I. Živić

Pattern recognition in damaged neural networks

We have studied the effect of various kinds of damaging that may occur in a neural network whose synaptic bonds have been trained (before damaging) so as to preserve a definite number of patterns. We have used the Hopfield model of the neural network, and applied the Hebbian rule of training (learning). We have studied networks with 600 elements (neurons) and investigated several types of damaging, by performing very extensive numerical investigation. Thus, we have demonstrated that there is no difference between symmetric and asymmetric damaging of bonds. Besides, it turns out that the worst damaging of synaptic bonds is the one that starts with ruining the strongest bonds, whereas in the opposite case, that is, in the case of damaging that starts with ruining the weakest bonds, the learnt patterns remain preserved even for a large percentage of extinguished bonds.

Statistical mechanics of polymer chains grafted to adsorbing boundaries of fractal lattices embedded in three-dimensional space

We study the adsorption problem of linear polymers, immersed in a good solvent, when the container of the polymer–solvent system is taken to be a member of the Sierpinski gasket (SG) family of fractals, embedded in the three-dimensional Euclidean space. Members of the SG family are enumerated by an integer b (2≤b<∞), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the surface critical exponents γ11,γ1, and γs which, within the self-avoiding walk model (SAW) of polymer chain, are associated with the numbers of all possible SAWs with both, one, and no ends grafted to the adsorbing surface (adsorbing boundary), respectively. By applying the exact renormalization group method, for 2≤b≤4, we have obtained specific values for these exponents, for various types of polymer conformations. To extend the obtained sequences of exact values for surface critical exponents, we have applied the Monte Carlo renormalization group method for fractals with 2≤b≤40. The obtained results show that all studied exponents are monotonically increasing functions of the parameter b, for all possible polymer states. We discuss mutual relations between the studied critical exponents, and compare their values with those found for other types of lattices, in order to attain a unified picture of the attacked problem.

Pulling self-interacting linear polymers on a family of fractal lattices embedded in three-dimensional space

We have studied the problem of force pulling self-interacting linear polymers situated in fractal containers that belong to the Sierpinski gasket (SG) family of fractals embedded in three-dimensional (3D) space. Each member of this family is labeled with an integer b (2 ≤ b ≤ ∞). The polymer chain is modeled by a self-avoiding walk (SAW) with one end anchored to one of the four boundary walls of the lattice, while the other (floating in the bulk of the fractal) is the position at which the force is acting. By applying an exact renormalization group (RG) method we have established the phase diagrams, including the critical force–temperature dependence, for fractals with b = 2,3 and 4. Also, for the same fractals, in all polymer phases, we examined the generating function G1 for the numbers of all possible SAWs with one end anchored to the boundary wall. We found that besides the usual power-law singularity of G1, governed by the critical exponent γ1, whose specific values are worked out for all cases studied, in some regimes the function G1 displays an essential singularity in its behavior.

Statistics of semiflexible self-avoiding trails on a family of two-dimensional compact fractals

We have applied the exact and Monte Carlo renormalization group (MCRG) method to study the statistics of semiflexible self-avoiding trails (SATs) on the family of plane-filling (PF) fractals. Each fractal of the family is compact, that is, the fractal dimension df is equal to 2 for all members of the PF family, which are enumerated by an odd integer b, . Varying values of the stiffness parameter s of trails from 1 to 0 (so that when s decreases the trail stiffness increases) we calculate exactly (for 3 ≤ b ≤ 7) and through the MCRG approach (for b ≤ 201) the sets of the critical exponents ν (associated with the mean squared end-to-end distances of SATs) and γ (associated with the total number of different SATs). Our results show that critical exponents are stiffness dependent functions, so that ν(s) is a monotonically decreasing function of s, for each studied b, whereas γ(s) displays a non-monotonic behavior for some values of b. On the other hand, by fixing the stiffness parameter s, our results show clearly that for highly flexible trails (with s = 1 and 0.9) ν is a non-monotonic function of b, while for stiffer SATs (with s ≤ 0.7) ν monotonically decreases with b. We also show that γ(b) increases with increasing b, independently of s. Finally, we compare the obtained SAT data with those obtained for the semiflexible self-avoiding walk (SAW) model on the same fractal family, and for both models we discuss behavior of the studied exponents in the fractal-to-Euclidean crossover region .

Critical behavior of interacting two-polymer system in a fractal solvent: an exact renormalization group approach

We study the polymer system consisting of two polymer chains situated in a fractal container that belongs to the three-dimensional Sierpinski gasket (3D SG) family of fractals. Each 3D SG fractal has four fractal impenetrable 2D surfaces, which are, in fact, 2D SG fractals. The two-polymer system is modeled by two interacting self-avoiding walks (SAW), one of them representing a 3D floating polymer, while the other corresponds to a chain confined to one of the four 2D SG boundaries (with no monomer in the bulk). We assume that the studied system is immersed in a poor solvent inducing the intra-chain interactions. For the inter-chain interactions we propose two models: in the first model (ASAW) the SAW chains are mutually avoiding, whereas in the second model (CSAW) chains can cross each other. By applying an exact renormalization group (RG) method, we establish the relevant phase diagrams for b = 2,3 and 4 members of the 3D SG fractal family for the model with avoiding SAWs, and for b = 2 and 3 fractals for the model with crossing SAWs. Also, at the appropriate transition fixed points we calculate the contact critical exponents, associated with the number of contacts between monomers of different chains. Throughout this paper we compare results obtained for the two models and discuss the impact of the topology of the underlying lattices on emerging phase diagrams.

Exact and Monte Carlo study of the two self‐avoiding random walks on the three‐dimensional Sierpinski lattices

We consider the phenomena of entanglement of the two interacting self‐avoiding walks (SAW) situated in a member of the three‐dimensional Sierpinski Gasket (SG) fractal family. We focus our attention to determine number of point contacts between the two SAW paths M, which turns out to be a set of power laws whose characteristics depend predominantly on the interactions between SAW steps. The phase diagrams have been establised and corresponding values of the contact critical exponents φ, associated with the two‐path mutual contacts, have been found.

On the number of contacts of a floating polymer chain cross-linked with a surface adsorbed chain on fractal structures

We study the problem of the interaction of a linear polymer chain, floating in fractal containers that belong to the three-dimensional Sierpinski gasket (3D SG) family of fractals, with a surface adsorbed linear polymer chain. Each member of the 3D SG fractal family has a fractal impenetrable 2D adsorbing surface, which appears to be 2D SG fractal. The two-polymer system is modelled by two mutually crossing self-avoiding walks. By applying the Monte Carlo renormalization group (MCRG) method, we calculate the critical exponents, , associated with the number of contacts of the 3D SG floating polymer chain, and the 2D SG adsorbed polymer chain, for a sequence of SG fractals with 2 ≤ b ≤ 40. Also, we propose a codimension additivity argument formula for , and compare its predictions with our reliable set of MCRG data. We find that decreases monotonically with increasing b, that is, with increase of the container fractal dimension. Finally, we discuss the relations between different contact exponents, and analyse their possible behaviour in the fractal-to-Euclidean crossover region b → ∞