The attractor—basin portrait of a cellular automaton

Local space-time structures, such as domains and the intervening dislocations, dominate a wide class of cellular automaton (CA) behavior. For such spatially-extended dynamics regular domains, vicinities, and attractors are introduced as organizing principles to identify the discretized analogs of attractors, basins, and separatrices: structures used in classifying dissipative continuous-state dynamical systems. We describe the attractor-basin portrait of nonlinear elementary CA rule 18, whose global dynamics is largely determined by a single regular attracting domain. The latter's basin is analyzed in terms of subbasin and portal structures associated with particle annihilation. The conclusion is that the computational complexity of such CA is more apparent than real. Transducer machines are constructed that automatically identify domain and dislocation structures in space-time, count the number of dislocations in a spatial pattern, and implement an isomorphism between rule 18 and rule 90. We use a transducer to trace dislocation trajectories, and confirm that in rule 18, isolated dislocation trajectories, as well as a dislocation gas, agree extremely well with the classical model of annihilating diffusive particles. The CA efficiently transforms randomness of an initial pattern ensemble into a random walk of dislocations in space-time.     

Hinnuliquinone,a bis-indolyl-2,5-dihydroxybenzoquinone pigment from nodulisphorium hinnuleum

A pigment, hinnuliquinone, isolated from the fungus $\text{Nodulisporium}$$\text{hinnuleum}$, has been shown to be 2,5-bis-[2-(1,1,-dimethyl-2-propenyl)1H-indol-3-yl-]-3,6-dihydroxy-2,5-cyclohexadiene-1,4-dione and to be biosynthesized from tryptophan and mevalonic acid.     

Analysis of a model for transport of charged particles in a random magnetic field

A model for the transport of charged particles in a random magnetic field is a Volterra integrodifferential equation with a long-range kernel. The integrodifferential equation is solved numerically with the method of Bellman, Kalaba, and Lockett (“Numerical Inversion of the Laplace Transform,” Elsevier, New York, 1966). The results are shown to be in excellent agreement with analytical asymptotic results.     

The relativistic spherical top

The classical theory of the free relativistic spherical top is first developed from a Lagrangian viewpoint. Our method allows the invariant mass to be an arbitrary function of the intrinsic spin. A canonical formalism is established following the approach suggested by Dirac for constrained Hamiltonian systems. There is a second arbitrary function in the theory, in addition to the usual one due to reparametrization invariance. The usual Newton-Wigner variables are supplemented by the Euler angles. The quantum theory of the free top is discussed. The classical theory is generalized to included charged tops with magnetic moments.     

A two-dimensional cationic lattice built from [Zn6(HPO4)2(PO4)2]2+ clusters

A unique cationic zinc phosphate cluster linked by neutral bifunctional rigid ligands to form a two dimensional framework was synthesized and structurally characterized.