Measuring the galaxy power spectrum and scale-scale correlations with multiresolution-decomposed covariance -- I. method

We present a method of measuring galaxy power spectrum based on the multiresolution analysis of the discrete wavelet transformation (DWT). Since the DWT representation has strong capability of suppressing the off-diagonal components of the covariance for selfsimilar clustering, the DWT covariance for popular models of the cold dark matter cosmogony generally is diagonal, or $j$(scale)-diagonal in the scale range, in which the second scale-scale correlations are weak. In this range, the DWT covariance gives a lossless estimation of the power spectrum, which is equal to the corresponding Fourier power spectrum banded with a logarithmical scaling. In the scale range, in which the scale-scale correlation is significant, the accuracy of a power spectrum detection depends on the scale-scale or band-band correlations. This is, for a precision measurements of the power spectrum, a measurement of the scale-scale or band-band correlations is needed. We show that the DWT covariance can be employed to measuring both the band-power spectrum and second order scale-scale correlation. We also present the DWT algorithm of the binning and Poisson sampling with real observational data. We show that the alias effect appeared in usual binning schemes can exactly be eliminated by the DWT binning. Since Poisson process possesses diagonal covariance in the DWT representation, the Poisson sampling and selection effects on the power spectrum and second order scale-scale correlation detection are suppressed into minimum. Moreover, the effect of the non-Gaussian features of the Poisson sampling can be calculated in this frame.Comment: AAS Latex file, 44 pages, accepted for publication in Ap

Multiresolution analysis in statistical mechanics. II. The wavelet transform as a basis for Monte Carlo simulations on lattices

In this paper, we extend our analysis of lattice systems using the wavelet transform to systems for which exact enumeration is impractical. For such systems, we illustrate a wavelet-accelerated Monte Carlo (WAMC) algorithm, which hierarchically coarse-grains a lattice model by computing the probability distribution for successively larger block spins. We demonstrate that although the method perturbs the system by changing its Hamiltonian and by allowing block spins to take on values not permitted for individual spins, the results obtained agree with the analytical results in the preceding paper, and ``converge'' to exact results obtained in the absence of coarse-graining. Additionally, we show that the decorrelation time for the WAMC is no worse than that of Metropolis Monte Carlo (MMC), and that scaling laws can be constructed from data performed in several short simulations to estimate the results that would be obtained from the original simulation. Although the algorithm is not asymptotically faster than traditional MMC, because of its hierarchical design, the new algorithm executes several orders of magnitude faster than a full simulation of the original problem. Consequently, the new method allows for rapid analysis of a phase diagram, allowing computational time to be focused on regions near phase transitions.Comment: 11 pages plus 7 figures in PNG format (downloadable separately

Discrepancy between sub-critical and fast rupture roughness: a cumulant analysis

We study the roughness of a crack interface in a sheet of paper. We distinguish between slow (sub-critical) and fast crack growth regimes. We show that the fracture roughness is different in the two regimes using a new method based on a multifractal formalism recently developed in the turbulence literature. Deviations from monofractality also appear to be different in both regimes

One-point Statistics of the Cosmic Density Field in Real and Redshift Spaces with A Multiresolutional Decomposition

In this paper, we develop a method of performing the one-point statistics of a perturbed density field with a multiresolutional decomposition based on the discrete wavelet transform (DWT). We establish the algorithm of the one-point variable and its moments in considering the effects of Poisson sampling and selection function. We also establish the mapping between the DWT one-point statistics in redshift space and real space, i.e. the algorithm for recovering the DWT one-point statistics from the redshift distortion of bulk velocity, velocity dispersion, and selection function. Numerical tests on N-body simulation samples show that this algorithm works well on scales from a few hundreds to a few Mpc/h for four popular cold dark matter models. Taking the advantage that the DWT one-point variable is dependent on both the scale and the shape (configuration) of decomposition modes, one can design estimators of the redshift distortion parameter (beta) from combinations of DWT modes. When the non-linear redshift distortion is not negligible, the beta estimator from quadrupole-to-monopole ratio is a function of scale. This estimator would not work without adding information about the scale-dependence, such as the power-spectrum index or the real-space correlation function of the random field. The DWT beta estimators, however, do not need such extra information. Numerical tests show that the proposed DWT estimators are able to determine beta robustly with less than 15% uncertainty in the redshift range 0 < z < 3.Comment: 39 pages, 12 figures, ApJ accepte

