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Understanding the consequences of the noncrossing constraint is one of the remaining challenges in the physics of walks and polymers. To address this problem, we performed molecular simulations for the separation of only two initially connected, overlapping polymer chains with interactions tuned such that they are nearly random walks. The separation time for a configuration strongly correlates with the number of monomer contacts between both chains. We obtain a broad distribution of separation times with a slowly decaying tail. Knots only play a role for those configurations that contribute to the tail of the distribution. In contrast, when starting from the same initial configuration but allowing for chain crossings, separation is qualitatively faster and the time distribution narrow. The simulation results are rationalized by analytical theory. A theory of contacts based on polymer fractality and criticality is presented, along with the expected effects of knots.

Colossal Magnetoresistance (CMR) in nonmagnetic semiconductors and magnetic materials has been investigated as a function of magnetic field, charge carriers concentration and temperature. Both types of materials demonstrated qualitative and quantitative coincidence of CMR dependence on magnetic field, charge carriers concentration and temperature. The findings support the CMR interpretation in the framework of the Excitonic Insulator (EI) model and transition of an insulating EI phase to conducting state under magnetic field for all types of materials under study. It is suggested that Jahn-Teller distortion caused by magnetic ions and external uniaxial stress could initiate EI phase formation

We study the collapse dynamics of a polyelectrolyte (PE) chain upon the change of solvent condition induced by a change of pH or solvent quality. The kinetics of a PE collapse differs from that of a neutral polymer because its initial conformation is naturally stretched due to the repulsive electrostatic interactions. If the solvent change leads to a complete neutralization of charge, the final configuration is one spherical globule (pearl). The chain conformation goes through an intermediate collinear conformation of pearls connected with strings. The dynamics is controlled by the motion of the two terminal pearls growing at the expense of small pearls in between. A slow coarse graining of other pearls in the middle does not affect the large-scale dynamics. The estimated total time for the collapse is tq = 0 ~ N4/3. In the case when the polymer remains charged after the change of solvent, the final structure is a stretched string of pearls. The collapse is a two-stage process: 1) relaxation to the stretched structure of pearls which are 2/5 of the equilibrium size, with a characteristic time tq ≠ 0 ~ N2 and 2) equilibration of pearl size and number, which requires activation to overcome energy barriers. The estimated activation energy barrier favorably agrees with the computer simulations by Limbach et al. (Europhys. Lett.60 (2002) 566).

We study different regimes in the dynamics of swelling of a long polymer chain from a compact globule. Depending on the age or degree of self-entanglement inside of the globule, we consider the limiting cases of “freshly formed” globule and “old” globule. Considering the density of polymer in a globule, which influences the mobility of solvent inside of the globule, we also distinguish “wet” and “dry” initial configurations. For each of these regimes, we discuss the leading dissipation mechanism governing the dynamics of swelling. Our theoretical predictions are nicely corroborated by large-scale molecular dynamics simulations.

We present a scaling theory describing the collapse of a homopolymer chain in poor solvent. At time t after the beginning of the collapse, the original Gaussian chain of length N is streamlined to form N/g segments of length R(t), each containing g ~ t monomers. These segments are statistical quantities representing cylinders of length R ~ t^{1/2} and diameter d ~ t^{1/4}, but structured out of stretched arrays of spherical globules. This prescription incorporates the capillary instability. We compare the time-dependent structure factor derived for our theory with that obtained from ultra-large-scale molecular dynamics simulation with explicit solvent. This is the first time such a detailed comparison of theoretical and simulation predictions of collapsing chain structure has been attempted. The favorable agreement between the theoretical and computed structure factors supports the picture of the coarse-graining process during polymer collapse.

We study the evolution of an elastic string, serving as model for a domain wall, into the pinned state at driving forces slightly below the depinning threshold force F c . We quantify the temporal evolution of the string by an activity function A ( t ) representing the fraction of active nodes at time t and find three distinct dynamic regimes. There is an initial stage of fast decay of the activity; in the second, intermediate, regime, an exponential decay of activity is observed; and, eventually, the fast collapse of the string towards its final pinned state results in decay in the activity with A r ∼( t p − t ) ψ , where t p is the pinning time in the finite system involved.

In dilute solutions of rodlike polyelectrolytes some counterions are distributed far from polyions while others are located in their vicinity in the regions of cylindrical symmetry of the electrostatic potential. For these cylindrical regions around rodlike polyelectrolytes we find an exact solution of the nonlinear Poisson-Boltzmann equation for the case of nonzero net charge in these regions. This exact solution implies three qualitatively different phases of counterion distribution around the polyions with second order phase transitions between these phases.

