Generating M-Indeterminate Probability Densities by Way of Quantum Mechanics

Probability densities that are not uniquely determined by their moments are said to be “moment-indeterminate,” or “M-indeterminate.” Determining whether or not a density is M-indeterminate, or how to generate an M-indeterminate density, is a challenging problem with a long history. Quantum mechanics is inherently probabilistic, yet the way in which probability densities are obtained is dramatically different in comparison with standard probability theory, involving complex wave functions and operators, among other aspects. Nevertheless, the end results are standard probabilistic quantities, such as expectation values, moments and probability density functions. We show that the quantum mechanics procedure to obtain densities leads to a simple method to generate an infinite number of M-indeterminate densities. Different self-adjoint operators can lead to new classes of M-indeterminate densities. Depending on the operator, the method can produce densities that are of the Stieltjes class or new formulations that are not of the Stieltjes class. As such, the method complements and extends existing approaches and opens up new avenues for further development. The method applies to continuous and discrete probability densities. A number of examples are given.

     

The finite element approximation of a coupled reaction-diffusion problem with non-Lipschitz nonlinearities

Summary. A coupled semilinear elliptic problem modelling an irreversible, isothermal chemical reaction is introduced, and discretised using the usual piecewise linear Galerkin finite element approximation. An interesting feature of the problem is that a reaction order of less than one gives rise to a "dead core" region. Initially, one reactant is assumed to be acting as a catalyst and is kept constant. It is shown that error bounds previously obtained for a scheme involving numerical integration can be improved upon by considering a quadratic regularisation of the nonlinear term. This technique is then applied to the full coupled problem, and optimal and error bounds are proved in the absence of quadrature. For a scheme involving numerical integration, bounds similar to those obtained for the catalyst problem are shown to hold. Received May 25, 1993 / Revised version received July 5, 1994     

Mechanism of the aromatic hydroxylation of thiophene by acid-catalyzed peracid oxidation

The oxidation of thiophene (1) with peracids in a strongly acidic environment yielded thiophen-2-one (4) as the product of an apparent direct hydroxylation of the thiophene aromatic ring together with the anticipated thiophene-S-oxide dimers, 2a,b, as the main products. Formation of the latter dimers can be rationalized in a straightforward manner by initial oxidation at the sulfur atom of thiophene (1) to yield thiophene-S-oxide followed by subsequent dimerization in a Diels-Alder type reaction. Trapping experiments in the presence of a competing dienophile indicated that thiophen-2-one (4) did not originate from the monomeric thiophene-S-oxide but was the product of an independent reaction pathway. The extent of thiophen-2-one (4) formation correlated with the acidity of the reaction medium and was suppressed in the presence of water, the latter presumably acting as a competing base. As evidenced by the use of 2,5-dideuterated thiophene (1-D), its mechanism of formation involved a 1,2-hydride shift, a feature commonly described in the peracid-mediated epoxidation of aromatic hydrocarbons and indicative for the occurrence of cationic intermediates. In agreement with all these observations we propose a mechanism involving initial protonation of thiophene followed by nucleophilic attack of the peracid in position 2 of the thiophene ring. Intramolecular epoxidation may lead to the formation of thiophene 2,3-epoxide as a highly reactive intermediate that then undergoes heterolytic ring opening and a 1,2-hydride shift to yield thiophen-2-one (4) after a final, acid-catalyzed, isomerization of the double bond.     

Mass spectra of some naturally occurring stictane triterpenoids and their trimethylsilyl derivatives

The mass spectra of ten di- and tri-oxygenated stictane triterpenoids and their trimethylsilyl derivatives have been studied in detail. Gas chromatography mass spectrometry of the TMS derivatives on OV-17 and OV-101 columns provided useful separations and identifications of mixtures of stictanes triterpenoids in lichen extracts. The major fragmentations involve ring C, but specific cleavages in rings A, B and E are also observed which allow the substitution pattern of the skeleton to be readily determined. The formation of an intense [M – C₅H₁₁?]⁺ ion for 22α-OTMS stictanes appears specific to their ring E configuration.     

Density-functional theory calculations of optical rotatory dispersion in the nonresonant and resonant frequency regions

The complex linear response function, which can be employed for calculations of second-order molecular properties in regions of strong absorption, is here extended to encompass the mixed electric-dipole-magnetic-dipole polarizability. The mixed electric-dipole-magnetic-dipole polarizability determines the optical rotation and, when absorption is taken into account, the full anomalous optical rotatory dispersion (ORD) spectra of chiral molecules can be calculated using first-principle quantum-chemical methods. Gauge-origin independence of the results is ensured through the use of London atomic orbitals. To illustrate the importance of taking the absorption process properly into account, we here apply this methodology to the study of the anomalous ORD of hydrogen peroxide, 3R-methylcyclohexanone, 4R-1,1-dimethyl-[3]-(1,2)-ferrocenophan-2-on, and the D(2) isomer of the C(84) fullerene.     

