Combinatorial Calculations Research Articles

The origin of the ultrametric topology of spin glasses

The author discusses the N to 0 limit of a variety of field theories, in particular those describing spin glasses. He shows how the convergence is naturally described using a p-adic norm. Replica symmetry breaking and combinatorial calculations in these theories make sense mathematically in the p-adic regime. The p-adic order parameter is viewed naturally as a topological central extension related to holonomy (like Berry's phase) from the quenched frustration in spin glasses.

Combinatorial calculations of topological defects in tetrafunctional polymer networks

The structure of real polymer networks admits many kinds of topological defects related to the way in which polymer chains are connected to crosslinks. The fractions of crosslink types have been derived from combinatorial considerations concerning the probabilities of various combinations of “structural elements” of crosslinks. 34 fractions of different crosslinks, possible in tetrafunctional systems, have been obtained as explicit functions of concentrations of seven structural elements — free-end chains, single-, double-, triple-, and quadruple-connected chains, loops, and unsaturated (void) functionalities. It was shown that different distributions (and different properties) result from end-to-end linking of precursor chains vs. vulcanization of long chains by bridging along their contours.

Realistic few-quasiparticle level densities in spherical nuclei.

State and level densities with fixed exciton numbers were estimated using combinatorial calculations. Shell-model orbitals were used as a basis from which all possible configurations were generated. The pairing interaction was taken into account by applying the BCS theory to each configuration. The spin and parity distributions were obtained. The role of the pairing interaction was understood and a method of treating pairing effects in practical reaction calculations is given. The spin and parity distributions were obtained and analyzed. The assumption of a Gaussian distribution for the spin projections was found to be valid for states containing at least four quasiparticles. The energy, spin, and parity distributions of states with low exciton numbers reveal strong fluctuations which are not predicted by closed formulae derived from the equidistant single-particle-level model. Effects of the restriction limiting configurations to only bound orbitals are discussed. The restriction decreases the possible spins of highly excited ``bound'' states compared to those of the unrestricted states.

Codes for the combinatorial calculation of few quasiparticle state densities in spherical and deformed nuclei

The codes for combinatorial calculation of few quasiparticle state and level densities are presented. The calculations are performed in the space of the shell-model single particle states and include pairing interaction explicitly. The spin and parity distributions of quasiparticle states are also provided. The calculations can be restricted to the states with all quasiparticles being bound.

Spin distribution of exciton levels for spherical nuclei

The spin distribution of levels with low excitation number was investigated by means of combinatorial calculations in the framework of a superconductivity model. In particular, the energy and exciton number dependence of spin cut-off par a meter and yrast lines were considered.

Combinatorial calculation of characteristic classes

Combinatorial calculation of characteristic classes

Combinatorial calculations of nuclear level densities, spin projections, and nuclear deformation

Calculations similar to those previously used for a spherical nucleus were performed at various deformations ranging in some cases from $\ensuremath{\delta}=\ensuremath{-}0.5$ to $\ensuremath{\delta}=1.0$. The states for each spin projection ($K$) were ordered according to energy to avoid crossing of states with same spin projection and parity. The corresponding energies of the ordered sets for each deformation were considered as the energy vs deformation for particular states. Energies at intermediate deformations were obtained using a cubic interpolation. Energy minima and their associated deformations were determined for each state. States often have more than one energy minimum indicating the presence of shape isomers as illustrated in the cases of $^{166}\mathrm{Ho}$ and $^{240}\mathrm{Pu}$. In the former, a prolate and an oblate minimum exist which persist at higher energies. In the latter, a second prolate minimum is observed where the fission isomer is expected. Differences in energy gaps are discernible in the various minima. A plot of the deformations corresponding to the energy minima of the lowest energy states for each spin projection (the yrast levels) vs the spin projection reveals that the model nucleus becomes more oblate as the spin projection increases. The change in oblateness contains discontinuities. A plot of these yrast deformations vs the yrast energies for various nuclides indicates an apparent independence of the kind or size of the nuclide. The "best" $G$'s (pairing parameters) for some nuclides were determined by fitting the level densities to the experimental values at the neutron binding energies. The results for deformed nuclei are compared with the earlier results obtained for spherical nuclei.[NUCLEAR STRUCTURE $^{40}\mathrm{Ca}$, $^{114}\mathrm{Cd}$, $^{166}\mathrm{Ho}$, $^{240}\mathrm{Pu}$; calculated levels vs spin projection and deformation, yrast levels, deduced pairing parameters. Nilsson levels, BCS pairing.]

The significance of shell corrections in the parameterization of numerical state density calculations

The nuclear state density of a non-interacting fermion system has been calculated directly from a set of single-particle Nilsson levels both as a function of energy and deformation by two numerical methods. The first method is a combinatorial calculation and is based on the repetitive use of recursion relations to expand the grand partition function. The second is based also on the grand partition function, however, in this method one uses the saddle-point approximation to deduce the relevant thermodynamical quantities. Both methods eliminate the unjustifiable assumption of the conventional theory of level densities that the single-particle level scheme can be represented by a continuous distribution. An approximate relationship between the Strutinsky shell correction and the effective zero of excitation energy extrapolated from the nuclear entropy in a region of high temperature has been demonstrated algebraically. A similar relationship is shown to exist numerically for excitation energies in the 40 to 60 MeV region. However, the entropy intercept depends on the temperature and is not equivalent to the Strutinsky shell correction for a non-uniform set of single-particle levels. In the calculations reported, we emphasize the importance of shell structure on the energy-dependent level density. The results of the calculations show that with a realistic set of single-particle orbitals, one can account qualitatively and in some cases quantitatively for the parameters found necessary to fit experimental data. Calculations reproduce qualitatively the observed values of a f a n found in measurements of the ratio of the fission width to neutron emission width and predict also the important role of the liquid-drop fission saddle-point deformation in the interpretation of medium-energy fission-fragment angular distributions.

Statistics of randomly branched polycondensates: Part 2 The application of Lagrange's expansion method to homodisperse fractions

The method of Lagrange's expansion is applied to the bivariate generating function (‘trail-weighting generating function’) previously introduced for combinatorial calculation in polymer science. The wide range of applicability of this method is illustrated by calculation of the moments of the molecular weight distribution function, the Stokes radius and the particle scattering factor of random f-functional branched molecules which are monodisperse (i.e. of fixed molecular weight). The branched molecules in our system are composed of the (random) isomeric mixtures of all possible monodisperse f-functional molecules. This is chemically more realistic than assuming a fixed number of branching units in a molecule, as in Zimm and Stockmayer's, or Kurata and Fukatsu's model. In our model, each branch unit is not necessarily f-functional but the maximum functionality of a branch unit is f, while in an extreme case, the molecule will be linear. Also, the route of the combinatorial calculation is more systematic and appreciably simpler, so that this work helps to clarify the competing effects of branching and polydispersity.

Aspects of Language Design for Combinatorial Computing

Experience in the area of combinatorial computing acquired by the Maniac group at Los Alamos has motivated language development in this direction. Features incorporated into the MADCAP language to aid in programming combinatorial calculations include notation for set synthesis and analysis, notation for variably nested iterations (backtracking) and notation for complex conditional statements. The paper also examines the relationship of such combinatorial language design to machine design. Prominent in this discussion are the bit manipulation (shifting, counting and searching) facilities of Maniac II.