We study the configurational statistics of a ring polyampholyte chain made of N randomly charged monomers with elementary charge .To a large extent, the overall structure of a polyampholyte is controlled by a total sum Q of all charges. When the total charge is smaller than , the polyampholyte has a compact globular structure. At charges larger than , the configuration has the form of a ring of small globules (beads) connected by strings. Between Q1 and Q2 we find a remarkable diversity of meta-stable configurations having the shapes of irregular clusters of small globules connected by the strings. We estimate the number of these configurations and the energy barriers between them. Between Q1 and Q2, the minimum energy configurations are completely controlled by randomness in the charge distribution along the chain. There are hysteresis effects in the shapes of the clusters. As the total charge increases, the linearly extended configurations become dynamically more preferable. When the charge decreases, the circular shapes are preferred. We remark on the probable connection with the multiple phase transitions observed in polyampholyte gels.

We present a simple minimal model which allows numerical and analytical study of a glass transition. This is a model of rigid rods with fixed centers of rotation. The rods can rotate freely but cannot cross each other. The ratio L of the length of the rods to the distance between the centers of rotation is the only parameter of this model. With increasing L we observed a sharp crossover to practically infinite relaxation times in 2D arrays of rods. In 3D we found a real transition to a completely frozen random state at $L_{\rm c}\cong 4.5$.

We study the configurational and dynamic properties of a long polymer chain during the process of penetration of the chain through the thin and porous membrane. The mobility of a polymer in this state depends strongly on the topological restrictions imposed by the membrane. The largest mobility have configurations in which the polymer chain crosses the membrane only once and diffuses through a single pore from one side of the membrane to the other. This configuration may have small statistical weight compared to other configurations with low mobility in which the polymer chain crosses the membrane many times. The relative weight of these configurations depends also on the chemical potential of the membrane. We find the configurations which give the main contribution to the transport and calculate the overall permeability of the membrane for Zimm and Rouse mechanisms of polymer dynamics.

We develop a scaling model for the dilute solution conformation of a uniformly charged polymer in a poor solvent. We find that there is a range of temperatures and charge densities for which the polymer has a necklace-like shape with compact beads joined by narrow strings. The free energy of a polyelectrolyte in this conformation is lower than in a cylindrical globule because the length of the necklace is larger than that of a cylinder and is proportional to the total charge on the chain. With changing charge on the chain or temperature, the polyelectrolyte undergoes a cascade of abrupt transitions between necklaces with different numbers of beads.

We present a simple scaling theory of the dynamics of a ring polymer in a gel. In the absence of excluded volume interactions, we predict that the translational diffusion coefficient D varies with the molecular weight M as D approximately M-2 and the longest relaxation time T scales as T approximately M5/2. The results of numerical simulations support these findings.

We discuss the elastic modulus G and swelling/deswelling behavior of networks as a function of their concentration 4 and their preparation state. Simple scaling ideas reproduce the prediction of James and Guth for the modulus of networks swollen in a 8 solvent (G - 4113) but lead to a new prediction in a good solvent (G - 4'/12). We also suggest that both fully swollen modulus and dry modulus are related to swelling in ways that are independent of the network preparation details. By cross-linking long chains at low concentration and removing the solvent, many temporary entanglements are formed that force each network strand into a double-folded treelike compact configuration. These deswollen networks are capable of stretching by enormous amounta (-1Oox) without breaking (superelastic) and have a much lower modulus than melt-cross-linked networks. The energy stored in temporary entanglements during deswelling is released upon stretching, leading to a weaker-than-linear dependence of stress u on elongation X in tension (a - All3).

It has been assumed until very recently that all long-range correlations are screened in three-dimensional melts of linear homopolymers on distances beyond the correlation length $\xi$ characterizing the decay of the density fluctuations. Summarizing simulation results obtained by means of a variant of the bond-fluctuation model with finite monomer excluded volume interactions and topology violating local and global Monte Carlo moves, we show that due to an interplay of the chain connectivity and the incompressibility constraint, both static and dynamical correlations arise on distances $r \gg \xi$. These correlations are scale-free and, surprisingly, do not depend explicitly on the compressibility of the solution. Both monodisperse and (essentially) Flory-distributed equilibrium polymers are considered., Comment: 60 pages, 49 figures

We present a simple description of configuration and dynamical properties of megabase DNA undergoing gel electrophoresis. The molecule moves in the field in the shape of a self-similar tree-like structure. The electrophoretic mobility of DNA in this regime is independent of its molecular weight.

