A noncommutative version of the matrix inversion formula

The cofactor expression (? 1)^i+j$\text{det}\text{(I ? B)}{}_{\mathrm{<mtext>j</mtext><mtext>?</mtext><mtext>i</mtext><mtext>?</mtext>}}\text{det}\text{(I ? B)}$ for the (i, j)-entry in the inverse of a matrix (I ? B) is proved to be equal to the corresponding entry of the series Σ_m≥0B^m, by using purely combinatorial methods: circuit monoid techniques and monomial rearrangements. Moreover, the identity is shown to hold in a non-commutative formal power series algebra.     

Studies on the titanium-catalyzed cyclopropanation of nitriles

The Ti-mediated reaction of Grignard reagents with nitriles was investigated with sub-stoichiometric amounts of titanium isopropoxide. Cyanoesters were converted to spirocyclopropanelactams in good yields using as low as 0.05 eq of Ti(O(i)Pr)4. Under similar conditions, cyanocarbonates led to spirocyclopropane oxazolidinones and/or aminocyclopropylcarbinols. A very short synthesis of the naturally occurring aminocyclopropanecarboxylic acid illustrates the usefulness of this methodology.     

A new approach to A,B-difunctionalisation of cyclodextrins using bulky 1,3-bis[bis(aryl)chloromethyl]benzenes as capping reagents

1,3-Bis[bis(4-tert-butylphenyl)chloromethyl]benzene and 1,3-bis[bis(4-anisyl)chloromethyl]benzene were employed as regioselective capping reagents for the preparation of C-6A,C-6B-bridged, permethylated alpha- and beta-CD derivatives; isolated yields up to 55% of proximally capped, methylated CDs were obtained, thus opening the way to the straightforward preparation of a wide range of A,B-functionalised CDs. As revealed by a single crystal X-ray diffraction study, the benzene-1,3-bis[bis(4-tert-butylphenyl)methyl] spacer is perfectly suited for A,B-capping of beta-cyclodextrin.     

Response of bare 1-3 piezocomposite array to localized electrical excitation

The theoretical response of a 1-3 piezocomposite plate submitted to localized electrical excitation was studied with the theory of guided waves. The theoretical modeling was based on the global matrix method, and the piezocomposite material was considered as a homogeneous medium. To validate the theoretical results, experimental displacement measurements were performed with an interferometric probe on two piezocomposite plates, one with a single element and one with an array of electrodes. The measured response on the single-element plate was mainly supported by the S0 and S3 modes of the plate. Homogenization limits of the composite in terms of frequency and wave number are defined on the basis of data from this sample. Within these limits, the piezocomposite material operates as a homogeneous medium, and comparison between theoretical and experimental results allows the equivalent electroacoustic parameters to be evaluated. A second sample was measured to study the effects of the electrode array on the electroacoustic response of the plate. Two kinds of electrical excitation were studied.     

Coherent combining in an Yb-doped double-core fiber laser

Coherent combining is demonstrated in a clad-pumped Yb-doped double-core fiber laser. A slope efficiency of more than 70% is achieved with 96% of the total output power in the fundamental mode of one of the two cores. This high combining efficiency is obtained when both cores are coupled via a biconical fused taper in a Michelson interferometer configuration.     

Easy preparation of the tris(2-fluoro-6-pyridylmethyl)amine ligand and instantaneous reaction of the corresponding dichloroferrous complex with molecular dioxygen: new access to dinuclear species

The tris(2-fluoro-6-pyridylmethyl)amine ligand, F3TPA, can easily be prepared by reaction of 2-fluoro-6-bromomethylpyridine with NH4Cl in the presence of NaOH. Complexation to FeCl2 affords the high-spin F3TPAFe(II)Cl2 complex, the X-ray structure of which is reported. The three fluorine substituents provide enough steric hindrance to force the tripod to coordinate in the tridentate mode, affording a trigonal bipyramidal iron center. This complex is thermally stable, and it reacts instantaneously with molecular dioxygen to afford the unsymmetrical micro-oxo dimer F3TPAFe(III)ClOFe(III)Cl3 as the major product, together with small amounts of the mixed salt [F3TPAFe(II)Cl]2, [Fe(III)2OCl6]. These two complexes have been isolated and characterized by X-ray diffraction analysis. A mechanism by which they are obtained is suggested and seems to parallel the well-known process of autoxidation of ferrous porphyrins.     

Tail Behavior of a Threshold Autoregressive Stochastic Volatility Model

We consider a threshold autoregressive stochastic volatility model where the driving noises are sequences of iid regularly random variables. We prove that both the right and the left tails of the marginal distribution of the log-volatility process (α_t)_t are regularly varying with tail exponent −α with α > 0. We also determine the exact values of the coefficients in the tail behaviour of the process (α_t)_t. AMS 2000 Subject Classification. Primary—62G32, 62PO5     

Nonlinear synthesis of input signals in ultrasonic experimental setups

A method is proposed to find the digital input signals to enter in nonlinear acoustical systems (power amplifiers, transducers, etc.) in order to obtain desired arbitrary response signals. The searched input signals are found by performing a Monte Carlo search guided by a simulated annealing process applied to a hidden model with a small number of parameters. The physical system is actually used in the optimization procedure, in a real-time manner, so that no theoretical model of the system response is required. The main aspects of the algorithm are described and illustrated with several examples.     

Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces

It is well known that the complexity of the Delaunay triangulation of $n$ points in $\RR ^d$, i.e., the number of its simplices, can be $\Omega (n^{\lceil {d}/{2}\rceil })$. In particular, in $\RR ^3$, the number of tetrahedra can be quadratic. Put another way, if the points are uniformly distributed in a cube or a ball, the expected complexity of the Delaunay triangulation is only linear. The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface. In this paper we bound the complexity of the Delaunay triangulation of points distributed on the boundary of a given polyhedron. Under a mild uniform sampling condition, we provide deterministic asymptotic bounds on the complexity of the three-dimensional Delaunay triangulation of the points when the sampling density increases. More precisely, we show that the complexity is $O(n^{1.8})$ for general polyhedral surfaces and $O(n\sqrt{n})$ for convex polyhedral surfaces. Our proof uses a geometric result of independent interest that states that the medial axis of a surface is well approximated by a subset of the Voronoi vertices of the sample points.