scholarly journals Information Theory for Non-Stationary Processes with Stationary Increments

Entropy ◽  
2019 ◽  
Vol 21 (12) ◽  
pp. 1223 ◽  
Author(s):  
Carlos Granero-Belinchón ◽  
Stéphane G. Roux ◽  
Nicolas B. Garnier
Keyword(s):  
Information Theory ◽  
Probability Distributions ◽  
Entropy Rate ◽  
Integration Time ◽  
Stationary Processes ◽  
Self Similarity ◽  
Single Realization ◽  
Self Similar ◽  
Non Gaussian ◽  
Do So

We describe how to analyze the wide class of non-stationary processes with stationary centered increments using Shannon information theory. To do so, we use a practical viewpoint and define ersatz quantities from time-averaged probability distributions. These ersatz versions of entropy, mutual information, and entropy rate can be estimated when only a single realization of the process is available. We abundantly illustrate our approach by analyzing Gaussian and non-Gaussian self-similar signals, as well as multi-fractal signals. Using Gaussian signals allows us to check that our approach is robust in the sense that all quantities behave as expected from analytical derivations. Using the stationarity (independence on the integration time) of the ersatz entropy rate, we show that this quantity is not only able to fine probe the self-similarity of the process, but also offers a new way to quantify the multi-fractality.

Zeta Functions of Discrete Self-similar Sets and Applications to Point Configuration and Sum–Product-Type Problems

2018 ◽  
Vol 2019 (24) ◽  
pp. 7733-7777
Author(s):  
Driss Essouabri ◽  
Ben Lichtin
Keyword(s):  
Zeta Functions ◽  
Product Type ◽  
Lattice Points ◽  
Self Similarity ◽  
Uniform Bounds ◽  
Point Configuration ◽  
Analytical Properties ◽  
Finite Set ◽  
Self Similar ◽  
Do So

Abstract In this paper we first prove analytical properties of zeta functions for discrete subsets of ${\mathbb{R}}^{n}$ that exhibit “self-similarity” with respect to an arbitrary finite set of (affine) similarities. We then show how such properties help solve Point Configuration resp. Sum–Product-Type problems over ${\mathbb{Z}}$. We do so by first extending a classic one-variable Tauberian theorem of Ingham to several variables to derive a nontrivial lower bound on the average of coefficients of an appropriate multivariate zeta function. We then combine this with well-known results from Diophantine Geometry that prove uniform bounds for the density of lattice points in families of algebraic hypersurfaces.

scholarly journals Estimation of Spectrum Requirements for Mobile Networks With Self-Similar Traffic, Handover, and Frequency Reuse

2010 ◽  
Vol 6 (4) ◽  
pp. 281-291
Author(s):  
Won Seok Yang ◽  
Eun Saem Yang ◽  
Hwa J. Kim ◽  
Dae K. Kim
Keyword(s):  
Mobile Networks ◽  
Probability Distributions ◽  
Frequency Reuse ◽  
Estimation Methods ◽  
Data Traffic ◽  
Load Spectrum ◽  
Self Similarity ◽  
Average Cell ◽  
Self Similar ◽  
The Impact

This paper considers self-similarity in data traffic, handover, and frequency reuse to estimate the spectrum requirements of mobile networks. An approximate average cell capacity subject to a delay requirement and self-similar traffic is presented. It is shown that handover traffic can be an additional load. Spectrum requirements are calculated based on carrier demand instead of spectral efficiency, as at least one carrier is necessary to transmit even 1 bit. The cell-split operation is considered under frequency reuse of one. Estimation methods are presented using cell traffic in two cases. First, a procedure is presented that estimates cell traffic from previous networks. Second, cell traffic is assumed to follow probability distributions. Numerical examples demonstrate the impact of self-similarity, handover, and the proportion of cell-split occurrences on the spectrum requirements.

scholarly journals Asymptotic behavior for quadratic variations of non-Gaussian multiparameter Hermite random fields

2019 ◽  
Vol 39 (2) ◽  
pp. 385-401
Author(s):  
Thi Thanh Diu Tran
Keyword(s):  
Similarity Index ◽  
Random Variable ◽  
Self Similarity ◽  
Similarity Parameter ◽  
Rosenblatt Process ◽  
Self Similar ◽  
Non Gaussian ◽  
Proper Renormalization ◽  
Quadratic Variations ◽  
Rosenblatt Sheet