Configurational (geometrical) averaging over all possible configurations is often used in the calculation of properties of polymer chains and branched structures. The author shows that for constrained systems more general consideration is needed which includes averaging over the dynamical degrees of freedom. As an example, the author has found nontrivial angle distribution functions for the simplest models of rigid monomers freely joined with each other. It is shown that a Flory-Stockmayer and percolation approach can be inapplicable for the description of the equilibrium solution of polymers.

We consider stress relaxation in a strongly segregated lamellar mesophase, where block copolymers are in the "brush" state with junction points confined to the interface between the adjacent lamellae and blocks stretched out away from it. If the molecular weight of the blocks is large enough, they entangle with their neighbors as well as with blocks from the opposite brush. The number of entanglements of one particular chain with the opposite brush varies from chain to chain even in the monodisperse system. We demonstrate that this dispersion of the number of entanglements leads to a very broad spectrum of relaxation times and to an effectively power-law-like stress relaxation function log G - (log t)" with CY = 1/2 (a = 2) in the strong (weak) chain-stretching limit. We analyze various disentanglement mechanisms for diblocks and triblocks and the onset of the diffusion of copolymers along the interface. Another relaxation mechanism is due to the displacement of a block across the interface into the "enemy" phase. We conclude that for highly entangled copolymers it will be an important mode of stress relaxation (especially for triblock copolymers). For asymptotically long times the relaxation of stress along the perfectly ordered lamella will be liquidlike. In a system with defects in domain structure the relaxation of stress is controlled by the processes of equilibration of excess density along the layers. For the lamellar mesophase we found G(t) - t 4. 2, while for the cylindrical mesophase, G(t) - t

A simple phenomenological approach is described for the superconducting transition in thin films and S-N superlattice structures. The principal ingredients in an approach based on the Ginzburg-Landau free energy are (a) a pairbreaking energy at the surface (or interface) and (b) a proximity-induced free energy for the order parameter in the normal phase. The shift in T c is found to be inversely proportional to the superconductor thickness in a moderate range of thicknesses.

Cisplatin has been known as an anticancer drug for a long time. It is therapeutically active upon binding to DNA. A double-bound cisplatin bends DNA into a localized kink. We model the elastic properties of cisplatin-DNA adducts at moderate tension (

Angular correlations in dense solutions and melts of flexible polymer chains are investigated with respect to the distance $r$ between the bonds by comparing quantitative predictions of perturbation calculations with numerical data obtained by Monte Carlo simulation of the bond-fluctuation model. We consider both monodisperse systems and grand-canonical (Flory-distributed) equilibrium polymers. Density effects are discussed as well as finite chain length corrections. The intrachain bond-bond correlation function $P(r)$ is shown to decay as $P(r) \sim 1/r^3$ for $\xi \ll r \ll \r^*$ with $\xi$ being the screening length of the density fluctuations and $r^* \sim N^{1/3}$ a novel length scale increasing slowly with (mean) chain length $N$., Comment: 17 pages, 5 figures, accepted for publication at Macromolecules

On the basis of recent experimental results obtained using a model biological system, we argue that well-known physical phenomena like wetting and percolation may play a significant role in morphogenetic changes which occur during development. We also present results of scanning microscopy studies on the structure of the system, which support our theoretical conclusions.

Cisplatin was incidentally discovered to suppress cell division and became one of the most successful antitumor drugs. It is therapeutically active upon binding to DNA and locally kinking it. We demonstrate that after a bimodal modeling, the degree of platination of a single DNA molecule can be consistently and reliably estimated from elasticity measurements performed with magnetic tweezers. We predicted and measured for the first time two separate persistence lengths of kinked DNA at high and low tensions. We also directly observed that the degree of platination of DNA strongly depends on the concentration of sodium chloride as required for cisplatin's intracellular activity. Our study shows that micromanipulation techniques accurately reveal the degree of chemical modification of DNA which can be used for a new type of structure-sensitive biosensors.

