Distinct Exponents Research Articles

MULTIFRACTAL ANALYSIS OF TEMPERATURE TIME SERIES: DATA FROM BOREHOLES IN KAMCHATKA

In the present work, we focus on the multifractal structure of the microtemperature time series monitored at depth in boreholes, namely in two holes drilled in Kamchatka (Russia). Two monitoring series were performed for approximately two weeks with a 5-second reading interval; thus, each series contains about 230,000 data points. The observed temperatures displayed sharp gradients and large fluctuations over all observed time ranges, reflecting fine structure of the heat transfer process in the shallow subsurface. Recurrence plot (RP) technique was applied for detecting hidden rhythms that generated the time series. The most characteristic feature of the RPs is their web-like structure, indicating quasi-periodic occurrence of the sharp changes. The spectral and the local growth of the second moment techniques were used for distinguishing between the potentially different heat transfer processes. Both spectra show "red noise" behavior, however, exhibit distinct scaling exponents for different frequency domains from near 1.33 for low frequencies to near 3 at the high frequency end of the spectra. Local growth of the second moment technique has revealed the presence of temperature forming process with the characteristic time of approximately 1 minute, that smoothes out generally anti-persistent behavior on shorter time scales, but has a little effect on longer time scales. The investigation was accomplished by the calculation of universal multifractal indices, which characterize temperature fluctuation upon scales in the range from minutes to weeks. Both time series show very similar multifractal behavior. The Hurst exponent, characterizing the degree of non-stationarity, equals to 0.18–0.20, the measure of intermittence C1 amounts to 0.10, and the α-index, characterizing the degree of multifractality, amounts to 1.32 and 1.24 for both boreholes. We speculate on the origin of these common features intervening all over the observed range of time scales in a borehole.

Reinforcement effect of carbon nanofillers in an epoxy resin system: Rheology, molecular dynamics, and mechanical studies

AbstractThe reinforcing effect of carbon nanoparticles in an epoxy resin has been estimated with different approaches based on rheology, molecular dynamics (evaluated by differential scanning calorimetry, dielectric relaxation spectroscopy, and thermally stimulated depolarization current), and dynamic mechanical analysis. Carbon particles aggregate as the volume increases and form a fractal structure in the matrix polymer. The dispersion microstructure has been characterized by its viscoelastic properties and relaxation time spectrum. The scaling of the storage modulus and yield stress with the volume fraction of carbon shows two distinct exponents and has thus been used to determine the critical carbon volume fraction of the network formation (Φ*) for the carbon/epoxy dispersions. At nanofiller concentrations greater than Φ*, the overall mobility of the polymer chains is restricted in both dispersions and solid nanocomposites. Therefore, (1) the relaxation spectrum of the dispersions is strongly shifted toward longer times, (2) the glass‐transition temperature is increased and (3) the relaxation strength of both the secondary (β) and primary (α) relaxations increases in the nanocomposites, with respect to the pure polymer matrix. The dispersion microstructure, consisting of fractal flocs and formed above Φ*, is proposed to play the main role in the reinforcement of nanocomposites. Moreover, the network structure and the interface polymer layer (bond layer), surrounding nanoparticles, increases the relaxation strength and slows the cooperative α relaxation, and this results in an improvement of the mechanical properties. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 522–533, 2005

Composition Fluctuations, Phase Behavior, and Complex Formation in Poly(vinyl methyl ether)/D2O Investigated by Small-Angle Neutron Scattering

Small-angle neutron scattering measurements are presented as a function of temperature and composition for homogeneous mixtures of poly(methyl vinyl ether) and D2O. The experimental data are analyzed to give values of the second-order compositional derivative of the Gibbs energy and the Ornstein−Zernike correlation length. From the experimental data the LCST spinodal temperatures are estimated and values for the parameters in a temperature- and composition-dependent extended Flory−Huggins (F−H) interaction function are determined. Using the extended interaction function the predicted miscibility behavior is in qualitative agreement with the experimental data, and importantly, the remarkable bimodal phase behavior is predicted. In the composition interval 0.75 ≤ wPVME ≤ 0.85 the Ornstein−Zernike correlation lengths follow the mean field sum rule, i.e., ξ2 ≈ [(∂2ΔG/(NkT))/∂φ22]-1. However, in the composition range 0.1 ≤ wPVME ≤ 0.7 the correlation lengths are still proportional to the second-order compositional derivative of the Gibbs energy but the data cluster on separate power laws with distinct exponents. Finally, the experimental data do not support the existence of a stable molecular complex at the investigated temperatures and compositions. Even at the lowest investigated temperature the energy required to induce typical Ornstein−Zernike-like concentration fluctuations is smaller than the thermal energy. Therefore, in the investigated temperature interval it must be concluded that the strength of the specific interactions between D2O and PVME is too weak to speak about complex formation.