Time domain study of frequency-power correlation in spin-torque oscillators

This paper describes a numerical experiment, based on full micromagnetic simulations of current-driven magnetization dynamics in nanoscale spin valves, to identify the origins of spectral linewidth broadening in spin torque oscillators. Our numerical results show two qualitatively different regimes of magnetization dynamics at zero temperature: regular (single-mode precessional dynamics) and chaotic. In the regular regime, the dependence of the oscillator integrated power on frequency is linear, and consequently the dynamics is well described by the analytical theory of current-driven magnetization dynamics for moderate amplitudes of oscillations. We observe that for higher oscillator amplitudes, the functional dependence of the oscillator integrated power as a function of frequency is not a single-valued function and can be described numerically via introduction of nonlinear oscillator power. For a range of currents in the regular regime, the oscillator spectral linewidth is a linear function of temperature. In the chaotic regime found at large current values, the linewidth is not described by the analytical theory. In this regime we observe the oscillator linewidth broadening, which originates from sudden jumps of frequency of the oscillator arising from random domain wall nucleation and propagation through the sample. This intermittent behavior is revealed through a wavelet analysis that gives superior description of the frequency jumps compared to several other techniques.Comment: 11 pages, 4 figures to appear in PR

A survey of parallel algorithms for fractal image compression

This paper presents a short survey of the key research work that has been undertaken in the application of parallel algorithms for Fractal image compression. The interest in fractal image compression techniques stems from their ability to achieve high compression ratios whilst maintaining a very high quality in the reconstructed image. The main drawback of this compression method is the very high computational cost that is associated with the encoding phase. Consequently, there has been significant interest in exploiting parallel computing architectures in order to speed up this phase, whilst still maintaining the advantageous features of the approach. This paper presents a brief introduction to fractal image compression, including the iterated function system theory upon which it is based, and then reviews the different techniques that have been, and can be, applied in order to parallelize the compression algorithm

The intermittent behavior and hierarchical clustering of the cosmic mass field

The hierarchical clustering model of the cosmic mass field is examined in the context of intermittency. We show that the mass field satisfying the correlation hierarchy $\xi_n\simeq Q_n(\xi_2)^{n-1}$ is intermittent if $\kappa < d$, where $d$ is the dimension of the field, and $\kappa$ is the power-law index of the non-linear power spectrum in the discrete wavelet transform (DWT) representation. We also find that a field with singular clustering can be described by hierarchical clustering models with scale-dependent coefficients $Q_n$ and that this scale-dependence is completely determined by the intermittent exponent and $\kappa$. Moreover, the singular exponents of a field can be calculated by the asymptotic behavior of $Q_n$ when $n$ is large. Applying this result to the transmitted flux of HS1700 Ly$\alpha$ forests, we find that the underlying mass field of the Ly$\alpha$ forests is significantly intermittent. On physical scales less than about 2.0 h$^{-1}$ Mpc, the observed intermittent behavior is qualitatively different from the prediction of the hierarchical clustering with constant $Q_n$. The observations, however, do show the existence of an asymptotic value for the singular exponents. Therefore, the mass field can be described by the hierarchical clustering model with scale-dependent $Q_n$. The singular exponent indicates that the cosmic mass field at redshift $\sim 2$ is weakly singular at least on physical scales as small as 10 h$^{-1}$ kpc.Comment: AAS Latex file, 33 pages,5 figures included, accepted for publication in Ap

On adaptive wavelet estimation of a class of weighted densities

We investigate the estimation of a weighted density taking the form $g=w(F)f$, where $f$ denotes an unknown density, $F$ the associated distribution function and $w$ is a known (non-negative) weight. Such a class encompasses many examples, including those arising in order statistics or when $g$ is related to the maximum or the minimum of $N$ (random or fixed) independent and identically distributed (\iid) random variables. We here construct a new adaptive non-parametric estimator for $g$ based on a plug-in approach and the wavelets methodology. For a wide class of models, we prove that it attains fast rates of convergence under the $\mathbb{L}_p$ risk with $p\ge 1$ (not only for $p = 2$ corresponding to the mean integrated squared error) over Besov balls. The theoretical findings are illustrated through several simulations

Multiresolution analysis in statistical mechanics. I. Using wavelets to calculate thermodynamic properties

The wavelet transform, a family of orthonormal bases, is introduced as a technique for performing multiresolution analysis in statistical mechanics. The wavelet transform is a hierarchical technique designed to separate data sets into sets representing local averages and local differences. Although one-to-one transformations of data sets are possible, the advantage of the wavelet transform is as an approximation scheme for the efficient calculation of thermodynamic and ensemble properties. Even under the most drastic of approximations, the resulting errors in the values obtained for average absolute magnetization, free energy, and heat capacity are on the order of 10%, with a corresponding computational efficiency gain of two orders of magnitude for a system such as a $4\times 4$ Ising lattice. In addition, the errors in the results tend toward zero in the neighborhood of fixed points, as determined by renormalization group theory.Comment: 13 pages plus 7 figures (PNG

A Stochastic Description for Extremal Dynamics

We show that extremal dynamics is very well modelled by the "Linear Fractional Stable Motion" (LFSM), a stochastic process entirely defined by two exponents that take into account spatio-temporal correlations in the distribution of active sites. We demonstrate this numerically and analytically using well-known properties of the LFSM. Further, we use this correspondence to write an exact expressions for an n-point correlation function as well as an equation of fractional order for interface growth in extremal dynamics.Comment: 4 pages LaTex, 3 figures .ep