On Producing Multiple Solutions Using Repeated Trials

The number of trials that is required by an algorithm to produce a given fraction of the problem solutions with a specified level of confidence is analyzed. The analysis indicates that the number of trials required to find a large fraction of the solutions rapidly decreases as the number of solutions obtained on each trial by an algorithm increases. In applications where multiple solutions are sought, this decrease in the number of trials could potentially offset the additional computational cost of algorithms that produce multiple solutions on a single trial. The analysis framework presented is used to compare the efficiency of a homotopy algorithm to that of a Newton method by measuring both the number of trials and the number of calculations required to obtain a specified fraction of the solutions.     

Energy-minimizing deformations of elastic sheets with bending stiffness

Necessary conditions for energy-minimizing deformations are derived for a theory of sheets in which the strain energy function depends on the second derivatives of the deformation as well as its first derivatives. All of these conditions are extensions of well-known necessary conditions in classical calculus of variations. The interpretation of some of these conditions as material stability conditions is explained.     

Photophysical properties of borondipyrromethene analogues in solution

The photophysical properties of seven new 8-(p-substituted)phenyl analogues of 4,4-difluoro-3,5-dimethyl-8-(aryl)-4-bora-3a,4a-diaza-s-indacene (derivatives of the well-known fluorophore BODIPY) in several solvents have been studied by means of absorption and steady-state and time-resolved fluorimetry. For each compound, the fluorescence quantum yield and lifetime are lower in solvents with higher polarity owing to an increase in the rate of nonradiative deactivation. Increasing the electron withdrawing strength of the p-substituent on the phenyl group in position 8 also leads to lower fluorescence quantum yields and lifetimes. When the p-substituent on the phenyl group in position 8 is a tertiary amine [8-(4-piperidinophenyl), 8-(4-N,N-dimethylaminophenyl), and 8-(4-morpholinophenyl)], the low quantum yields of these compounds in more polar solvents can be rationalized by the inversion of the energy levels of an apolar, highly fluorescent and a polar, nonfluorescent excited state, where charge transfer from the tertiary amine to the BODIPY unit occurs. These amine analogues can be protonated at low pH in aqueous solution. Fluorescence titrations yielded pK(a) values of their conjugate ammonium salts which are in agreement with the electron donating tendency of the amine group: piperidino (4.15) > dimethylamino (2.37) > morpholino (1.47), with the pK(a) values in parentheses. The rate constant of radiative deactivation (k(f)) is the same for all compounds in all solvents studied (k(f) = 1.4 x 10(8) s(-1)).     

Methylene-azaphosphirane as a reactive intermediate

Reaction of the transient phosphinidene complexes R-P=W(CO)5 with N-substituted-diphenylketenimines leads unexpectedly to the novel 2-aminophosphindoles, as confirmed by an X-ray crystal structure determined for one of the derivatives. Experimental evidence for a methylene-azaphosphirane intermediate was found by using the iron-complexed phosphinidene iPr2N-P=Fe(CO)4, which affords the 2-aminophosphindole together with the novel methylene-2,3-dihydro-1H-benzo[1,3]azaphosphole. Analysis of the reaction pathways with DFT indicates that the initially formed methylene-azaphosphirane yields both phosphorus heterocycles by way of a [1,5]- or [1,3]-sigmatropic shift, respectively, followed by a H-shift. Strain underlies both rearrangements, which causes these remarkably selective conversions that can be tuned by changing the substituents.     

On the use of the Bayliss-McRae dispersion model in NMR solvent effects

The use of the Bayliss-McRae theory on the solvent induced electronic frequency shifts for NMR dispersion shifts is criticized. It is suggested that the NMR shifts should actually be proportional to the square of the Bayliss-McRae function. It is shown that the methane gas-to-liquid shifts in eleven halo-methanes as solvents are indeed proportional to this squared function; ?σ_m(CH₄) = 9.62 (n₂²?1)²/(2n₂²+1)² ppm, where n₂ is the refractive index of the solvent. The relation between this solvent factor and several existing continuum models for NMR medium shifts is discussed.     

Synthesis of 1H-imidazo[1,2-a]imidazole

The synthesis and structure analysis of the unknown 1H-imidazo[1,2-a]imidazole ( 1 ) is described. The preparation involves alkylation of 2-aminoimidazole with bromoacetaldehyde diethyl acetal and subsequent hydrolysis and cyclization with hydrochloric acid. The structure was characterized by mass spectrometry and by ¹H-, ¹⁵N- and ¹³C-nmr of 1 and by ¹H-nmr of its 1-benzyl derivative 8 . An independent synthesis of 8 was accomplished via cyclization of 2-(N-dichloroethyl-N-benzyl)aminoimidazole ( 11 ).     

Langmuir-Blodgett films of a clay mineral and ruthenium(II) complexes with a noncentrosymmetric structure

Mono- and multilayers of amphiphilic [Ru(phen)(2)(dcC12bpy)](2+) (phen = 1,10-phenanthroline, dcC12bpy = 4,4'-caboxyl-2,2'-bipyridyl didodecyl ester) hybridized with a clay mineral have been prepared by a modified Langmuir-Blodgett method, and their structures and properties have been investigated. Formation of a hybrid monolayer of the Ru(II) complex cations and the clay platelets at an air-clay suspension interface was confirmed by surface pressure-molecular area (pi-A) isotherm measurement and atomic force microscopic (AFM) observation. Multilayers were fabricated by depositing the hybrid monolayers onto glass substrates. The absorbance at 492 nm due to the Ru(II) complex cation in the multilayer increased linearly with the increase in the layer number, indicating layer-by-layer deposition of the hybrid monolayers. Because no increase in the second-harmonic generation (SHG) signal from the multilayers against the layer number was observed, the orientation of the Ru(II) complex cations in the layer would be disturbed. The hydrophilic surface of the transferred hybrid monolayer can be converted to a hydrophobic surface by dipping it in an aqueous solution of octadecylammonium chloride (ODAH(+)Cl(-)). The multilayers modified with ODAH(+) showed a quadratic relation between the SHG intensity and the layer number. This means that the Ru(II) complex cations in the multilayer are successfully oriented in a noncentrosymmetric way by the conversion of the surface property. Both a racemic mixture and an enantiomer of the Ru(II) complex cations were employed to examine the chiral effect on the film properties. The chiral contribution to the SHG signal was enhanced in the multilayer modified with ODAH(+).