Atomistic molecular dynamics simulations of chemical force microscopy

Chemical force microscopy and related force measurement techniques have emerged as powerful tools for studying fundamental interactions central to understanding adhesion and tribology at the molecular scale. However, detailed interpretation of these interactions requires knowledge of chemical and physical processes occurring in the region of the tip-sample junction that experiments cannot provide, such as atomic-scale motions and distribution of forces. In an effort to address some of these open issues, atomistic molecular dynamics simulations were performed modeling a chemical force microscope stylus covered with a planar C12 alkylthiolate self-assembled monolayer (SAM) interacting with a solid wall. A complete loading-unloading sequence was simulated under conditions of near-constant equilibrium, approximating the case of infinitely slow tip motion. In the absence of the solid wall, the stylus film existed in a fluid state with structural and dynamic properties similar to those of the analogous planar SAM at an elevated temperature. When the wall was brought into contact with the stylus and pressed against it, a series of reversible changes occurred culminating with solidification of the SAM film at the largest compressive force. During loading, the chemical composition of the contact changed, as much of the film's interior was exposed to the wall. At all tip heights, the distribution of forces within the contact zone was uneven and subject to large local fluctuations. Analysis using the Johnson-Kendall-Roberts, Derjaguin-Muller-Toporov, and Hertz contacts mechanics models revealed significant deviations from the simulation results, with the JKR model providing best overall agreement. Some of the discrepancies found would be overlooked in an actual experiment, where, unlike the simulations, contact area is not separately known, possibly producing a misleading or incorrect interpretation of experimental results. These shortcomings may be improved upon by using a model that correctly accounts for the finite thickness of the compliant components and nonlinear elastic effects.     

Exploratory Investigations Probing a Preparatively Versatile, Pyridinium Salt Photoelectrocyclization-Solvolytic Aziridine Ring Opening Sequence

A novel pyridinium salt photoelectrocyclization-nucleophilic bicyclic aziridine ring opening reaction sequence has been investigated in order to determine its preparative potential. N-Alkylpyridinium perchlorates were found to undergo photoinduced electrocyclization upon irradiation in nucleophilic solvents, such as H(2)O and MeOH, to efficiently produce 6-alkyl-6-azabicyclo[3.1.0]hex-2-en-4-yl alcohols and ethers. The bicyclic aziridine photoproducts react with a number of different nucleophiles (e.g., H(2)O, MeOH, AcOH, AcSH) under acid-catalyzed conditions to produce 5-(nucleophile-substituted)-4-(alkylamino)cyclopenten-3-yl alcohols and ethers. The aziridine ring opening processes are both regioselective and stereoselective, yielding trans,trans-trisubstituted cyclopentenes exclusively, apparently as a consequence of the operation of an SN(2) mechanism. The effects of C-alkyl substitution on the regiochemistry of the pyridinium cation photocyclization reaction were briefly probed, and a method was developed to produce trans,cis-trisubstituted cyclopentenes by use of this tandem preparative sequence.     

Heat capacities of binary cycloalkane mixtures at 298.15 K

Volumetric heat capacities of the six binary mixtures formed from cyclopentane, cyclohexane, cycloheptane and cyclooctane were determined at 298.15 K in a Picker flow microcalorimeter. Excess heat capacities obtained from the results are compred with the temperature variation of excess enthalpies from the literature.     

A one-pot multicomponent coupling reaction for the stereocontrolled synthesis of (Z)-trisubstituted allylic alcohols

In this Communication, we outline a new one-pot, multicomponent coupling reaction that allows easy access to (Z)-trisubstituted allylic alcohols. Our strategy is based on E to Z isomerization of the 1-bromo-1-dialkylvinylborane upon reaction with dialkylzinc reagents, and subsequent transmetalation to give (Z)-trisubstituted vinylzinc species. In situ trapping of the reactive vinylzinc intermediates with aldehydes furnished a series of (Z)-trisubstituted allylic alcohols. This method represents a viable alternative to the Still-Gennari modification of the HWE olefination reaction, and it has the advantage that it allows coupling of larger fragments.