Arithmetic multivariate Descartes' rule

Let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] be any number field or p-adic field and consider F :=( f 1 , . . . , f k ) where f 1 , . . . , f k ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /][ x 1 , . . . , x n ] and no more than μ distinct exponent vectors occur in the monomial term expansions of the f i . We prove that F has no more than 1 + ( C n (μ - n ) 3 log (μ - n )) n geometrically isolated roots in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] n , where C is an explicit and effectively computable constant depending only on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]. This gives a significantly sharper arithmetic analogue of Khovanski's Theorem on Real Fewnomials and a higher-dimensional generalization of an earlier result of Hendrik W. Lenstra, Jr. for the special case of a single univariate polynomial. We also present some further refinements of our new bounds and a higher-dimensional generalization of a bound of Lipshitz on p -adic complex roots. Connections to non-Archimedean amoebae and computational complexity (including additive complexity and solving for the geometrically isolated rational roots) are discussed along the way. We thus provide the foundations for an effective arithmetic analogue of fewnomial theory.

A stochastic modeling approach for characterizing the spatial structure of L band radiobrightness temperature imagery

This study focuses on the statistical characterization of the spatial structure of L band microwave radiobrightness temperature fields retrieved during the Southern Great Plains hydrology experiment of 1997 (SGP97). It is found that the radiobrightness temperature observations of interest can be considered nonstationary scaling processes that exhibit persistence or long memory. This implies that the spatial dependence of observations decays very slowly at large separation distances such that models with exponentially decaying autocorrelation (e.g., autoregressive moving average models or exponential models commonly used in geostatistics) are not appropriate. It is further shown that the radiobrightness temperature fields retrieved during SGP97 exhibit distinct scaling behaviors along the horizontal (west to east) and vertical (south to north) directions. A two‐dimensional implementation of the fractionally integrated moving average (FIMA) time series model is shown capable of capturing the spatial autocovariance structure of the observations. The results presented evince that the FIMA paradigm allows for robust estimation of distinct scaling exponents along the horizontal and vertical directions both in stationary and nonstationary situations. Comparisons to alternative heuristic methods for determining the scaling exponent(s) further demonstrate that FIMA models yield estimates with superior accuracy. Additionally, within the FIMA framework it is possible to jointly and accurately model the spatial dependence of radiobrightness temperature for observations separated by short and long distances.

Scaling and universality in the micro-structure of urban space

We present a broad, phenomenological picture of the distribution of the length of open space linear segments, l, derived from maps of 36 cities in 14 different countries. By scaling the Zipf plot of l, we obtain two master curves for a sample of cities, which are not a function of city size. We show that a third class of cities is not easily classifiable into these two universality classes. The cumulative distribution of l displays power-law tails with two distinct exponents, α B =2 and α R =3. We suggest a link between our data and the possibility of observing and modelling urban geometric structures using Lévy processes.

Multiscale resolution of fluidized‐bed pressure fluctuations

AbstractPressure fluctuation signals measured from four different axial locations in a bubbling bed 0.3 m in diameter and 3 m in height were analyzed using multiple approaches, including wavelet transform, Hurst analysis, multiscale resolution, and time‐delay embedding. After examining decomposition residuals using different compact support Daubechies wavelets, the Daubechies second‐order wavelet was chosen as an optimal wavelet for decomposing pressure signals. Hurst analysis of the decomposed signals shows that the measured pressure fluctuations can be resolved to three characteristic scales: bifractal mesoscale signals with two distinct Hurst exponents; monofractal micro‐ and macroscale signals with only one characteristic Hurst exponent. Energy profiles of the three scale components confirm that the measured pressure signals mainly reflect the mesoscale component. Time‐delay embedding analysis of three scale signals demonstrates that the microscale dynamics is more complex than the mesoscale dynamics, and the mesoscale dynamics is more complex than the macroscale dynamics. That this result cannot be found solely from Hurst analysis shows the importance of integrating multiple approaches for characterizing the complexity of fluidized systems.

