Table of contents

Modern approaches to semantic analysis if reformulated as Hilbert-space problems reveal formal structures known from quantum mechanics. A similar situation is found in distributed representations of cognitive structures developed for the purpose of neural networks. We take a closer look at similarities and differences between the above two fields and quantum information theory.

We show that the conditionally exactly solvable potential of Dutt et al (1995 J. Phys. A: Math. Gen.28 L107) and the exactly solvable potential from which it is derived form a dual system.

Dimensional reduction occurs when the critical behaviour of one system can be related to that of another system in a lower dimension. We show that this occurs for directed branched polymers (DBP) by giving an exact relationship between DBP models in D + 1 dimensions and repulsive gases at negative activity in D dimensions. This implies relations between exponents of the two models: γ(D + 1) = α(D) (the exponent describing the singularity of the pressure), and ν_⊥(D + 1) = ν(D) (the correlation length exponent of the repulsive gas). It also leads to the relation θ(D + 1) = 1 + σ(D), where σ(D) is the Yang–Lee edge exponent. We derive exact expressions for the number of DBP of size N in two dimensions.

We consider dynamics generated by Hamiltonians with three degrees of freedom and symmetries. It is shown that locally, away from a possible saddle equilibrium, some codimension-1 invariant manifold may exist. They are stable/unstable manifolds of a codimension-2 hyperbolic invariant manifold. This structure appears when some periodic orbits constitutive of the Arnold web have bifurcated and become linearly unstable. This result generalizes the existence of normally hyperbolic invariant manifolds and their codimension-1 stable/unstable manifolds in the vicinity of an unstable ⊗ (stable)² equilibrium point.

The probability distribution of percolation thresholds in finite lattices was first believed to follow a normal Gaussian behaviour. With increasing computer power and more efficient simulational techniques, this belief turned to a stretched exponential behaviour, instead. Here, based on a further improvement of the Monte Carlo data, we show evidence on square lattices that this question is numerically not yet answered at all.

A two-parameter, probabilistic growth model for partition polygon clusters is introduced and exact results obtained relating to the area moments and the area probability distribution. In particular, the scaling behaviour in the presence of asymmetry between growth along the two principal axes is discussed. Variants of the model are also examined, including the extension to rooted stack polygons. An interesting application relates to characterizing the asymptotic behaviour of the cumulative customer waiting time distribution in a particular discrete-time queue.

The probability of the first entrance to the negative semi-axis for a one-dimensional discrete Ornstein–Uhlenbeck (O-U) process is studied in this work. The discrete O-U process is a simple generalization of the random walk and many of its statistics may be calculated using essentially the same formalism. In particular, the case in which Sparre-Andersen's theorem applies for normal random walks is considered, and it is shown that the universal features of the first passage probability do not extend to the discrete O-U process. Finally, an explicit expression for the generating function of the probability of first entrance to the negative real axis at step n is calculated and analysed for a particular choice of the step distribution.

A cellular automaton model of traffic flow taking into account velocity anticipation is introduced. The strength of anticipation can be varied to describe different driving schemes. We find a new phase separation into a free-flow regime and a so-called v-platoon in an intermediate density regime. In a v-platoon all cars move with velocity v and have vanishing headway. The velocity v of a platoon only depends on the strength of anticipation. At high densities, a congested state characterized by the coexistence of a 0-platoon with several v-platoons is reached. The results are not only relevant for automated highway systems, but also help to elucidate the effects of anticipation that play an essential role in realistic traffic models. From a physics point of view the model is interesting because it exhibits phase separation with a condensed phase in which particles move coherently with finite velocity coexisting with either a non-condensed (free-flow) phase or another condensed phase that is non-moving.

In our work we have reconsidered the old problem of diffusion at the boundary of an ultrametric tree from a 'number theoretic' point of view. Namely, we use the modular functions (in particular, the Dedekind η-function) to construct the 'continuous' analogue of the Cayley tree isometrically embedded in the Poincaré upper half-plane. Later we work with this continuous Cayley tree as with a standard function of a complex variable. In the framework of our approach, the results of Ogielsky and Stein on dynamics in ultrametric spaces are reproduced semi-analytically or semi-numerically. The speculation on the new 'geometrical' interpretation of replica n → 0 limit is proposed.

We study a complex Ginzburg–Landau (CGL) equation perturbed by a random force which is white in time and smooth in the space variable x. Assuming that dim x ⩽ 4, we prove that this equation has a unique solution and discuss its asymptotic in time properties. Next we consider the case when the random force is proportional to the square root of the viscosity and study the behaviour of stationary solutions as the viscosity goes to zero. We show that, under this limit, a subsequence of solutions in question converges to a nontrivial stationary process formed by global strong solutions of the nonlinear Schrödinger equation.

We devised a novel measure that dynamically evaluates temporal interdependences between two coupled units based on the properties of the distributions of their relative interevent intervals. We investigate its properties on the system of two coupled non-identical Rössler oscillators and a system of non-identical Hindmarsh–Rose models of thalamocortical neurons and show that the measure highlights the properties of phase synchronization observed in those two systems. We postulate that the observed properties of the phase lag, in conjunction with the experimentally observed activity-dependent synaptic modification in the neural systems, may drive the changes of the direction of information flow in a neural network, and thus the measure can play an important role in assessing those changes.

A symmetry reduction of a PDEs system, describing the expansive growth of a benign tumour, is obtained via a group analysis approach. The presence in the model of three arbitrary functions suggests the use of Lie symmetries by using the weak equivalence transformations. An invariant classification is given which allows us to reduce the initial PDEs system to an ODEs system. Numerical simulations show a realistic enough description of the physical process.

We construct a polynomial differential system that admits a given set of invariant algebraic curves. For such a system we solve the Darboux problem (the existence of the Darboux first integral), the Poincaré problem (the existence of an upper bound for the degree of invariant algebraic curve) and study Hilbert's 16th problem for algebraic limit cycles (the existence of an upper bound for the number of algebraic limit cycles).

We discuss a phase-space description of the photon number distribution of non-classical states which is based on Husimi's Q(α) function and does not rely on the WKB approximation. We illustrate this approach using the examples of displaced number states and two photon coherent states and show it to provide an efficient method for computing and interpreting the photon number distribution. This result is interesting in particular for the two photon coherent states which, for high squeezing, have the probabilities of even and odd photon numbers oscillating independently.

We present a systematic discussion of supersymmetric solutions of 2D dilaton supergravity. In particular those solutions which retain at least half of the supersymmetries are ground states with respect to the bosonic Casimir function (essentially the ADM mass). Nevertheless, by tuning the prepotential appropriately, black-hole solutions may emerge with an arbitrary number of Killing horizons. The absence of dilatino and gravitino hair is proved. Moreover, the impossibility of supersymmetric dS ground states and of nonextremal black holes is confirmed, even in the presence of a dilaton. In these derivations, knowledge of the general analytic solution of 2D dilaton supergravity plays an important role. The latter result is addressed in the more general context of gPSMs which have no supergravity interpretation.

Finally it is demonstrated that the inclusion of non-minimally coupled matter, a step which is already nontrivial by itself, does not change these features in an essential way.