N -method. Applying the same optimization strategy as in the experiments, good qualitative agreement between computations and experiments was found. The optimization algorithm has been applied to computer models where the relation between suction flow rates and transition location is described by an empirical analytical function. This shows that the controller can in principle be applied to systems with more than two suction panels.Nomenclature b transition location with zero suction - d desired transition location - e(k) error signal - k iteration index - p rms pressure - p _ref reference rms pressure - r sum of the reference pressure - u streamwise velocity - u _e external velocity - inviscid external velocity - A wave amplitude - F( ) cost function - I identity matrix - N maximum amplification factor - P projection matrix - R Reynolds number - Re Reynolds number based on the boundary-layer thickness - R matrix of weights - Tu turbulence level - vector of suction flow rates - v normal velocity - v _wall suction velocity at the surface - x streamwise coordinates - x _m microphone location - x _T(k) measured transition location - y normal coordinate - y(k) sum of the measured pressures - w(k) noise - plate length - r +i _i - free stream velocity - * displacement thickness - gradient vector - Lagrange multiplier - controller gain - disturbance stream function - disturbance amplitude - wave frequency = complex wave number     

Carbon Nano-Flakes Produced by an Inductively Coupled Thermal Plasma System for Catalyst Applications

Carbon material was produced using an inductively coupled thermal plasma torch system of 35 kW and a conical shape reactor. The carbon nanopowders were obtained by plasma decomposition of methane at various flow rates and show a uniform microstructure throughout the reactor. The product has a crystalline graphitic structure, with a stacking of between 6 and 16 planes and a nano-flake morphology with particles dimensions of approximately 100 nm long, 50 nm wide and 5 nm thick. Nitrogen was also introduced in some synthesis experiments along with the methane precursor using flow rates of 0.1 and 0.2 slpm. The resulting product has the same structural properties and the nitrogen is incorporated into the graphitic structure through pyridinic type bonds.     

Design of efficient ambipolar host materials for organic blue electrophosphorescence: theoretical characterization of hosts based on carbazole derivatives

Density functional theory calculations were carried out to investigate the electronic structures of representative ambipolar hosts for blue electroluminescence, based on two carbazole end groups and meta-terphenyl (mTP)-like bridges. The bridge molecular segments include mTP, 2,6-bisphenylpyridine, 3,5-bisphenylpyridine, and 2,6-bisphenylpyrimidine. While the ionization potentials and electron affinities of these molecules are mainly determined by their hole- and electron-transport subunits, respectively, each subunit impacts the electronic properties of the other upon their binding, mainly in an inductive way. Importantly, the lowest triplet state of the hosts is determined to be confined into the mTP-like bridges since these are the subunits with lowest individual triplet energy. Extension of the phenyl-based π-conjugated system via meta linkages is found to be effective in modulating the electron affinity value while maintaining a high triplet energy.     

Nonribosomal peptide synthesis in animals: the cyclodipeptide synthase of Nematostella

Cyclodipeptide synthases (CDPSs) are small enzymes structurally related to class-I aminoacyl-tRNA synthetases (aaRSs). They divert aminoacylated tRNAs from their canonical role in ribosomal protein synthesis, for cyclodipeptide formation. All the CDPSs experimentally characterized to date are?bacterial. We show here that a predicted CDPS from the sea anemone Nematostella vectensis is an active CDPS catalyzing the formation of various cyclodipeptides, preferentially containing tryptophan. Our findings demonstrate that eukaryotes encode active CDPSs and suggest that all CDPSs have?a similar aminoacyl-tRNA synthetase-like architecture and ping-pong mechanism. They also raise questions about the biological roles of the cyclodipeptides produced in bacteria and eukaryotes.     

Water behavior in mesoporous materials as studied by NMR relaxometry

Water in mesoporous materials possessing a two-dimensional hexagonal structure has been studied by the variation of its NMR longitudinal relaxation time T(1) as a function of the static magnetic field value, or equivalently of the NMR measurement frequency. This technique, dubbed relaxometry, has been applied from 5 kHz (measurement frequency) up to 400 MHz with various instruments including a variable-field spectrometer operating between 8 and 90 MHz. Moreover, the range 0-5 kHz could be investigated by transverse relaxation, T(2) denoting the corresponding relaxation time, and relaxation in the rotating frame, T(1ρ) denoting the corresponding relaxation time. Measurements of proton relaxation rates (inverse of relaxation times) have been performed with H(2)O and HOD (residual protons of heavy water) at water volumes of 80%, 60%, and 40% relative to the porous volume. Comparison between H(2)O and HOD shows clearly that, above 1 MHz where both sets of data are superposed, relaxation is purely intermolecular and due to paramagnetic relaxation (dipolar interactions of water protons with unpaired electrons of paramagnetic entities). Below 1 MHz, it is possible to subtract the intermolecular contribution (given by HOD data) from H(2)O data so that one is left with intramolecular relaxation which is solely due to water reorientational motions. The analysis of these low-frequency data (in terms of Lorentzian functions) reveals two types of water within the pores: one interacting strongly with the surface and the other corresponding to a second layer. High-frequency data, which arise from paramagnetic relaxation, exhibit again two types of water. Due to their correlation times, one type is assigned to relatively free water within the pores while the other type corresponds to bulk (interparticular) water. Their proportions, given as a function of the volume fraction, are consistent with the above assignments.