Scaling spectra and return times of dynamical systems

The grand canonical version of the spectrum of singularities formalism is presented, relying naturally upon certain Markov transition graphs. The structure of a graph is simply determined by the close return times of the dynamical system described. Thus, an intimate connection exists between the shape of the singularity curve and a small but interesting set of dynamical properties.     

Superlinearly convergent affine scaling interior trust-region method for linear constrained LC<Superscript>1</Superscript> minimization

We extend the classical affine scaling interior trust region algorithm for the linear constrained smooth minimization problem to the nonsmooth case where the gradient of objective function is only locally Lipschitzian. We propose and analyze a new affine scaling trust-region method in association with nonmonotonic interior backtracking line search technique for solving the linear constrained LC1 optimization where the second-order derivative of the objective function is explicitly required to be locally Lipschitzian. The general trust region subproblem in the proposed algorithm is defined by minimizing an augmented affine scaling quadratic model which requires both first and second order information of the objective function subject only to an affine scaling ellipsoidal constraint in a null subspace of the augmented equality constraints. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required. Applications of the algorithm to some nonsmooth optimization problems are discussed.     

Anomalous Scaling of Surface Growth Equations with Spatially and Temporally Correlated Noise

Based on the scaling idea of local slopes by Lopez et al. [Phys. Rev. Lett. 94 （2005） 166103], we investigate anomalous dynamic scaling of （d ＋ 1）-dimensional surface growth equations with spatially and temporally correlated noise. The growth equations studied include the Kardar-Parisi-Zhang （KPZ）, Sun-Guo-Grant （SGG）, and Lai-Das Sarma-Villain （LDV） equations. The anomalous scaling exponents in both the weak- and strong-coupling regions are obtained, respectively.     

Kq, Scaling of ac susceptibility and the nonlinear response function of high-temperature superconductors

The amplitude-dependent ac susceptibility of high-temperature superconductors is shown to obey some empirical scaling relations. We try to analyze this behavior by extending a dc nonlinear response function of mixed state to the ac cases. The derived equations for critical current and ac susceptibility ξ(T) agree with the scaling relations of experimental data.     

A hybrid method for multidimensional scaling using city-block distances

The problem of multidimensional scaling with city-block distances in the embedding space is reduced to a two level optimization problem consisting of a combinatorial problem at the upper level and a quadratic programming problem at the lower level. A hybrid method is proposed combining randomized search for the upper level problem with a standard quadratic programming algorithm for the lower level problem. Several algorithms for the combinatorial problem have been tested and an evolutionary global search algorithm has been proved most suitable. An experimental code of the proposed hybrid multidimensional scaling algorithm is developed and tested using several test problems of two- and three-dimensional scaling.     

Inverse problems of aggregation processes

The coagulation frequency is the key ingredient in the population balance (Smoluchowski) equation of coagulation kinetics. An inverse problem is formulated to extract the coagulation frequency from transient size distributions when these distributions are self-similar. Two numerical examples illustrate the procedure. The first demonstrates the inverse problem for the recovery of singular coagulation frequencies, while the second shows the procedure when self-similarity is approximate. Transient droplet coagulation experiments in a turbulent flow field have been performed. The resulting size distributions are observed to be self-similar. The inverse problem is used to determine the drop coagulation frequency. This frequency shows significant deviation from the coagulation frequencies derived from simple models of drop-drop interactions in a turbulent flow field.     

Model investigation of non-thermal phase transition in high energy collisions

The non-thermal phase transition in high energy collisions is studied in detail in the framework of random cascade model. The relation between the characteristic parameter A_p, of phase transition and the rank q of moment is obtained using Monte Carlo simulation, and the existence of two phases in self-similar cascading multiparticle systems is shown. The relation between the critical point qc of phase transition on the fluctuation parameter a is obtained and compared with the experimental results from NA22. The same study is carried out also by analytical calculation under central limit approximation. The range of validity of the central limit approximation is discussed.     

Universality in two‐dimensional enhancement percolation

We consider a type of dependent percolation introduced in 2 , where it is shown that certain “enhancements” of independent (Bernoulli) percolation, called essential, make the percolation critical probability strictly smaller. In this study we first prove that, for two‐dimensional enhancements with a natural monotonicity property, being essential is also a necessary condition to shift the critical point. We then show that (some) critical exponents and the scaling limit of crossing probabilities of a two‐dimensional percolation process are unchanged if the process is subjected to a monotonic enhancement that is not essential. This proves a form of universality for all dependent percolation models obtained via a monotonic enhancement (of Bernoulli percolation) that does not shift the critical point. For the case of site percolation on the triangular lattice, we also prove a stronger form of universality by showing that the full scaling limit 12 , 13 is not affected by any monotonic enhancement that does not shift the critical point. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

The trajectory scaling function for period doubling bifurcations in flows

We extend the trajectory scaling function as defined for maps to flows whose dynamics is governed by ordinary differential equations. The results are obtained for the Duffing oscillator and are expected to be the same for other dissipative flows as well.