Spiral flames

We describe computations of periodic and meandering spiral patterns in a reaction-diffusion model of flames.     

Phase transitions in phylogeny

We apply the theory of Markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies.

We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least , and the net transition on each edge is bounded by . Motivated by a conjecture by M. Steel, we show that if 1$">, then for balanced trees, the topology of the underlying tree, having leaves, can be reconstructed from samples (characters) at the leaves. On the other hand, we show that if , then there exist topologies which require at least samples for reconstruction.

Our results are the first rigorous results to establish the role of phase transitions for Markov random fields on trees, as studied in probability, statistical physics and information theory, for the study of phylogenies in mathematical biology.

     

Translational Symmetry-Breaking for Spiral Waves

Summary. Spiral waves are observed in numerous physical situations, ranging from Belousov-Zhabotinsky (BZ) chemical reactions, to cardiac tissue, to slime-mold aggregates. Mathematical models with Euclidean symmetry have recently been developed to describe the dynamic behavior (for example, meandering) of spiral waves in excitable media. However, no physical experiment is ever infinite in spatial extent, so the Euclidean symmetry is only approximate. Experiments on spiral waves show that inhomogeneities can anchor spirals and that boundary effects (for example, boundary drifting) become very important when the size of the spiral core is comparable to the size of the reacting medium. Spiral anchoring and boundary drifting cannot be explained by the Euclidean model alone. In this paper, we investigate the effects on spiral wave dynamics of breaking the translation symmetry while keeping the rotation symmetry. This is accomplished by introducing a small perturbation in the five-dimensional center bundle equations (describing Hopf bifurcation from one-armed spiral waves) which is SO(2)-equivariant but not equivariant under translations. We then study the effects of this perturbation on rigid spiral rotation, on quasi-periodic meandering and on drifting. Received August 24, 1999; accepted August 8, 2000 Online publication October 11, 2000     

Bifurcations and Dynamics of Spiral Waves

(N) . In this article, it is shown that the dynamics near meandering spiral waves or other patterns is determined by a finite-dimensional vector field that has a certain skew-product structure over the group SESE(N) . This generalizes our earlier work on center-manifold theory near rigidly rotating spiral waves to meandering spirals. In particular, for meandering spirals, it is much more sophisticated to extract the aforementioned skew-product structure since spatio-temporal rather than only spatial symmetries have to be accounted for. Another difficulty is that the action of the Euclidean symmetry group on the underlying function space is not differentiable, and in fact may be discontinuous. Using this center-manifold reduction, Hopf bifurcations and periodic forcing of spiral waves are then investigated. The results explain the transitions to patterns with two or more temporal frequencies that have been observed in various experiments and numerical simulations. Received December 8, 1997; accepted May 19, 1996     

Phase transitions in project scheduling

Researchers in the area of artificial intelligence have recently shown that many NP-complete problems exhibit phase transitions. Often, problem instances change from being easy to being hard to solve to again being easy to solve when certain of their characteristics are modified. Most often the transitions are sharp, but sometimes they are rather continuous in the order parameters that are characteristic of the system as a whole. To the best of our knowledge, no evidence has been provided so far that similar phase transitions occur in NP-hard scheduling problems. In this paper we report on the existence of phase transitions in various resource-constrained project scheduling problems. We discuss the use of network complexity measures and resource parameters as potential order parameters. We show that while the network complexity measures seem to reveal continuous easy-hard or hard-easy phase transitions, the resource parameters exhibit a relatively sharp easy-hard-easy transition behaviour.     

Second-order metropolitan urban phase transitions

The morphology evolution of Metropolitan Urban Areas constituted by different Central Business Districts is studied in this paper. For this matter, we propose a stochastic model which combines an initial percolation setting followed by a diffusion-limited aggregation mechanism. Our model mimics better than either case (percolation or diffusion-limited aggregation) the Metropolitan Urban Areas formation progress. We argue that the Metropolitan Urban Areas case introduced in this paper, grows in such a way that undergoes a non-equilibrium second-order phase transition during this process. This conclusion is supported by a fractal dimension and configurational entropy analysis, as well as by studying an empirical case.     

Phase transitions with semi-diffuse interfaces

In this paper, we examine new “phase-field” models with semi-diffuse interfaces. These models have the property that the −1/+1 planar phase transitions take place over a finite interval. The models also support multiple interface solutions with interfaces centered at arbitrary points L₁<L₂<?<L_N. These solutions correspond to local minima of an entropy functional (see (3.3) and (3.7)) rather than saddle points and are dynamically stable. The classical models have no such exact solutions but they do support solutions with N equally spaced transition points where the order parameter transitions between values p_min(N) and p_max(N) satisfying −1<p_min(N)<0<p_max(N)<1. These solutions of the classical model are saddle points of the entropy functional associated with those models and are not dynamically stable.     

Phase transitions in multi-phase media

Two problems on phase transitions in a continuous medium are considered. The first problem deals with an elastic medium admitting more than two phases. Necessary conditions for equilibrium states are derived. The dependence of equilibrium states on the surface tension coefficients and temperature is studied for one model of a three-phase elastic medium such that each phase has a quadratic energy density. The second problem deals with phase transitions under some restrictions on the vector field under consideration. These restrictions imply that this vector field is solenoidal and its normal component vanishes on the boundary of the interfaces of phases. The equilibrium equations are deduced. Bibliography: 5 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 120–170.     

The Chen-Type of the Spiral Surfaces

We show that a spiral surface M in E³ is of finite type if and only if M is minimal Also, the plane is the only spiral surface in E³ whose the Gauss map G is of finite type, or satisfies the condition ΔG = ΛG, where Λ ∈ R^3×3.     

Systematics of magnetic dipole transitions

An empirical approach to distinguish betweenl-allowed andl-forbidden magnetic dipole transitions is made. The reduced lifetimes of these transitions are calculated and their variation with neutron number is studied. It is observed that their general tendency is to remain constant whilel-forbidden andl-allowed odd-proton magnetic dipole transitions are distinguishable from their reduced lifetimes, it is not so in the case of odd-neutron transitions. A slight increase in reduced lifetime with neutron number is observed for fixed proton number.     

Homological type of geometric transitions

The present paper gives an account and quantifies the change in topology induced by small and type II geometric transitions, by introducing the notion of the homological type of a geometric transition. The obtained results agree with, and go further than, most results and estimates, given to date by several authors, both in mathematical and physical literature.     

Physical nonlinearity with microstructural transitions

A physical model of a deformable body with a network structure composed of individual elements linked by valence bonds is considered. The effect of dissociated valence bonds on the irreversible deformation is investigated using Fermi-Dirac statistics. The physical stress-strain relation obtained is a generalization of certain known constitutive equations.Bulgarian Academy of Sciences, Sofia. Translated from Mekhanika Polimerov, No. 1, pp. 36–44, January–February, 1973.     

Geometry of quasiminimal phase transitions

We consider the quasiminima of the energy functional