Characterizing Valiant's algebraic complexity classes

Valiant introduced 20 years ago an algebraic complexity theory to study the complexity of polynomial families. The basic computation model used is the arithmetic circuit, which makes these classes very easy to define and open to combinatorial techniques. In this paper we gather known results and new techniques under a unifying theme, namely the restrictions imposed upon the gates of the circuit, building a hierarchy from formulas to circuits. As a consequence we get simpler proofs for known results such as the equality of the classes VNP and _VNPe${\mathrm{VNP}}_{\mathrm{e}}$ or the completeness of the Determinant for VQP, and new results such as a characterization of the classes VQP and VP (which we can also apply to the Boolean class LOGCFL) or a full answer to a conjecture in Bürgisser's book [Completeness and reduction in algebraic complexity theory, Algorithms and Computation in Mathematics, vol. 7, Springer, Berlin, 2000]. We also show that for circuits of polynomial depth and unbounded size these models all have the same expressive power and can be used to characterize a uniform version of VNP.     

Functionalized analogues of Tröger's base: scope and limitations of a general synthetic procedure and facile, predictable method for the separation of enantiomers

A major stumbling block in the applications of enantiomerically pure Tröger's base analogues is their poor availability. We have therefore developed a facile method for the enantioseparation of functionalized Tröger's base analogues possessing various substitution patterns. The systematic separation of a library comprising 36 representatives on the commercially available Whelk O1 chiral stationary phase provided valuable information on structure-enantioselectivity relationships. A mechanistic explanation of observed relationships allows one to predict whether or not enantioseparation of a given, perhaps yet unknown derivative of Tröger's base will be feasible. In addition, we provide a detailed report on the scope and limitations of the general synthetic protocol employing anilines and paraformaldehyde in CF₃COOH, as well as some considerations concerning the mechanism of formation of Tröger's base analogues.     

Trapped-ion cell with improved dc potential harmonicity for FT-ICR MS

The trapped-ion cell is a key component critical for optimal performance in Fourier transform ion cyclotron resonance (FT-ICR) mass spectrometry (MS). To extend the performance of FT-ICR MS, we have developed a new cell design that is capable of generating a DC trapping potential which closely approaches that of an ideal Penning trap, i.e., a 3D axial quadrupolar potential distribution. The new cell design was built upon an open cylindrical geometry, supplemented with two pairs of cylindrical compensation segments. Electric potential calculations for trial cell geometries were aimed at minimizing spatial variations of the radial electric field divided by radius. The resulting cell proportions and compensation voltages delivered practically constant effective ion cyclotron frequency that was independent of ion radial and axial positions. Our customized 12 tesla FT-ICR instrument was upgraded with the new cell, and the performance was characterized for a range of ion excitation power and ion populations. Operating the compensated cell at increased postexcitation radii, approximately 0.7 of the cell inner radius, resulted in improved mass measurement accuracy together with increased signal intensity. Under these same operating conditions the noncompensated open cell configuration exhibited peak splitting and reduced signal life time. Mass accuracy tests using 11 calibrants covering a wide m/z range reproducibly produced under 0.05 ppm RMS precision of the internal calibration for reduced ion populations and the optimal excitation radius. Conditions of increased ion population resulted in a twofold improvement in mass accuracy compared with the noncompensated cell, due to the larger achievable excitation radii and correspondingly lower space charge related perturbations of the calibration law.     

On the complexity of the multivariate resultant

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n$n$ homogeneous equations in n$n$ variables is satisfiable (the resultant is a polynomial in the system’s coefficients which vanishes if and only if the system is satisfiable). In this paper, we investigate the complexity of computing the multivariate resultant.     

Molten salt synthesis of a mixed-valent lanthanide(III/IV) oxychloride with an unprecedented Sillen X(2)(4) structure: Ce1.3Nd0.7O3Cl

A new cerium neodymium oxychloride, Ce1.3Nd0.7O3Cl, has been synthesized by precipitation in a LiCl–CaCl2 molten salt by humid argon sparging. Chemical and structure characterization have been undertaken by powder X-ray diffraction, scanning electron microscopy, high-temperature X-ray diffraction, thermogravimetric analysis, and X-ray photoelectron scattering. This oxychloride crystallizes in space group P4/nmm, a = 3.9848(3) ? and c = 12.467(2) ?, in a new Sillen-type phase represented by the symbol X(2)(4) where “quadruple” fluorite-type layers [M4O6], containing Ce(IV) in “inner” sublayers and both CeIII and NdIII in “outer” sublayers, alternate with double-halide ion sheets. The structure is also described as a stacking of LnOCl and fluorite-type blocks and constitutes the term n = 2 of a possible series (MO2)n(NdOCl)2.     

Characterization of photoproducts of m-THPP in aqueous solution

This study examined the nature of photoproducts after pulse laser irradiation (647.5 nm) of 5,10,15,20-tetrakis(meso-hydroxyphenyl)porphyrin (m-THPP) (10 micromol/L) in ethanol-water (1/99, vol/vol) solution. Spectroscopic measurements (UV-visible absorption and fluorescence) and mass spectrometry techniques (matrix-assisted laser desorption-ionization [MALDI] coupled with time-of-flight mass spectrometer [TOF-MS] or tandem time of flight mass spectrometer [TOF/TOF-MS]) were used to follow photomodifications. Spectroscopic measurements evidenced photomodification as the main process after m-THPP irradiation. Three oxidized photoproducts at m/z 693.25, 695.24 and 713.25 were characterized by MS. After prolonged irradiation new isotopic distributions were registered at m/z 1355.41, 2031.57, 2707.80 and 3383.98 with MALDI-TOF-MS and TOF/TOF-MS. These new photoproducts were attributed to covalent oligomeric structures as dimer, trimer, tetramer and pentamer of m-THPP.     

MALDI-TOF mass spectrometric analysis for the characterization of the 5,10,15,20-tetrakis- (m-hydroxyphenyl)bacteriochlorin (m-THPBC) photoproducts in biological environment

Photoproducts formation upon irradiation (739 nm) of 5,10,15,20-tetrakis(m-hydroxyphenyl)bacteriochlorin (m-THPBC) in phosphate buffer saline (PBS) supplemented with human serum albumin (HSA) were studied by means of absorption spectroscopy and MALDI-TOF mass spectrometry. The experiments were performed with a freshly prepared PBS-HSA solution of m-THPBC and with a PBS-HSA m-THPBC solution incubated for 6 h at 37 degrees C. The incubation of m-THPBC solution leads to the dye monomerisation, whereas in the freshly prepared solution, m-THPBC is under an aggregated form. Regardless of the incubation condition, photobleaching experiments carried out by absorption spectroscopy demonstrate the degradation of the photosensitizer and its phototransformation in m-THPC. Moreover, m-THPC was the sole photoproduct detected using absorption spectroscopy. Together with a degradation of m-THPBC and formation of m-THPC, MALDI-TOF mass spectrometry evidenced several other photoinduced modifications. Photoproducts such as dihydroxy m-THPBC and dihydroxy m-THPC were detected in both conditions; however, the formation of hydroxylated photoproducts was significantly greater in incubated solution. In addition, small molecules arising from the degradation of the photosensitizer and identified as dipyrin derivatives and dipyrrolic synthon were observed.