Abelian networks III: The critical group

The critical group of an abelian network is a finite abelian group that governs the behavior of the network on large inputs. It generalizes the sandpile group of a graph. We show that the critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network. We exhibit the critical group as a quotient of a free abelian group by a subgroup containing the image of the Laplacian, with equality in the case that the network is rectangular. We generalize Dhar’s burning algorithm to abelian networks and estimate the running time of an abelian network on an arbitrary input up to a constant additive error.     

The Divisible Sandpile at Critical Density

The divisible sandpile starts with i.i.d. random variables (“masses”) at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses ≤  1. The process stabilizes almost surely if m < 1 and it almost surely does not stabilize if m > 1, where m is the mean mass per vertex. The main result of this paper is that in the critical case m = 1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete bi-Laplacian Gaussian field.     

Asymmetric hydrogenations for the synthesis of Boc-protected 4-alkylprolinols and prolines

The utility of 4-substituted prolinols and their corresponding prolines in peptides, peptidomimetics, and natural products has motivated researchers to find new and efficient routes for their preparation. Herein, we report a general approach to the synthesis of Boc-protected 4-alkylprolinols and prolines via a divergent asymmetric hydrogenation strategy. Intermediate exocyclic olefins were prepared by Wittig-type reactions with ketone 6 and subjected to hydroxyl and sterically directed reductions. The Crabtree catalyst (Ir[COD]PyPCy(3)PF(6)) proved to be highly effective in diastereoselective hydrogenations to give trans-substituted pyrrolidines (9). Good facial selectivities were also observed in heterogeneous hydrogenations with Raney-nickel to obtain cis-substituted pyrrolidines (11). Employing this strategy, we describe the synthesis of novel prolinol and proline-based building blocks for incorporation into biologically relevant peptidomimetics.     

Copolyesteramides—III. Anionic copolymers of ε-caprolactam with ε-caprolactone. Crystalline character and mechanical properties

The crystalline character of anionic copolymers of ε-caprolactam with ε-caprolactone and of some alternatingly-sequenced analogues has been studied and investigations made of the mechanical and dynamic mechanical properties of the anionic copolymers. The anionic copolymers are crystalline over the whole range of compositions. Depending on the proportions of NH(CH₂)₅CO and O(CH₂)₅CO units and the thermal history of the copolymers, the crystalline phases present are either wholly of polyamide type or composed of co-existing and mutually incompatible polyamide and polyester entities. However, the constituents appear to be miscible in the amorphous phase. The mechanical properties of the copolymers (like their crystalline melting temperatures) change discontinuously with composition, showing minima in the values of initial modulus, yield stress and breaking stress at ca 25–40% amide-group content where dual crystallinity exists. In addition to their variation with composition, the properties are also affected by changes in the procedure of anionic copolymerisation.     

Classical and quantum mechanical systems of Toda-lattice type

Solutions to the classical periodic and non-periodic Toda lattice type Hamiltonian systems are expressed in terms of an Iwasawa-type factorization of a large Lie group. The scattering of these systems is determined in the non-periodic case. For the generalized periodic Toda lattices a generalization of Kostant's formula is obtained using standard representations of affine Lie groups.Research partially supported by NSF Grant MCS 83-01582Research partially supported by NSF Grant MCS 79-03153     

Reversible "irreversible" inhibition of chymotrypsin using nanoparticle receptors

Anionically functionalized amphiphilic nanoparticles efficiently inhibit chymotrypsin through electrostatic binding followed by protein denaturation. We demonstrate the ability to disrupt this "irreversible" inhibition of chymotrypsin through modification of the nanoparticle surface using cationic surfactants. Up to 50% of original chymotrypsin activity is rescued upon long-chain surfactant addition. Dynamic light-scattering studies demonstrate that chymotrypsin is released from the nanoparticle surface. The conformation of the rescued chymotrypsin was characterized by fluorescence and fluorescence anisotropy, indicating that chymotrypsin regains a high degree of native structure upon surfactant addition.     