Electronic transport properties of SiGe alloys and heterostructures grown by Sb assisted MBE

Electronic transport properties have been measured in thin pseudomorphic SiGe alloys and heterostructures, grown by surfactant mediated epitaxy (SME), in order to investigate the unintentional doping effects caused by Sb, used as surfactant. In alloy films, we measured at 300 K, a Hall mobility ( μ H=20 cm 2 V −1 s −1) at least one order of magnitude lower than the undoped material ( μ H=100 cm 2 V −1 s −1), grown in similar conditions, but without surfactant. At low temperatures (40 K< T<75 K) we obtained μ H values of about 3.6×10 4 cm 2 V −1 s −1 for SME grown material and up to 5.5×10 4 cm 2 V −1 s −1, for undoped ones. Above 75 K, all the alloy samples showed a μ H decrease with the temperature steeply than the T −1.5 dependence. Moreover, in undoped materials we observed two distinct power law exponents. In SME (Si mGe n) p heterostructures we measured μ H values generally lower than 100 cm 2 V −1 s −1 at 300 K, due to the Sb high doping level. The strong doping effect caused by Sb in SME SiGe films suggests the use of different surfactant elements or alternative growth techniques in order to fabricate Si–Ge heterostructure based devices.

Invariant measure and Lyapunov exponents for birational maps of P^2

In this paper we construct and study a natural invariant measure for a birational self-map of the complex projective plane. Our main hypothesis - that the birational map be "separating" - is a condition on the indeterminacy set of the map. We prove that the measure is mixing and that it has distinct Lyapunov exponents. Under a further hypothesis on the indeterminacy set we show that the measure is hyperbolic in the sense of Pesin theory. In this case, we also prove that saddle periodic points are dense in the support of the measure.

Particle Adhesion to Highly Compliant Substrates: Anomalous Power-Law Dependence of the Contact Radius on Particle Radius

The adhesion-induced contact radius and meniscus height between soda-lime glass particles and three highly compliant polyurethane substrates (Young's modulus between 4 kPa and 0.7 MPa) were measured using scanning electron microscopy as a function of particle radius. Two distinct power-law exponents were found. A value of about 2/3, which is in good agreement with the predictions of the JKR theory (Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc., London Ser. A 1971, 324, 301), was found for larger particles. However, the experimentally obtained exponent for smaller particles was found to be approximately 3/4. The present results are compared with the predictions of the large-scale deformation model of Maugis (Maugis, D. Langmuir 1995, 11, 679) as well as a self-consistent surface-energy model. Although both models contain some interesting features, neither theory can fully account for the observed behavior.

Weak and strong dynamic scaling in a one-dimensional driven coupled-field model: effects of kinematic waves.

We study the coupled dynamics of the displacement fields in a one-dimensional coupled-field model for drifting crystals, first proposed by Lahiri and Ramaswamy [Phys. Rev. Lett. 79, 1150 (1997)]. We present some exact results for the steady state and the current in the lattice version of the model for a special subspace in the parameter space, within the region where the model displays kinematic waves. We use these results to construct the effective continuum equations corresponding to the lattice model. These equations decouple at the linear level in terms of the eigenmodes. We examine the long-time, large-distance properties of the correlation functions of the eigenmodes by using symmetry arguments, Monte Carlo simulations, and self-consistent mode-coupling methods. For most parameter values, the scaling exponents of the Kardar-Parisi-Zhang equation are obtained. However, for certain symmetry-determined values of the coupling constants the two eigenmodes, although nonlinearly coupled, are characterized by two distinct dynamic exponents. We discuss the possible application of the dynamic renormalization group in this context.

Evidence of two distinct dynamic critical exponents in connection with vortex physics.

