Polymerizable derivatives of long-chain fatty acids. IV. Vinyl esters

Contrary to some published reports, the vinyl esters of saturated fatty acids polymerize readily and rapidly. Vinyl oleate, when present in excess of 5%, and oxygen exert marked retarding effects. Techniques are described for the free-radical-initiated polymerization of the vinyl esters of caprylic, capric, lauric, myristic, palmitic, and stearic acids in bulk, dispersion, solution, and emulsion. Some data are given for polymerization in the presence of chain-transfer agents, such as carbon tetrachloride, dodecylmercaptan, and ethylbenzene. Conditions are reported for obtaining degrees of polymerization from about 2 (when chain-transfer agents are employed) to 10,000 (weight average). The weight average degree of polymerization increases markedly as the conversion increases, particularly above 80%. Even up to extremely high conversions, soluble polymers are obtained in most cases. Solubility characteristics, transition point data, molecular weights (osmometric and light-scattering), and isolation and purification techniques are also reported.     

Invariants of knot diagrams

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams. J. Hass was partially supported by an NSF grant.     

OPTIMAL FISH HARVESTING FOR A POPULATION MODELED BY A NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATION

As the human population continues to grow, there is a need for better management of our natural resources in order for our planet to be able to produce enough to sustain us. One important resource we must consider is marine fish populations. We use the tool of optimal control to investigate harvesting strategies for maximizing yield of a fish population in a heterogeneous, finite domain. We determine whether these solutions include no‐take marine reserves as part of the optimal solution. The fishery stock is modeled using a nonlinear, parabolic partial differential equation with logistic growth, movement by diffusion and advection, and with Robin boundary conditions. The objective for the problem is to find the harvest rate that maximizes the discounted yield. Optimal harvesting strategies are found numerically.     

Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions

Algorithms are developed, based on topological principles, to evaluate the boundary and “internal structure” of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree   to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the “internal” envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in R³${\mathbb{R}}^3$, and other forms of geometrical convolution, are briefly discussed.     

Nuclear polarization and signs of quadrupole moments of high-spin Gd isomers

Time-differential perturbed-angular-distribution (TDPAD) experiments have been performed on polarized intermediate- and high-spin isomers ${}^{144}\text{Gd}\text{(10}{}^{\mathrm{+}}\text{)}\text{and}{}^{147}\text{Gd}\text{(}\text{13}\text{2}{}^{\mathrm{+}}\text{,}\text{27}\text{2}{}^{\mathrm{?}}\text{,}\text{49}\text{2}{}^{\mathrm{+}}\text{)}$. Excited Gd nuclei populated by $\text{(}{}^{28}\text{Si}\text{, x}\text{n}\text{)}$ reactions were polarized by the tilted-multifoil technique and implanted in a Gd single-crystal host. Observation of the subsequent electric quadrupole interaction with the known electric field gradient yielded negative values of the sign of the deformation for all isomers studied. The magnitude of the induced nuclear polarization P_I was also determined from the TDPAD data and compared to model calculations in order to deduce the average atomic angular momentum and polarization of Gd ions at v/c ～ 0.018.     

Nuclear polarization of high-spin levels and the sign of the quadrupole moment of 54Fe(10+)

The determination of the sign of the quadrupole deformation at high spins requires an observation of the electric quadrupole interaction of polarized isomers in single crystals of non-cubic metallic hosts. The ⁵⁴Fe(10⁺) isomer was polarized, subsequent to its population by the (¹²C, p2n) reaction, by passage through an array of tilted carbon foils. The isomers were then recoil implanted into single crystals of zinc and cadmium and the tune differential modulations of the angular distribution of decay γ-rays were observed. Nuclear polarization values of P_I = 0.08(3)?0.18(5) were found for 13–17 polarizing foils, respectively, and a positive sign of the quadrupole moment was deduced. An improved value was established for the quadrupole moment: $\text{Q[}{}^{54}\text{Fe}\text{(10}{}^{\mathrm{+}}\text{)] = + 29.7(4) e ·}\text{fm}{}^2$, in agreement with current shell-model predictions.     

The Size of Spanning Disks for Polygonal Curves

Abstract. For each integer n ≥ 0 , there is a closed, unknotted, polygonal curve K _n in R ³ having less than 10n+9 edges, with the property that any piecewise-linear triangulated disk spanning the curve contains at least 2 ^n-1 triangles.     

The unimolecular and collisionally activated decompositions of fluorine-substituted tetraphenyltetrahedrane ions on the microsecond time-scale

Unimolecular and collision-induced decomposition products of [C₄(C₆H₅)₂(C₆H₄F)₂]⁺˙ generated from four unsymmetrical sources include [C₁₄H₁₀]⁺˙ and [C₁₄H₈F₂]⁺˙ and so provide evidence for a tetrahedral intermediate. Other decompositions show substantial influence of the position of the ρ-fluorophenyl ring on ion energy distributions. This influence may be related to the reported absence of peaks diagnostic for the tetrahedral intermediate from the spectrum of the equivalent ion from the appropriate ¹³C-labeled analog. Alternatively the difference in spectra can be correlated with lifetimes of ions.