Saturation transfer spectroscopy: signals sensitive to very slow molecular reorientation

A density matrix-transition rate matrix formalism is employed to compute the eight unique electron paramagnetic resonance signals at the first two harmonics of the applied Zeeman modulation. Calculations are carried out for conditions appropriate for investigating spin-labeled biomolecules with partially saturating microwave fields and Zeeman modulation amplitudes comparable to resonance linewidths. Spectra are computed for rotational correlation times ranging from 10^?7 to 10^?3 s and for modulation frequencies of 50 and 100 kHz. These simulations indicate that of the eight signals the in-phase dispersion signal at the first harmonic of the modulation and the out-of-phase dispersion signal at the first harmonic afford the best sensitivity to molecular motion and the largest signal amplitudes. It is suggested that study of these signals is the method of choice for monitoring slowly tumbling spin labels when signal-to-noise considerations are critical. The conclusions derived from computer simulations are borne out by experimental measurements performed on 10^?3 M solutions of the steroid spin label 17β-hydroxy-4′,4′-dimethylspiro-[5α-androstrance-3,2′-oxazolidin]-3′-oxyl in sec-butylbenzene.     

Theoretical evaluation of the origin of the regio- and stereoselectivity in the Diels-Alder reactions of dialkylvinylboranes: studies on the reactions of vinylborane, dimethylvinylborane, and vinyl-9-BBN with trans-piperylene and isoprene

Ab initio and DFT calculations have been performed to study the origin of the regio- and stereoselectivity of the Diels-Alder reactions of dialkylvinylboranes with substituted dienes. B3LYP/6-31G energies of the transition structures for the reactions of dimethylvinylborane and vinyl-9-BBN with trans-piperylene and isoprene yielded calculated ratios which are in very good agreement with experimental values. Nonclassical carbon-boron [4+3] secondary orbital interactions seem to account for the high endo stereoselectivity of these reactions. However, C-B interactions become less important when the bulkiness of the alkyl groups attached to boron increases. Both endo and exo transition structures for the reactions of dimethylvinylborane and vinyl-9-BBN adopt classical [4+2] character. This study also extends Singleton's investigation on butadiene to regioselectivity. FMO theory has been used to rationalize the lack of regioselectivity in the reactions of dimethylvinylborane. The anomalous meta regioselectivity of the Diels-Alder reaction of vinyl-9-BBN with trans-piperylene is mainly caused by steric effects.     

Synthesis and crystal structure of tetra- and hexanuclear uranium(IV) complexes with hexadentate compartmental Schiff-base ligands

Treatment of UCl4 with the hexadentate Schiff bases H2Li in thf gave the expected [ULiCl2(thf)] complexes [H2Li=N,N'-bis(3-methoxysalicylidene)-R and R = 2,2-dimethyl-1,3-propanediamine (i= 1), R = 1,3-propanediamine (i= 2), R = 2-amino-benzylamine (i= 3), R = 2-methyl-1,2-propanediamine (i= 4), R = 1,2-phenylenediamine (i= 5)]. The crystal structure of [UL4Cl2(thf)] (4) shows the metal in a quite perfect pentagonal bipyramidal configuration, with the two Cl atoms in apical positions. Reaction of UCl4 with H4Li in pyridine did not afford the mononuclear products [U(H2Li)Cl2(py)x] but gave instead polynuclear complexes [H4Li=N,N'-bis(3-hydroxysalicylidene)-R and R = 1,3-propanediamine (i= 6), R = 2-amino-benzylamine (i= 7) or R = 2-methyl-1,2-propanediamine (i= 8)]. In the presence of H4L6 and H4L7 in pyridine, UCl4 was transformed in a serendipitous and reproducible manner into the tetranuclear U(iv) complexes [Hpy]2[U4(L6)2(H2L6)2Cl6] (6a) and [Hpy]2[U4(L7)2(H2L7)2Cl6][U4(L7)2(H2L7)2Cl4(py)2] (7), respectively. Treatment of UCl4 with [Zn(H2L6)] led to the formation of the neutral compound [U4(L6)2(H2L6)2Cl4(py)2] (6b). The hexanuclear complex [Hpy]2[U6(L8)4Cl10(py)4] (8) was obtained by reaction of UCl4 and H4L8. The centrosymmetric crystal structures of 6a.2HpyCl.2py, 6b.6py, 7.16py and 8.6py illustrate the potential of Schiff bases as associating ligands for the design of polynuclear assemblies.