The dynamic critical exponent z is determined from numerical simulations for the three-dimensional (3D) lattice Coulomb gas (LCG) and the 3D XY models with relaxational dynamics. It is suggested that the dynamics is characterized by two distinct dynamic critical indices z0 and z related to the divergence of the relaxation time tau by tau proportional to xi(z0) and tau proportional to k(-z), where xi is the correlation length and k the wave vector. The values determined are z0 approximately 1.5 and z approximately 1 for the 3D LCG and z0 approximately 1.5 and z approximately 2 for the 3D XY model. Comparisons with other results are discussed.

Extremal Dynamics Model on Evolving Networks

We investigate an extremal dynamics model of evolution with variable number of units. Because of addition and removal of the units, the topology of the network evolves and the network splits into several clusters. The activity is mostly concentrated in the largest cluster. The time dependence of the number of units exhibits intermittent structure. The self-organized criticality is manifested by a power-law distribution of forward avalanches, but two regimes with distinct exponents $\ensuremath{\tau}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.98\ifmmode\pm\else\textpm\fi{}0.04$ and ${\ensuremath{\tau}}^{\ensuremath{'}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.65\ifmmode\pm\else\textpm\fi{}0.05$ are found. The distribution of extinction sizes obeys a power law with exponent $2.32\ifmmode\pm\else\textpm\fi{}0.05$.

Mesoscopic behavior near a two-dimensional metal-insulator transition

We study conductance fluctuations in a two-dimensional electron gas as a function of chemical potential (or gate voltage) from the strongly insulating to the metallic regime. Power spectra of the fluctuations decay with two distinct exponents (1/v_l and 1/v_h). For conductivity $\sigma\sim 0.1 e^{2}/h$, we find a third exponent (1/v_i) in the shortest samples, and non-monotonic dependence of v_i and v_l on \sigma. We study the dependence of v_i, v_l, v_h, and the variances of corresponding fluctuations on \sigma, sample size, and temperature. The anomalies near $\sigma\simeq 0.1 e^{2}/h$ indicate that the dielectric response and screening length are critically behaved, i.e. that Coulomb correlations dominate the physics.

Three-dimensional thirring model for small Nf

We formulate the three-dimensional Thirring model on a space-time lattice and study it for various even numbers of fermion flavours N f by Monte Carlo simulation. We find clear evidence for spontaneous chiral symmetry breaking at strong coupling, contradicting the predictions of the 1 N f expansion. The critical point appears to correspond to an ultraviolet fixed point of the renormalisation group; a fit to a RG-inspired equation of state in the vicinity of the fixed point yields distinct critical exponents for N f = 2 and N f = 4, while no fit is found for N f = 6, suggesting there is a critical number N fc < 6 beyond which no chiral symmetry breaking occurs. The spectrum of the N f = 2 theory is studied; the states examined vary sharply but continuously across the transition.

Mesoscopic behavior near a two-dimensional metal-insulator transition

We study conductance fluctuations in a two-dimensional electron system as a function of chemical potential (or gate voltage) from the strongly insulating to the metallic regime. Power spectra of the fluctuations decay with two distinct exponents ${(1/v}_{l}$ and ${1/v}_{h}).$ For conductivity $\ensuremath{\sigma}\ensuremath{\sim}{0.1e}^{2}/h,$ we find a third exponent ${(1/v}_{i})$ in the shortest samples, and nonmonotonic dependence of ${v}_{i}$ and ${v}_{l}$ on \ensuremath{\sigma}. We study the dependence of ${v}_{i},$ ${v}_{l},$ ${v}_{h},$ and the variances of corresponding fluctuations on \ensuremath{\sigma}, sample size, and temperature. The anomalies near $\ensuremath{\sigma}\ensuremath{\simeq}{0.1e}^{2}/h$ indicate that the dielectric response and screening length are critically behaved, i.e., that Coulomb correlations dominate the physics.

