The dynamical origin of the universality classes of spatiotemporal intermittency

Two universality classes of spatiotemporal intermittency are seen in the spreading and non-spreading regimes of the sine circle map lattice, spatiotemporal intermittency of the directed percolation class, and spatial intermittency, not of the DP class, where the temporal behavior is regular. The transition between the two classes maps to a probabilistic to deterministic transition of the equivalent cellular automaton of the model, and is seen to have its dynamic origin in an attractor-widening crisis.     

New universality classes in “Percolative” dynamics

We study a site analogue of directed percolation. Random trajectories are generated and their critical behavior is studied. The critical behavior corresponds to that of simple percolation in some of the parameter space, but elsewhere the exponents reveal new universality classes. As a byproduct, we use the model to make an improved estimate of the percolation hull exponents and to calculate the site percolation probability for the square lattice.     

On computing the entropy of the Henon attractor

In a recent article D. Ruelle [inLecture Notes in Physics, No. 80 (Springer, Berlin, 1978)] has conjectured that for the Hénon attractor its measure theoretic entropy should be equal to its characteristic exponent. This result is known to be true for systems which satisfy Smale's Axiom A. In this article we report the results of our computations which suggest that Ruelle's conjecture may be true for the Hénon attractor. Further, in our study we are confronted with fundamental questions which suggest that certain existence theorems from ergodic theory are not sufficient from a computational point of view.     

On the fractal dimension of the Henon attractor

We re-measured the fractal dimension of the Hénon attractor by direct box-counting. We paid special attention to (a) optimal speed and use of storage, and (b) systematic corrections due to the finiteness of the number of iterations. Covering with grids of up to 9600 × 9600 boxes, we observe that the number N(?, n) of boxes visited after n iterations obeys a scaling law N(?, ∞) - N(?, n) ≈ const × ?^-αn^-β (for n → ∞) with α = 2.42 ± 0.15, β = 0.89 ± 0.03. Using this extrapolate to n → ∞, we obtain D = 1.28 ± 0.01 in disagreement with previous box-counting estimates, but in agreement with a recent indirect evaluation.     

On the universality of string theory

String theory is accused by some of its critics to be a purely abstract mathematical discipline, having lost the contact to the simple yet deeply rooted questions which physics provided until the beginning of this century. We argue that, in contrary, there are indications that string theory might be linked to a fundamental principle of a quantum computational character. In addition, the nature of this principle can possibly provide some new insight into the question of universality of string theory (string theory as the “theory of everything”).     

On the universality of Einstein equations

It is proved that a Lagrangian field theory based on a linear connection in space-time is equivalent to Einstein's general relativity interacting with additional matter fields.     

Laplacian growth and diffusion limited aggregation: different universality classes

It had been conjectured that diffusion limited aggregates and Laplacian growth patterns (with small surface tension) are in the same universality class. Using iterated conformal maps we construct a one-parameter family of fractal growth patterns with a continuously varying fractal dimension. This family can be used to bound the dimension of Laplacian growth patterns from below. The bound value is higher than the dimension of diffusion limited aggregates, showing that the two problems belong to two different universality classes.     

Crossover between universality classes in the statistics of rare events in disordered conductors

The crossover from orthogonal to the unitary universality classes in the distribution of the anomalously localized states (ALS) in two-dimensional disordered conductors is traced as a function of magnetic field. We demonstrate that the microscopic origin of the crossover is the change in the symmetry of the underlying disorder configurations that are responsible for ALS. These disorder configurations are of weak magnitude (compared to the Fermi energy) and of small size (compared to the mean free path). We find their shape explicitly by means of the direct optimal fluctuation method.     

On the universality of quark jet fragmentation

Universality of inclusive fragmentation density functions in lepton induced processes (ep, νp, e⁺e^?) and purely hadronic processes is discussed from the point of view of the Topological Expansion/Dual Unitarization Scheme. It is shown that planar, single jet dominated processes have universal inclusive distributions and average multiplicities. In multi-jet processes, treated in a simple approximation, 〈x〉 is inversely proportional to the number N of jets and the magnitude of the seagull effect increases as N².     

On universality for area-preserving maps of the plane

We present detailed evidence that one-parameter families of area-preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter. We relate this behaviour to a fixed point equation of the form

$\text{Λ}{}^{\mathrm{?1}}\text{°Φ°Φ°Λ = Φ}$

and

$\text{det}\text{DΦ = 1}$

, Φ:$\text{R}$²→$\text{R}$². In particular we argue that the scaling transformation Λ:$\text{R}$²→$\text{R}$² is conjugate to the transformation Λ₀:(x, y)→(λx, μy), with λ² ≠ μ, and in fact λ² >μ. We present some numerical evidence that

$\text{δ = 8.721}$

…,

$\text{?}\text{1}\text{λ}\text{= 4.018}$

…,

$\text{1}\text{μ}\text{= 16.36}$

…, where δ is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods 2^k described above.     

Jamming transition and new percolation universality classes in particulate systems with attraction

We numerically study the jamming transition in particulate systems with attraction by investigating their mechanical response at zero temperature (T=0). We find three regimes of mechanical behavior separated by two critical transitions--connectivity and rigidity percolation. The transitions belong to different universality classes than their lattice counterparts, due to force balance constraints. We also find that these transitions are unchanged at low temperatures and resemble gelation transitions in experiments on colloidal and silica gels.     

Cascaded Fresnel holographic image encryption scheme based on a constrained optimization algorithm and Henon map

We propose an image encryption scheme using chaotic phase masks and cascaded Fresnel transform holography based on a constrained optimization algorithm. In the proposed encryption scheme, the chaotic phase masks are generated by Henon map, and the initial conditions and parameters of Henon map serve as the main secret keys during the encryption and decryption process. With the help of multiple chaotic phase masks, the original image can be encrypted into the form of a hologram. The constrained optimization algorithm makes it possible to retrieve the original image from only single frame hologram. The use of chaotic phase masks makes the key management and transmission become very convenient. In addition, the geometric parameters of optical system serve as the additional keys, which can improve the security level of the proposed scheme. Comprehensive security analysis performed on the proposed encryption scheme demonstrates that the scheme has high resistance against various potential attacks. Moreover, the proposed encryption scheme can be used to encrypt video information. And simulations performed on a video in AVI format have also verified the feasibility of the scheme for video encryption.