Multipoint laser Doppler vibrometry using holographic optical elements and a CMOS digital camera

A laser Doppler vibrometer (LDV) is described in which holographic optical elements are used to provide the interferometer reference and object illumination beams. A complementary metal-oxide semiconductor camera, incorporating a digital signal processor, is used to carry out real-time signal processing of the interferometer output to allow multipoint LDV to be implemented.     

Comments on generalized Heron polynomials and Robbins’ conjectures

Heron’s formula for a triangle gives a polynomial for the square of its area in terms of the lengths of its three sides. There is a very similar formula, due to Brahmagupta, for the area of a cyclic quadrilateral in terms of the lengths of its four sides. (A polygon is cyclic if its vertices lie on a circle.) In both cases if A is the area of the polygon, (4A)² is a polynomial function of the square in the lengths of its edges. David Robbins in [D.P. Robbins, Areas of polygons inscribed in a circle, Discrete Comput. Geom. 12 (2) (1994) 223-236. MR 95g:51027; David P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly 102 (6) (1995) 523-530. MR 96k:51024] showed that for any cyclic polygon with n edges, (4A)² satisfies a polynomial whose coefficients are themselves polynomials in the edge lengths, and he calculated this polynomial for n=5 and n=6. He conjectured the degree of this polynomial for all n, and recently Igor Pak and Maksym Fedorchuk [Maksym Fedorchuk, Igor Pak, Rigidity and polynomial invariants of convex polytopes, Duke Math. J. 129 (2) (2005) 371-404. MR 2006f:52015] have shown that this conjecture of Robbins is true. Robbins also conjectured that his polynomial is monic, and that was shown in [V.V. Varfolomeev, Inscribed polygons and Heron polynomials (Russian. Russian summary), Mat. Sb. 194 (3) (2003) 3-24. MR 2004d:51014]. A short independent proof will be shown here.     

Structural consequences of the one-electron reduction of d4 [Mo(CO)2(eta-PhC[triple bond]CPh)Tp']+ and the electronic structure of the d5 radicals [M(CO)L(eta-MeC[triple bond]CMe)Tp'] {L = CO and P(OCH2)3CEt}

Reduction of [M(CO)2(eta-RC[triple bond]CR')Tp']X {Tp' = hydrotris(3,5-dimethylpyrazolyl)borate, M = Mo, X = [PF6]-, R = R' = Ph, C6H4OMe-4 or Me; R = Ph, R' = H; M = W, X = [BF4]-, R = R' = Ph or Me; R = Ph, R' = H} with [Co(eta-C5H5)2] gave paramagnetic [M(CO)2(eta-RC[triple bond]CR')Tp'], characterised by IR and ESR spectroscopy. X-Ray structural studies on the redox pair [Mo(CO)2(eta-PhC[triple bond]CPh)Tp'] and [Mo(CO)2(eta-PhC[triple bond]CPh)Tp'][PF6] showed that oxidation is accompanied by a lengthening of the C[triple bond]C bond and shortening of the Mo-C(alkyne) bonds, consistent with removal of an electron from an orbital antibonding with respect to the Mo-alkyne bond, and with conversion of the alkyne from a three- to a four-electron donor. Reduction of [Mo(CO)(NCMe)(eta-MeC[triple bond]CMe)Tp'][PF6] with [Co(eta-C5H5)2] in CH2Cl2 gives [MoCl(CO)(eta-MeC[triple bond]CMe)Tp'], via nitrile substitution in [Mo(CO)(NCMe)(eta-MeC[triple bond]CMe)Tp'], whereas a similar reaction with [M(CO){P(OCH2)3CEt}(eta-MeC[triple bond]CMe)Tp']+ (M = Mo or W) gives the phosphite-containing radicals [M(CO){P(OCH2)3CEt}(eta-MeC[triple bond]CMe)Tp']. ESR spectroscopic studies and DFT calculations on [M(CO)L(eta-MeC[triple bond]CMe)Tp'] {M = Mo or W, L = CO or P(OCH2)3CEt} show the SOMO of the neutral d5 species (the LUMO of the d4 cations) to be largely d(yz) in character although much more delocalised in the W complexes. Non-coincidence effects between the g and metal hyperfine matrices in the Mo spectra indicate hybridisation of the metal d-orbitals in the SOMO, consistent with a rotation of the coordinated alkyne about the M-C2 axis.     

High-spin structure of yrast-band in 78Kr

Lifetime of levels up to 22⁺, have been measured in ⁷⁸Kr and an oblate shape is assigned to the ground state using the CSM and the configuration dependent shell correction calculations. Calculations further show that ⁷⁸Kr is highly γ-soft nucleus. The experimental Q _t values coupled with theoretical calculations indicate an oblate shape for ⁷⁸Kr at low spins and triaxial shape at higher spins     

Numerical analysis of four-wave mixing between picosecond mode-locked laser pulses in a tensile-strained bulk SOA

A numerical model of four-wave mixing between picosecond pulses in a tensile-strained bulk semiconductor optical amplifier is presented. The model utilizes a modified Schrödinger equation to model pulse propagation. The Schrödinger equation parameters such as the material gain first and second order dispersion, linewidth enhancement factors and optical loss coefficient were obtained using a previously developed steady-state model. The predicted four-wave mixing pulse characteristics show reasonably good agreement with experimental pulse characteristics obtained using Frequency Resolved Optical Gating. In particular simulations predict a large increase in the pulse pedestals of the FWM converted pulse, which is verified by experiment.     

Blowing Up Polygonal Linkages

{Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewise-differentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the well-studied carpenters rule conjecture.     

Bi- and poly-metallic cyanide-bridged complexes of the redox-active cyanomanganese nitrosyl unit [Mn(CN)(PR3)(NO)(eta-C5H4Me)

Cationic nitrile complexes and neutral halide and cyanide complexes, with the general formula [MnL1L2(NO)(eta-C5H4Me)]z, undergo one-electron oxidation at a Pt electrode in CH2Cl2. Linear plots of oxidation potential, Eo', vs. nu(NO) or the Lever parameters, EL, for L1 and L2, allow Eo' to be estimated for unknown analogues. In the presence of TlPF6, [MnIL'(NO)(eta-C5H4Me)] reacts with [Mn(CN)L(NO)(eta-C5H4Me)] to give [(eta5-C5H4Me)(ON)LMn(mu-CN)MnL'(NO)(eta5-C5H4Me)][PF6] which undergoes two reversible one-electron oxidations; DeltaE, the difference between the potentials for the two processes, differs significantly for stable cyanide-bridged linkage isomers. Novel pentametallic complexes such as [Mn[(mu-NC)Mn(CNBut)(NO)(eta5-C5H4Me)]4(OEt2)][PF6]2 and [Mn[(mu-NC)Mn(CNXyl)(NO)(eta5-C5H4Me)]4(NO3-O,O')][PF6], containing a trigonal bipyramidal and a distorted octahedral Mn(II) centre, respectively, result either from slow decomposition of the binuclear cyanide-bridged species or from the reaction of anhydrous MnI2 with four equivalents of [Mn(CN)L(NO)(eta5-C5H4Me)] in the presence of TlPF6.