Zariski closure and the dimension of the Gaussian law of the product of random matrices. I

We prove the Central Limit Theorem for products of i.i.d. random matrices. The main aim is to find the dimension of the corresponding Gaussian law. It turns out that ifG is the Zariski closure of a group generated by the support of the distribution of our matrices, and ifG is semi-simple, then the dimension of the Gaussian law is equal to the dimension of the diagonal part of Cartan decomposition ofG. In this article we present a detailed exposition of results announced in [GGu]. For reasons explained in the introduction, this part is devoted to the case ofSL(m, ℝ) group. The general semi-simple Lie group will be considered in the second part of the work. The central limit theorem for products of independent random matrices is our main topic, and the results obtained complete to a large extent the general picture of the subject. The proofs rely on methods from two theories. One is the theory of asymptotic behaviour of products of random matrices itself. As usual, the existence of distinct Lyapunov exponents is the most important fact here. The other is the theory of algebraic groups. We want to point out that algebraic language and methods play a very important role in this paper. In fact, this mixture of methods has already been used for the study of Lyapunov exponents in [GM1, GM2, GR3]. We believe that it is impossible to avoid the algebraic approach if one aims to obtain complete and effective answers to natural problems arising in the theory of products of random matrices. In order also to present the general picture of the subject we describe several results which are well known. Some of these can be proven for stationary sequences of matrices, others are true also for infinite dimensional operators (see e.g. [BL, O, GM2, L, R]). But our main concern is with independent matrices, in which case very precise and constructive statements can be obtained.

Anisotropic scaling in threshold critical dynamics of driven directed lines.

The dynamical critical behavior of a single directed line driven in a random medium near the depinning threshold is studied both analytically (by renormalization group) and numerically, in the context of a flux line in a type-II superconductor with a bulk current J\ensuremath{\rightarrow}. In the absence of transverse fluctuations, the system reduces to recently studied models of interface depinning. In most cases, the presence of transverse fluctuations is found not to influence the critical exponents that describe longitudinal correlations. For a manifold with d=4-\ensuremath{\epsilon} internal dimensions, longitudinal fluctuations in an isotropic medium are described by a roughness exponent ${\mathrm{\ensuremath{\zeta}}}_{\mathrm{\ensuremath{\parallel}}}$=\ensuremath{\epsilon}/3 to all orders in \ensuremath{\epsilon}, and a dynamical exponent ${\mathit{z}}_{\mathrm{\ensuremath{\parallel}}}$=2-2\ensuremath{\epsilon}/9+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). Transverse fluctuations have a distinct and smaller roughness exponent ${\mathrm{\ensuremath{\zeta}}}_{\mathrm{\ensuremath{\perp}}}$=${\mathrm{\ensuremath{\zeta}}}_{\mathrm{\ensuremath{\parallel}}}$-d/2 for an isotropic medium. Furthermore, their relaxation is much slower, characterized by a dynamical exponent ${\mathit{z}}_{\mathrm{\ensuremath{\perp}}}$=${\mathit{z}}_{\mathrm{\ensuremath{\parallel}}}$+1/\ensuremath{\nu}, where \ensuremath{\nu}=1/(2-${\mathrm{\ensuremath{\zeta}}}_{\mathrm{\ensuremath{\parallel}}}$) is the correlation length exponent. The predicted exponents agree well with numerical results for a flux line in three dimensions. As in the case of interface depinning models, anisotropy leads to additional universality classes. A nonzero Hall angle, which has no analogue in the interface models, also affects the critical behavior. \textcopyright{} 1996 The American Physical Society.

Renormalization-group study of a hybrid driven diffusive system.

We consider a d-dimensional stochastic lattice gas of interacting particles, diffusing under the influence of a short-ranged, attractive Ising Hamiltonian and a ``hybrid'' external field which is a superposition of a uniform and an annealed random drive, acting in orthogonal subspaces of dimensions one and m, respectively. Driven into a nonequilibrium steady state, the half-filled system phase segregates via a continuous transition at a field-dependent critical temperature. Using renormalization-group techniques, we show that its critical behavior falls into a new universality class with upper critical dimension ${\mathit{d}}_{\mathit{c}}$=5-m, characterized by two distinct anisotropy exponents, which, like all other indices, are computed exactly to all orders in perturbation theory.

Multifractal magnetization on hierarchical lattices

A new approach is applied to show that the local magnetization of the ferromagnetic Ising model on hierarchical lattices has a multifractal structure at the critical point. Thef(α) function characterizing its multifractality is presented and discussed for the diamond hierarchical lattice. Distinct exact critical exponents for the average magnetization and for the local magnetization of the deepest sites are found. The average magnetization (as function of the temperature) is also calculated. The critical exponent of the susceptibility is estimated using finite-size scale arguments.