Barycentric coordinates for convex sets

In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green’s functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.      

Approximate dynamic programming algorithms for optimal dosage decisions in controlled ovarian hyperstimulation

In the controlled ovarian hyperstimulation (COH) treatment, clinicians monitor the patients’ physiological responses to gonadotropin administration to tradeoff between pregnancy probability and ovarian hyperstimulation syndrome (OHSS). We formulate the dosage control problem in the COH treatment as a stochastic dynamic program and design approximate dynamic programming (ADP) algorithms to overcome the well-known curses of dimensionality in Markov decision processes (MDP). Our numerical experiments indicate that the piecewise linear (PWL) approximation ADP algorithms can obtain policies that are very close to the one obtained by the MDP benchmark with significantly less solution time.     

Mixed-integer bilinear programming problems

This paper addresses a class of problems called mixed-integer bilinear programming problems. These problems are identical to the well known bilinear programming problems with the exception that one set of variables is restricted to be binary valued, and they arise in various production, location—allocation, and distribution application contexts. We first identify some special cases of this problem which are relatively more readily solvable, even though their continuous relaxations are still nonconvex. For the more general case, we employ a linearization technique and design a composite Lagrangian relaxation-implicit enumeration-cutting plane algorithm. Extensive computational experience is provided to test the efficacy of various algorithmic strategies and the effects of problem data on the computational effort of the proposed algorithm.     

Synthesis and Characterization of Sterically Encumbered Derivatives of Aluminum Hydrides and Halides: Assessment of Steric Properties of Bulky Terphenyl Ligands

The synthesis and characterization of several sterically encumbered monoterphenyl derivatives of aluminum halides and aluminum hydrides are described. These compounds are [2,6-Mes(2)C(6)H(3)AlH(3)LiOEt(2)](n)() (1), (Mes = 2,4,6-Me(3)C(6)H(2)-), 2,6-Mes(2)C(6)H(3)AlH(2)OEt(2) (2), [2,6-Mes(2)C(6)H(3)AlH(2)](2) (3), 2,6-Mes(2)C(6)H(3)AlCl(2)OEt(2) (4), [2,6-Mes(2)C(6)H(3)AlCl(3)LiOEt(2)](n)() (5), [2,6-Mes(2)C(6)H(3)AlCl(2)](2) (6), TriphAlBr(2)OEt(2) (7), (Triph = 2,4,6-Ph(3)C(6)H(2)-), [2,6-Trip(2)C(6)H(3)AlH(3)LiOEt(2)](2) (8) (Trip = 2,4,6-i-Pr(3)C(6)H(2)-), 2,6-Trip(2)C(6)H(3)AlH(2)OEt(2) (9), [2,6-Trip(2)C(6)H(3)AlH(2)](2) (10), 2,6-Trip(2)C(6)H(3)AlCl(2)OEt(2) (11), and the partially hydrolyzed derivative [2,6-Trip(2)C(6)H(3)Al(Cl)(0.68)(H)(0.32)(&mgr;-OH)](2).2C(6)H(6) (12). The structures of 2, 3a, 4, 6, 7, 9a, 10a, 10b, 11, and 12 were determined by X-ray crystallography. The structures of 3a, 9a, 10a, and 10b, are related to 3, 9, and 10, respectively, by partial occupation of chloride or hydride by hydroxide. The compounds were also characterized by (1)H, (13)C, (7)Li, and (27)Al NMR and IR spectroscopy. The major conclusions from the experimental data are that a single ortho terphenyl substituent of the kind reported here are not as effective as the ligand Mes (Mes = 2,4,6-t-Bu(3)C(6)H(2)-) in preventing further coordination and/or aggregation involving the aluminum centers. In effect, one terphenyl ligand is not as successful as a Mes substituent in masking the metal through agostic and/or steric effects.     