Non-classical behaviour, brought about by a confinement that imposes spatial constraints on molecules, is opening avenues to novel applications. For example, carbon nanotubes, which show rapid and selective transport of small molecules across the nanotubes, have significant potential as biological or chemical separation materials for organic solvents or gaseous molecules1,2,3,4,5. With polymers, when the dimensions of a confining volume are much less than the radius of gyration, a quantitative understanding of perturbations to chain dynamics due to geometric constraints remains a challenge6,7,8,9,10 and, with the development of nanofabrication processes, the dynamics of confined polymers have significant technological implications11,12,13,14,15,16,17. Here, we describe a weak molecular-weight-dependent mobility of polymers confined within nanoscopic cylindrical pores having diameters smaller than the dimension of the chains in the bulk. On the basis of the chain configuration along the pore axis, the measured mobility of polymers in the confined geometry is much higher than the mobility of the unconfined chain. With the emergence of nanofabrication processes based on polymer flow, the unexpected enhancement in flow and reduction in intermolecular entanglements are of significant importance in the design and execution of processing strategies.

We discuss theoretically and numerically the intramolecular form factor $F(q)$ in dense polymer systems. Following Flory's ideality hypothesis, chains in the melt adopt Gaussian configurations and their form factor is supposed to be given by Debye's formula. At striking variance to this, we obtain noticeable (up to 20%) non-monotonic deviations which can be traced back to the incompressibility of dense polymer solutions beyond a local scale. The Kratky plot ($q^2F(q)$ {\it vs.} wavevector $q$) does not exhibit the plateau expected for Gaussian chains in the intermediate $q$-range. One rather finds a significant decrease according to the correction $\delta(F^{-1}(q)) = q^3/32\rho$ that only depends on the concentration $\rho$ of the solution, but neither on the persistence length or the interaction strength., Comment: 29 pages, 11 figures, submitted to Macromolecules

It is well known that small neutral particles normally tend to aggregate due to the van der Waals forces. We discover a new universal long-range interaction between solid objects in polymer media that is directly opposite the van der Waals attraction. The new force could reverse the sign of the net interaction, possibly leading to the net repulsion. This universal repulsion comes from the subtracted soft fluctuation modes, which are not present in the real polymer system, but rather are in its ideal counterpart. The predicted effect has a deep relation to the classical Casimir interactions, providing an unusual example of fluctuation-induced repulsion instead of normal attraction. That is why it is referred to as the anti-Casimir effect. We also find that the correlation function of monomer units in a concentrated solution of infinite polymer chains follows a power-law rather than an exponential decay at large distances.

We present a theory for the reverse analysis on the sequence information of a single H/P two-letter random hetero-polymer (RHP) from its force-extension(f-z) curves during quasi static stretching. Upon stretching of a self-assembled RHP, it undergoes several structural transitions. The typical elastic response of a hetero-polymeric globule is a set of overlapping saw-tooth patterns. With consideration of the height and the position of the overlapping saw-tooth shape, we analyze the possibility of extracting the binding energies of the internal domains and the corresponding block sizes of the contributing conformations., Comment: 5 figures 7 pages

Upon aging, a collapsed long chain evolves from a crumpled state to a self-entangled globule which can be thought of as a large knot. Swelling of an equilibrium globule in good solvent is a two-step process: (i) fast swelling into an arrested stretched structure with conserved entanglement topology followed by (ii) slow disentanglement. Using computer simulation, we found both mass-mass (m-m) and entanglement-entanglement (e-e) power law correlations inside the swollen globule. The m-m correlations are characterized by a set of two exponents in agreement with a Flory-type argument. The e-e correlations are also characterized by two exponents, both of them larger (by $\ensuremath{\sim}0.3$) than the related m-m exponents. We interpret this difference as evidence of distance-dependent repulsion $E=\ensuremath{-}0.3\mathrm{ln} (\ensuremath{\rho}){k}_{B}T$ between entanglements sliding along the polymer chain.

Usually polyelectrolyte solutions present a peak in the polyion/polyion structure factor for some value of the momentum transfer (scattering vector). We describe a new regime in dense polymer solutions, typically above , where is proportional to the concentration c of the solution. It is however not observed for all polyelectrolytes but only for those of intermediate stiffness, usually together with an ultimate regime where is proportional to and local orientational order prevails. This regime is robust against sample preparation and stable over time. We can understand it as jammed with randomly oriented persistent segments. This may explain why fully developed nematic order is so difficult to observe in synthetic polyelectrolytes.

Melts of unconcatenated and unknotted polymer rings are a paradigm for soft matter ruled by topological interactions. We propose a description of a system of rings of length N as a collection of smaller polydisperse Gaussian loops, ranging from the entanglement length to the skeleton ring length , assembled in random trees. Individual rings in the melt are predicted to be marginally compact with a mean square radius of gyration . As a rule, simple power laws for asymptotically long rings come with sluggish crossovers. Experiments and computer simulations merely deal with crossover regimes typically extending to . The estimated crossover functions allow for a satisfactory fit of simulation data.

