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Single-chain simulations of densely branched comb polymers, or "molecular bottle-brushes" with side-chains attached to every (or every second) backbone monomer, were carried out by off-lattice Monte Carlo technique. A coarse-grained model, described by hard spheres connected by harmonic springs, was employed. Backbone lengths of up to 100 units were considered, and compared with the corresponding linear chains. The backbone molecular size was investigated as a function of its length at fixed arm size, and as a function of the arm size at fixed backbone length. The apparent swelling exponents obtained by a power-law fit were found to be larger than those for the corresponding linear polymers, indicative of stiffening of the comb backbone. The probability distribution function for the backbone end-to-end distance was also investigated for different backbone lengths and arm sizes. Analysis of this function yielded the critical exponents, which revealed an increase in the swelling exponent consistent with values found from the molecular size. The apparent persistence length of the backbone was also determined, and was found to increase with increasing branching density. Finally, the static structure factors of the whole bottle-brushes and of their backbones are discussed, which provides another consistent estimate of the swelling exponents.

ForewordThis paper presents the analysis of a periodically fbrced second order nonlinear dynamical system describing predator-prey communities. Six different seasonality mechanisms are identified and compared in terms of bifurcation diagrams. The analysis is carried out by means of an interactive package which detects Hopf, flip and fold bifurcations curves as well as codimension two bifurcation points. The results are in agreement with the general theory of periodically perturbed Hopf bifurcations. This work shows that complex environmental issues can be highlighted by suitably combining basic results of nonlinear system theory and powerful numerical techniques. Moreover, the two classical routes to chaos, namely, torus destruction and cascade of period doublings, are numerically detected. Since in the case of constant parameters the model cannot have multiple attractors, catastrophes, and chaos, the results support the conjecture that seasons can very easily give rise to complex population dynamics.

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Single-chain Monte Carlo simulations were carried out, in continuous space, of polymers with various topologies (branched and linear) in the good solvent. Using an inherently flexible beadand-spring model, the backbone of linear polymers with either linear or dendritic side-groups attached was found to be elongated, indicative of an induced stiffness. This "topological stiffness" was compared to the "intrinsic stiffness" of semiflexible linear polymers in terms of various observables. Semiflexible comb polymers, which contained both types of stiffness, were also considered.

We study conformations assumed by single diblock star copolymers in a poor solvent by means of the Gaussian variational theory and Monte Carlo simulation in continuous space. Cases of stars with internal and external hydrophobic blocks are analysed. While in the former case the collapsed state has an obvious micellar shape, the latter case exhibits two nontrivial conformational structures. Apart from the equilibrium state of a globular hydrophobic core with hydrophilic daisy loops, one also finds here a metastable state of outstretched hydrophilic blocks with hydrophobic subglobules at their ends. Such a state appears to be rather long‐lived during the kinetics of collapse of a swollen star. The plots of monomer densities and other observables computed by means of both techniques are found to be in good agreement with each other.

New mimetic discretizations of diffusion-type equations (for instance, equations modeling single phase Darcy flow in porous media) on unstructured polygonal meshes are derived. The first order convergence rate for the fluid velocity and the second-order convergence rate for the pressure on polygonal, locally refined and non-matching meshes are demonstrated with numerical experiments.

We present results of Monte Carlo study of the monomer-monomer correlation functions, static structure factor and asphericity characteristics of a single homopolymer in the coil and globular states for three distinct architectures of the chain: ring, open and star. To rationalise the results we introduce the dimensionless correlation functions rescaled via the corresponding mean-squared distances between monomers. For flexible chains with some architectures these functions exhibit a large degree of universality by falling onto a single or several distinct master curves. In the repulsive regime, where a stretched exponential times a power law form (de Cloizeaux scaling) can be applied, the corresponding exponents δ and θ have been obtained. The exponent δ = 1/ν is found to be universal for flexible strongly repulsive coils and in agreement with the theoretical prediction from improved higher-order Borel-resummed renormalisation group calculations. The short-distance exponents θυ of an open flexible chain are in a good agreement with the theoretical predictions in the strongly repulsive regime also. However, increasing the Kuhn length in relation to the monomer size leads to their fast cross-over towards the Gaussian behaviour. Likewise, a strong sensitivity of various exponents θij on the stiffness of the chain, or on the number of arms in star polymers, is observed. The correlation functions in the globular state are found to have a more complicated oscillating behaviour and their degree of universality has been reviewed. Average shapes of the polymers in terms of the asphericity characteristics, as well as the universal behaviour in the static structure factors, have been also investigated.PACS numbers: 36.20.Ey, 61.25.Hq