Synthesis, structure and magnetic behavior of five-coordinate bis(iminopyrrolyl) complexes of cobalt(II) containing PMe3 and THF ligands

The new bis-iminopyrrolyl five-coordinate Co(II) complexes [Co(kappa (2) N, N'-NC 4H 3C(R)N-2,6- (i)Pr 2C 6H 3) 2(PMe 3)] (R = H 3a; Me 3b) were synthesized in high yields (ca. 80-90%), using THF and diethyl ether as solvents, respectively, by (a) treatment of CoCl 2(PMe 3) 2 with the corresponding iminopyrrolyl Na salts ( Ie or If) or (b) reaction of anhydrous CoCl 2 and PMe 3 with Ie or If. A third route was tested, involving the addition of excesses of PMe 3 to the complexes [Co(kappa (2) N, N'-NC 4H 3C(R)N-2,6- (i)Pr 2C 6H 3) 2] (R = H 1e; Me 1f), which was only successful for the synthesis of 3a, in lower yields (ca. 30%). The synthesis of 3b in THF was unfruitful because of the kinetic competition of the solvent, giving rise to mixtures of 1f and its coordinated THF adduct 4b. The synthesis of the new bis-iminopyrrolyl five-coordinate Co(II) complexes [Co(kappa (2) N, N'-NC 4H 3C(R)N-2,6- (i)Pr 2C 6H 3) 2(THF)] (R = H 4a; Me 4b) were carried out in high yields (ca. 80-90%) by the reaction of CoCl 2(THF) 1.5 with the corresponding iminopyrrolyl Na salt. All the compounds have been characterized by X-ray diffraction, with 3a and 3b showing axially compressed trigonal bipyramidal geometry (with the PMe 3 ligand lying on the equatorial plane), whereas complexes 4a and 4b exhibit distorted square pyramidal geometries with the THF molecule occupying the axial position. Complex 4a shows clearly a compressed geometry, but for complex 4b, two polymorphs were characterized, exhibiting molecules with different Co-O (THF) bond lengths, one of them being compatible with an elongated form. Magnetic measurements either in the solid or in the liquid phases indicate that complexes 3a and 3b have low-spin ground states ( S = 1/2). In toluene solution, the geometry is fully confirmed by EPR data, which further indicates a d x (2) - y (2) /d xy ground state. However, compounds 4a and 4b behave unusually because they show magnetic moments that are compatible with high-spin ground states ( S = 3/2) in the solid state, but conform to low-spin ground states ( S = 1/2) when both complexes are dissolved in toluene solutions. The low-spin ground states in toluene solution are confirmed by EPR spectroscopy, which further supports, for complexes 4a and 4b, an axially elongated square pyramidal geometry and a d z (2) ground state. Thus the change in the ground-state and, consequently, in the geometry of complexes 4a and 4b from solid state to toluene solution might be a consequence of the elongation of the Co-O(THF) bond length. DFT studies performed on complexes 3 and 4 corroborate their different structure and magnetic behaviors and verify, for the latter complexes, the structural differences observed in the solid state and in toluene solution.     

Versatile coordination chemistry towards multifunctional carbon nanotube nanohybrids

Dispersible single-walled carbon nanotubes grafted with poly(4-vinylpyridine), SWNT-PVP, were tested in coordination assays with zinc tetraphenylporphyrin (ZnP). Kinetic and spectroscopic evidence corroborates the successful formation of a SWNT-PVPZnP nanohybrid. Within this SWNT-PVPZnP nanohybrid, static electron-transfer quenching (2.0+/-0.1) x 10(9) s(-1) converts the photoexcited-ZnP chromophore into a radical-ion-pair state with a microsecond lifetime, namely one-electron oxidized-ZnP and reduced-SWNT.     

Nature of cationic poly(propylenimine) dendrimers in aqueous solutions as studied using versatile indicator dyes