Recent computational studies on melts of nonconcatenated rings suggest compact configurations of fractal dimension d f = 3. This begs the question of whether the irregular surfaces of these compact rings may be characterized by a fractal surface dimension d s < 3. We revisit the scaling analysis of the form factor by Halverson et al. [J. Chem. Phys.134, 204904 (2011)] implying d s ≈ 2.8. Our analysis suggests that this conclusion might be due to the application of the Generalized Porod Law at large wavevectors where length scales other than the total chain size do matter. We present an alternative “decorated Gaussian loop” model which does not require d s < 3.

We present a new model for the motion of a megabase-long DNA molecule undergoing gel electrophoresis. We assume that the dynamics of large segments of DNA is almost deterministic and can be described by a set of simple mechanical equations. This allows the numerical study of gel electrophoresis of ultra-high molecular weight DNA. A strong electric field forces DNA in a gel into a tree-like structure with branches-loops of different sizes. We determined the loop-size distribution function. This distribution has a power law form, confirming the hypothesis of the statistical self-similarity of a moving polymer. We find periodic configuration changes in the motion of a circular polymer, with the average period proportional to the molecular weight. During the period, a polymer goes through three distinct phases: a simple V-shape configuration, a growing tree, and a decaying tree. For a linear polymer this periodicity is much less pronounced because of additional perturbations to the dynamics caused by free ends. A circular polymer stays in a simple V-shaped configuration about 30% of the time, independent of molecular weight (10% for a linear polymer).

BACKGROUND Pineal region meningiomas are extremely rare tumors and comprise about 8% of tumors of this region. Two cases of large pineal region meningiomas in young males are presented. METHODS Computed tomography (CT) scan and cerebral angiography were used to evaluate the patients preoperatively. Both patients were operated on through an occipital transtentorial approach either as a single- or two-stage procedure. RESULTS Preoperative work-up revealed that both tumors received their main blood supply from the posterior lateral choroidal arteries and were therefore originating from the velum interpositum cerebri (superior tela choroidea). Complete tumor removal was effected in the first patient using a right occipital transtentorial approach. In the second patient, after partial removal using the same approach, complete resection was achieved at a second stage via the left occipital transtentorial approach 3 months later. CONCLUSION Cerebral angiography, which must include vertebral angiography, is important in the preoperative surgical planning of these tumors. The occipital transtentorial approach provides good access. However, a two-stage procedure using right and left sides may be necessary for large meningiomas receiving a bilateral blood supply from both posterior lateral choroidal arteries. Diagnosis, the surgical approach, and methods of tumor excision are discussed.

Corrections to chain ideality have been demonstrated recently for polymer melts in the bulk and in ultrathin films. It has been shown that the effect of incomplete screening is stronger in the latter. We show here that the deviation from ideality is even stronger in thin capillaries. Describing the crossover from the free bulk to the confined regime as the radius of the capillary decreases below the typical coil radius we make connection to the so far disconnected work by Brochard and de Gennes (J. Phys. (Paris), Lett., 40 (1979) 399) predicting chain segregation in very thin capillaries. Due to the generalized Porod scattering of the segregated chains, the Kratky representation of the intrachain structure factor reveals a plateau for all regimes although the chains become swollen with increasing confinement.

A chain reptating through a high molecular weight polymer melt leaves a trace in the form of elastic distortions of an entanglement network. Neighboring chains can partially relax these elastic distortions by following this trace with some accuracy. Thus the memory of chain configuration is reproduced by other chains when the first one leaves the tube after time τ dif . The number of diffusion steps of duration τ dif required to completely lose the memory of chain configuration grows with the molecular weight M of the chain as τ relax /τ dif ∼M 1/3 in agreement with experiments, where Tτ relax is the relaxation time of the melt

Capillary rise of polymer in thin tubes ≈40 nm in diameter (comparable to the size of individual polymer molecule) shows anomalous molecular-weight dependence (Shin K. et al., Nat. Mater., 6 (2007) 961). We suggest that this dependence reveals breakdown of macroscopic hydrodynamics characterized by the bulk viscosity and propose a new microscopic mechanism for transport along a thin capillary based on the reptation model. The crossover capillary diameter from microscopic to macroscopic flow D*≈70 nm for 105 molecular-weight polyethylene and is proportional to the molecular weight. We discuss how a new transport mechanism might affect boundary conditions in a capillary when undulations of the walls prevent the slipage of the flow. We discuss the relevance of nonlinear effects and show, that they might be observed at the very onset of capillary rise flow.

