Hypoellipticity on the Heisenberg group

Let P be a left-invariant differential operator on the Heisenberg group Hⁿ, P homogeneous with respect to the dilations on Hⁿ. We show that a necessary and sufficient condition for the hypoellipticity of P is that π(P) be an injective operator for every irreducible unitary representation π of Hⁿ (except the trivial representation). Furthermore, hypoellipticity is preserved if the homogeneous operator P is perturbed by terms of lower order of homogeneity. (Homogeneity means homogeneity with respect to dilations of Hⁿ.) It is also shown that if P is homogeneous, left-invariant and hypoelliptic on Hⁿ, then its formal adjoint is hypoelliptic.     

Recommendations for measuring second-order rate constants and kinetic solvent isotope effects on acid-catalyzed reactions

Kinetic solvent isotope effects (KSIE) were measured for the hydrolyses of acetals of benzaldehydes in aqueous solutions covering the pH (pD) range of 1–6. For p-methoxybenzaldehyde diethyl acetal, k/k = 1.8–3.1, depending on the procedure used to calculate the KSIE and on the pH (pD) range used as the basis for k(k). It is shown that this variation is an experimental artifact, and is a characteristic of KSIE measurements in general. It is recommended that k be calculated from a least-squares fit of data to the equation k_obs = k[L⁺], and that the KSIE be reported as k/k. The limitation remains, however, that the KSIE measured for a variety of substances over quite different pH (pD) ranges may not be comparable to more than ?20%. The source of these observations is discussed in terms of small changes in the activity coefficient ratios (a specific salt effect), including the solvent isotope effect on the activity coefficient ratio [eq. (3)].     

Altering the polymorphic product distribution in a solid-state dehydration process by rapid sample rotation in a solid-state NMR probe

There is currently considerable interest in the phenomenon of polymorphism in organic molecular solids, and a key issue in this field is to understand the experimental techniques and procedures that may be employed to obtain new polymorphic forms of a given molecule. This paper demonstrates that the polymorphic form of a material (sodium acetate) obtained by a solid-state dehydration process (starting from sodium acetate trihydrate) can be altered by carrying out the dehydration process under conditions of rapid (several kilohertz) sample rotation in a solid-state magic-angle-spinning NMR probe. This observation suggests a new opportunity to influence the outcome of solid-state dehydration/desolvation processes and, in particular, to alter the polymorphic form of the product obtained.     

Electron exchange between alpha-Keggin tungstoaluminates and a well-defined cluster-anion probe for studies in electron transfer

Fully oxidized alpha-AlIIIW12O40(5-) (1ox), and one-electron-reduced alpha-AlIIIW12O40(6-) (1red), are well-behaved (stable and free of ion pairing) over a wide range of pH and ionic-strength values at room temperature in water. Having established this, 27Al NMR spectroscopy is used to measure rates of electron exchange between 1ox (27Al NMR: 72.2 ppm relative to Al(H2O)63+; nu(1/2) = 0.77 Hz) and 1red (74.1 ppm; nu(1/2) = 0.76 Hz). Bimolecular rate constants, k, are obtained from line broadening in 27Al NMR signals as ionic strength, mu, is increased by addition of NaCl at the slow-exchange limit of the NMR time scale. The dependence of k on is plotted using the extended Debye-Hückel equation: log k = log k0 + 2alphaz1z2mu(1/2)/(1 + betarnu(1/2)), where z1 and z2 are the charges of 1ox and 1red, alpha and beta are constants, and r, the distance of closest contact, is fixed at 1.12 nm, the crystallographic diameter of a Keggin anion. Although not derived for highly charged ions, this equation gives a straight line (R2 = 0.996), whose slope gives a charge product, z1z2, of 29 +/- 2, statistically identical to the theoretical value of 30. Extrapolation to mu = 0 gives a rate constant k11 of (6.5 +/- 1.5) x 10(-3) M(-1) s(-1), more than 7 orders of magnitude smaller than the rate constant [(1.1 +/- 0.2) x 10(5) M(-1) s(-1)] determined by 31P NMR for self-exchange between P(V)W12O40(3-) and its one-electron-reduced form, P(V)W12O40(4-). Sutin's semiclassical model reveals that this dramatic difference arises from the large negative charges of 1ox and 1red. These results, including independent verification of k11, recommend 1red as a well-behaved electron donor for investigating outer-sphere electron transfer to molecules or nanostructures in water, while addressing a larger issue, the prediction of collision rates between uniformly charged nanospheres, for which 1ox and 1red provide a working model.     

Dynamic yielding, shear thinning, and stress rheology of polymer-particle suspensions and gels

The nonlinear rheological version of our barrier hopping theory for particle-polymer suspensions and gels has been employed to study the effect of steady shear and constant stress on the alpha relaxation time, yielding process, viscosity, and non-Newtonian flow curves. The role of particle volume fraction, polymer-particle size asymmetry ratio, and polymer concentration have been systematically explored. The dynamic yield stress decreases in a polymer-concentration- and volume-fraction-dependent manner that can be described as apparent power laws with effective exponents that monotonically increase with observation time. Stress- or shear-induced thinning of the viscosity becomes more abrupt with increasing magnitude of the quiescent viscosity. Flow curves show an intermediate shear rate dependence of an effective power-law form, becoming more solidlike with increasing depletion attraction. The influence of polymer concentration, particle volume fraction, and polymer-particle size asymmetry ratio on all properties is controlled to a first approximation by how far the system is from the gelation boundary of ideal mode-coupling theory (MCT). This emphasizes the importance of the MCT nonergodicity transition despite its ultimate destruction by activated barrier hopping processes. Comparison of the theoretical results with limited experimental studies is encouraging.     

A Maximum Principle for an Explicit Finite Difference Scheme Approximating the Hodgkin-Huxley Model

We analyze an explicit finite difference scheme for the general form of the Hodgkin-Huxley model, which is a nonlinear partial differential equation coupled to a set of ODEs. The system of equations describes propagation of an electrical signal in excitable cells. We prove that the numerical solution is bounded in the L^∞-norm and L² converges to a unique solution. The L^∞-bound, which is the key point of our analysis, is proved by showing that the discrete solutions are invariant in a physically relevant bounded region. For the convergence proof we use the compactness method. AMS subject classification (2000) 65F20     

Chemical constituents of the defensive secretion of the striped skunk (Mephitis mephitis)

The volatile components in the musk of the striped skunk (Mephitis mephitis) were analyzed by GC-MS. Several thiols (mercaptans), thioethers (sulfides), and disulfides were identified.     

Kinetics of polymerization of styrene initiated by substituted benzoyl peroxides. IV. Decomposition induced by substituted diethylanilines

Equilibrium complex formation between symmetrically substituted benzoyl peroxides and ring-substituted N,N-diethylanilines has been quantitatively measured. From the results the rate constant for the unimolecular decomposition of the complexes, producing an initiating free radical, is found to be approximately 10¹² exp {(–25 X 10³)/RT}. The data obtained are consistent with a previously postulated mechanism for radical production in the presence of vinyl monomers.