Aqueous solutions of four cationic poly(propylenimine) low-generation dendrimers of different architecture and hydrophobicity have been examined as media for acid-base reactions of indicator dyes. The cationic dendrimers in solution can be considered as oligomers of cationic polyelectrolytes, or surfactant-like species, able to form micelles through self-association or sometimes even as unimolecular micelles. The dendrimers influence the ionization constants, tautomeric equilibria, and absorption/emission/excitation spectra of indicator dyes. The p K a values of the majority of the indicator dyes decrease in dendrimer solutions, often by 1-2 p K a units, similar to effects registered in micellar solutions of cationic surfactants. Analogously, the shifts of absorption band maxima indicate that the microenvironments of the dyes bound to the dendrimers are less polar than in water. However, some spectral effects denote the specificity of the dendrimers. The greatest difference between the dendrimers and spherical surfactant micelles is revealed by kinetic processes, especially of bromophenol blue alkaline fading in a dendrimer solution but not in a micellar surfactant solution. Within the dendrimer series, the most significant differences were observed for substances possessing n-dodecyl tails on the one hand and those without such hydrophobic portions on the other. For the last-named, the decrease in p K a's of indicators, band shifts of their anions, and in particular displacement of tautomeric equilibria compared with aqueous solutions are much smaller than for more hydrophobic dendrimers.     

Two-Photon Excitation of 4'-Hydroxymethyl-4,5',8-Trimethylpsoralen

Abstract— Psoralens are a class of pharmaceutical agents commonly used to treat several cutaneous disorders. When irradiated with a mode-locked titanium: sapphire (Ti: sapphire) laser tuned to 730 nm, an aqueous solution of 4'-hydroxymethyl-4,5',8-trimethylpsoralen (HMT) emits blue light. The emission spectrum is centered at 452 nm and is identical to that obtained by one-photon excitation with UVA excitation, and its magnitude depends quad-ratically on the intensity of laser excitation. These results suggest that two-photon excitation occurs to a potentially photochemically active state. To estimate the two-photon absorption cross section, it was first necessary to measure the emission quantum yield of HMT using 365 nm excitation at room temperature that resulted in a value of 0.045 ± 0.007. The two-photon absorption cross section of HMT at 730 nm is therefore estimated to be 20 ± 10⁻⁵⁰ cm⁴ s (20 Göppert-Mayer). The excited-state photophysics and photochemistry of psoralens suggest potential applications to cutaneous phototherapy in diseases such as psoriasis and dystrophic epidermolysis bullosa.     

Synthesis of 2-pentafluorosulfanylnaphthalene

A three step synthesis of 2-pentafluorosulfanylnaphthalene is reported. Initial addition of SF₅Cl to benzobarralene was followed by elimination to form 2-pentafluorosulfanylbenzobarralene. Heating of this compound with 3,6-bis-(2-pyridyl)-1,2,4,5-tetrazine led to elimination of the ethylene bridge via a sequence of cycloadditions and retro-cycloadditions to form the title compound.     

Transition state analysis of Vibrio cholerae sialidase-catalyzed hydrolyses of natural substrate analogues

A series of isotopically labeled natural substrate analogues (phenyl 5-N-acetyl-α-d-neuraminyl-(2→3)-β-d-galactopyranosyl-(1→4)-1-thio-β-d-glucopyranoside; Neu5Acα2,3LacβSPh, and the corresponding 2→6 isomer) were prepared chemoenzymatically in order to characterize, by use of multiple kinetic isotope effect (KIE) measurements, the glycosylation transition states for Vibrio cholerae sialidase-catalyzed hydrolysis reactions. The derived KIEs for Neu5Acα2,3LacβSPh for the ring oxygen ((18)V/K), leaving group oxygen ((18)V/K), C3-S deuterium ((D)V/K(S)) and C3-R deuterium ((D)V/K(R)) are 1.029 ± 0.002, 0.983 ± 0.001, 1.034 ± 0.002, and 1.043 ± 0.002, respectively. In addition, the KIEs for Neu5Acα2,6βSPh for C3-S deuterium ((D)V/K(S)) and C3-R deuterium ((D)V/K(R)) are 1.021 ± 0.001 and 1.049 ± 0.001, respectively. The glycosylation transition state structures for both Neu5Acα2,3LacβSPh and Neu5Acα2,6LacβSPh were modeled computationally using the experimental KIE values as goodness of fit criteria. Both transition states are late with largely cleaved glycosidic bonds coupled to pyranosyl ring flattening ((4)H(5) half-chair conformation) with little or no nucleophilic involvement of the enzymatic tyrosine residue. Notably, the transition state for the catalyzed hydrolysis of Neu5Acα2,6βSPh appears to incorporate a lesser degree of general-acid catalysis, relative to the 2,3-isomer.