Counting acyclic digraphs by sources and sinks

We count labeled acyclic digraphs according to the number sources, sinks, and edges.     

Electron spin resonance study of free radicals in the radiation grafting of butadiene to poly(vinyl chloride)

The kinetics of the disappearance of propagating butadiene radicals produced in the grafting of butadiene to poly(vinyl chloride) was followed by electron spin resonance measurements. The radical species observed in the temperature range 213–298°K is the conjugated allylic radical: The reaction conditions were adjusted to provide a highly viscous medium in which the propagation reaction took place. Under these conditions the radicals were stabilized and a precise ESR spectrum was obtained. Arrhenius plots of the second-order decay rate constants yield an activation energy of 5 kcal/mole. This suggests that the bimolecular termination reaction is controlled by the segmental motion of the grafted side chains.     

An invariant imbedding method for computation of eigenlengths of singular two-point boundary-value problems

In this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular TPBVP with the singularity at the origin. The invariant imbedding yields a first-order nonlinear equation called a Riccati equation and also gives the initial conditions at the origin for this equation. With the aid of Theorem 8 in Section 3 we numerically integrate the Riccati equation to “blowup” which gives our computed eigenlength.In closing, we would like to comment on the numerical merits of the integration-to-blowup technique. On the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results. The feature of a variable step size is essential as the value of z approaches the actual eigenlength. However, it is desirable to have a priori estimate or bounds of the eigenlength similar to those of Boland and Nelson [2] for the nonsingular case. The singular system, however, presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds. Hopefully an investigation of the matrix R(z) will yield these results.     

When the dual of an ideal is a ring

Let R be a commutative integral domain with identity with quotient field K, and let I be a nonzero ideal of R. We analyze several general and particular instances when I^–1 is a subring of K. We then apply some of our results to show that certain non-maximal prime ideals in Prüfer domains are divisorial.     

Depth-first search as a combinatorial correspondence

A depth first search algorithm is used to establish the connection between labeled connected graphs and inversions of trees.     

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.     

Invariant imbedding applied to homogeneous two-point boundary-value problems with a singularity of the first kind

It is shown that previous work of Elder can be used to extend the version of invariant imbedding due to M. R. Scott to homogeneous (vector) differential systems having a singularity of the first kind. The boundary conditions considered consist of existence (finite) at the singularity and specified values for some subset of the dependent variables at a second point. The important special case of a second-order equation is discussed in some detail. Computational considerations are discussed and numerical examples are presented.     

Affinity capillary electrophoresis applied to the studies of interactions of a member of heat shock protein family with an immunosuppressant

The bioaffinity of receptor-ligand interactions is investigated by determining the binding constant (association constant or dissociation constant) of the resulting complex utilizing affinity capillary electrophoresis (ACE). The ACE binding assay was established with a potent immunosuppressant, deoxyspergualin (DSG), that binds specifically to Hsc70, a constitutive or cognate member of heat shock protein 70 (Hsp70) family. Quantitative determination of binding constants under different running buffer systems provide comparative results. The association constants for the interaction between Hsc70 protein and DSG were found to be 5.7·10⁴ M⁻¹ in a buffer with pH 6.95 and 6.3·10⁴ M⁻¹ in a buffer with pH 5.30 (or corresponding dissociation constants, 18 and 16 μM, respectively) based on Scatchard analyses. Binding of DSG to a synthetic peptide, SINPDEAVAYGAAV-QAAILSGDK, one of the DSG-binding fragments found from tryptic digest of Hsc70 protein, provides further detailed information for the understanding of Hsc70 binding domain. The applicability of using coated capillaries was also evaluated for probing Hsc70-DSG interaction.