A number of experiments1–3 have reported measurements of the transition temperature and other properties of superlattices made from high Tc superconductors. These superlattices consist of M layers of YBaCu3O7 (Tc = 90 K), separated by N layers of PrBaCu3O7, which is a semiconductor. They find3 that Tc(M,N) for small M decreases with N and then saturates to an N independent value. It increases with increasing M for all N. For large N, when the superconducting layers can be considered independent, the transition temperature Tc(M) increases linearly with M and saturates to the bulk Tc.

It is shown that critical behavior of systems with self-organized criticality can be explained as a Goldstone-mode phenomenon. The nonlinear interaction of the Goldstone mode causes the nontrivial critical exponents. Two more models are introduced with self-organized criticality features.

A cornerstone of modern polymer physics is the `Flory ideality hypothesis' which states that a chain in a polymer melt adopts `ideal' random-walk-like conformations. Here we revisit theoretically and numerically this pivotal assumption and demonstrate that there are noticeable deviations from ideality. The deviations come from the interplay of chain connectivity and the incompressibility of the melt, leading to an effective repulsion between chain segments of all sizes $s$. The amplitude of this repulsion increases with decreasing $s$ where chain segments become more and more swollen. We illustrate this swelling by an analysis of the form factor $F(q)$, i.e. the scattered intensity at wavevector $q$ resulting from intramolecular interferences of a chain. A `Kratky plot' of $q^2F(q)$ {\em vs.} $q$ does not exhibit the plateau for intermediate wavevectors characteristic of ideal chains. One rather finds a conspicuous depression of the plateau, $\delta(F^{-1}(q)) = |q|^3/32\rho$, which increases with $q$ and only depends on the monomer density $\rho$., Comment: 4 pages, 4 figures, EPL, accepted January 2007

A generalization of the percolation problem is proposed for disordered systems without a centre of inversion. Critical exponents are calculated in 5-ϵ dimensions. The structure of the infinite directed cluster is discussed.

Arguments are given that a full description of any realistic model of randomly distributed conducting sticks in an insulating matrix must incorporate the history of system preparation. Different regimes of composite formation produce different dependences of the percolation threshold value on the width-to-length ratio in : rho c approximately in 0, rho c approximately in 1 or even rho c approximately in 2.

It is noted that the 2D self-avoiding walk problem admits the introduction of two pseudoscalar changes. These charges describe the left-right asymmetry of the probability of the next step near the points of possible selfintersections. The renorm-group equations are derived for the most general case of five charges. It is shown that the pseudoscalar charges lead to the absence of stable fixed points.

Gelation during the branched radical post-polymerization of a mixture of bi- and tetra-functional monomers is considered. Depending on the ratio of the concentrations of bi- and tetrafunctional monomers and the initiator the gel is formed either on contact of growing coils of linear macromolecules or as a result of significant mutual overlap of the growing macromolecules. There thus exist two different conditions of gel formation, indicating two different types of MMD near the gel point.

It is shown that the renormalisation of the 'true' self-avoiding walk of Amit et al. (1982) is its upper critical dimensionality two involves at least two (possibly three) coupling constants. Fixed points to first order in epsilon =2-D are identified and the asymptotic behaviour at D=2 is discussed.

The Flory approximation (1969) for the self-avoiding chain problem is compared with a conventional perturbation theory expansion. While in perturbation theory each term is averaged over the unperturbed set of configurations, the Flory approximation is equivalent to the perturbation theory with the averaging over the stretched set of configurations. This imposes restrictions on the integration domain in higher-order terms and they can be treated self consistently. The accuracy delta nu / nu of the Flory approximation for self-avoiding chain problems is estimated to be 10-1-10-2 for 1

Random Boolean networks on nearest-neighbour d-dimensional lattices are argued to belong to the universality class of directed percolation with quenched disorder in d+1 dimensions. Hansen's computer simulations for d=4 are thus interpreted as being at the upper critical dimension.

It is shown that the root-mean-square distance R2(12)/ for the self-avoiding walk problem is related to the number of steps N as R2 approximately N2 nu with nu =2/3 rather than the 'self-obvious' nu =1. This coincides with the mean-field theory result obtained by Pietronero (1983) which erroneously was not extended to the case d=1.