Over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2 + √ 2/2. Over an alphabet of size 4 we construct an infinite balanced word with critical exponent (5 + √ 5)/4. Over larger alphabets, we give some candidates for balanced words (found computationally) having small critical exponents. We also explore a method for proving these results using the automated t...

We simulate the kinetic Ashkin-Teller model with both ordered and disordered initial states, evolving in contact with a heat-bath at the critical temperature. The power law scaling behaviour for the magnetic order and electric order are observed in the early time stage. The values of the critical exponent θ vary along the critical line. Another dynamical exponent z is also obtained in the process.

We show that the Hertz phi(4) theory of quantum criticality is incomplete as it misses anomalous nonlocal contributions to the interaction vertices. For antiferromagnetic quantum transitions, we found that the theory is renormalizable only if the dynamical exponent z=2. The upper critical dimension is still d=4 - z=2; however, the number of marginal vertices at d=2 is infinite. As a result, the...

The short-time critical dynamics of the two-dimensional eight-state random-bond Potts model is investigated with large-scale Monte Carlo simulations. Dynamic relaxation starting from a disordered and an ordered state is carefully analyzed. The continuous phase transition induced by disorder is studied, and both the dynamic and static critical exponents are estimated. The static exponent beta/nu...

This problem has positive solutions for ǫ > 0 (with pqǫ > 1) and no non-trivial solution for ǫ ≤ 0. We study the asymptotic behaviour of least energy solutions as ǫ → 0. These solutions are shown to blow-up at exactly one point, and the location of this point is characterized. In addition, the shape and exact rates for blowing up are given. Résumé. Considéré le problème −∆uǫ = v p ǫ vǫ > 0 en Ω...

We consider the problem: −div(p∇u) = u + λu, u > 0 in Ω, u = 0 on ∂Ω. Where Ω is a bounded domain in IR, n ≥ 3, p : Ω̄ −→ IR is a given positive weight such that p ∈ H(Ω) ∩ C(Ω̄), λ is a real constant and q = 2n n−2 . We study the effect of the behavior of p near its minima and the impact of the geometry of domain on the existence of solutions for the above problem.

A new summability method was tested to calculate the critical exponent ν of the localization length for the symplectic case derived from the non-linear σ-model. Although we used the same series as Hikami and others, unlike them we were able to resum the series in two-dimensions (2D) and obtain the result ν ∼ 1. Values of ν in 2+ ε dimensions seem to saturate the Harris inequality up to ε = 0.2....

where B1 = {x ∈ RN | |x| < 1} is the unit ball in RN (N ≥ 3), λ, μ > 0, 2∗ := 2N/(N − 2). When μ < 0, this problem has been considered by many authors recently (cf. [5, 6, 7, 8]). But when μ > 0, this problem has not been considered as far as we know. In fact, the existence of nontrivial solution for (1.1) when μ > 0 is an open problem which was imposed in [7]. In this paper, we get the followi...

We analyze the geometry of the species– and genotypesize distribution in evolving and adapting populations of single-stranded self-replicating genomes: here programs in the Avida world. We find that a scale-free distribution (power law) emerges in complex landscapes that achieve a separation of two fundamental time scales: the relaxation time (time for population to return to equilibrium after ...