Let Zt q,H t∈[0,1]d denote a d-parameter Hermite random field of order q ≥ 1 and self-similarity parameter H = H₁, . . . ,Hd ∈  ½, 1d. This process is H-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion q = 1, d = 1, fractional Brownian sheet q = 1, d ≥ 2, the Rosenblatt process q = 2, d = 1 as well as the Rosenblatt sheet q = 2, d ≥ 2. For any q ≥ 2, d ≥ 1 and H ∈ ½, 1d we show in this paper that a proper renormalization of the quadratic variation of Zq,H converges in L2Ω to a standard d-parameter Rosenblatt random variable with self-similarity index H' = 1 + 2H − 2/q.

scholarly journals Extended power-law scaling of air permeabilities measured on a block of tuff

2012 ◽  
Vol 16 (1) ◽  
pp. 29-42 ◽  
Author(s):  
M. Siena ◽  
A. Guadagnini ◽  
M. Riva ◽  
S. P. Neuman
Keyword(s):  
Power Law ◽  
Structure Functions ◽  
Self Similarity ◽  
Scaling Exponents ◽  
Multi Scale ◽  
Sample Structure ◽  
Heavy Tailed ◽  
Non Gaussian ◽  
Nonlinear Fashion ◽  
Extended Power

Abstract. We use three methods to identify power-law scaling of multi-scale log air permeability data collected by Tidwell and Wilson on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on q-th-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents, ξ(q), of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. The finding by Tidwell and Wilson that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power-law scaling of our data, as well as of non-Gaussian heavy-tailed signals subordinated to tfBm, are extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders.

scholarly journals Onset of the Mach Reflection of Zel’dovich–von Neumann–Döring Detonations

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 314
Author(s):  
Tianyu Jing ◽  
Huilan Ren ◽  
Jian Li
Keyword(s):  
Hot Spot ◽  
Mach Reflection ◽  
Near Field ◽  
The Self ◽  
Small Scale ◽  
Mach Stem ◽  
Self Similarity ◽  
Von Neumann ◽  
Similarity Problem ◽  
Self Similar

The present study investigates the similarity problem associated with the onset of the Mach reflection of Zel’dovich–von Neumann–Döring (ZND) detonations in the near field. The results reveal that the self-similarity in the frozen-limit regime is strictly valid only within a small scale, i.e., of the order of the induction length. The Mach reflection becomes non-self-similar during the transition of the Mach stem from “frozen” to “reactive” by coupling with the reaction zone. The triple-point trajectory first rises from the self-similar result due to compressive waves generated by the “hot spot”, and then decays after establishment of the reactive Mach stem. It is also found, by removing the restriction, that the frozen limit can be extended to a much larger distance than expected. The obtained results elucidate the physical origin of the onset of Mach reflection with chemical reactions, which has previously been observed in both experiments and numerical simulations.

scholarly journals Depressurization-Induced Nucleation in the “Polylactide-Carbon Dioxide” System: Self-Similarity of the Bubble Embryos Expansion

Polymers ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1115
Author(s):  
Dmitry Zimnyakov ◽  
Marina Alonova ◽  
Ekaterina Ushakova
Keyword(s):  
Initial Rate ◽  
Initial Pressure ◽  
External Pressure ◽  
Self Similarity ◽  
Embryo Formation ◽  
Linear Behavior ◽  
Joint Influence ◽  
External Pressures ◽  
Adiabatic Cooling ◽  
Self Similar

Self-similar expansion of bubble embryos in a plasticized polymer under quasi-isothermal depressurization is examined using the experimental data on expansion rates of embryos in the CO2-plasticized d,l-polylactide and modeling the results. The CO2 initial pressure varied from 5 to 14 MPa, and the depressurization rate was 5 × 10−3 MPa/s. The constant temperature in experiments was in a range from 310 to 338 K. The initial rate of embryos expansion varied from ≈0.1 to ≈10 µm/s, with a decrease in the current external pressure. While modeling, a non-linear behavior of CO2 isotherms near the critical point was taken into account. The modeled data agree satisfactorily with the experimental results. The effect of a remarkable increase in the expansion rate at a decreasing external pressure is interpreted in terms of competing effects, including a decrease in the internal pressure, an increase in the polymer viscosity, and an increase in the embryo radius at the time of embryo formation. The vanishing probability of finding the steadily expanding embryos for external pressures around the CO2 critical pressure is interpreted in terms of a joint influence of the quasi-adiabatic cooling and high compressibility of CO2 in the embryos.

scholarly journals Self-similar resistive circuits as fractal-like structures

2017 ◽  
Vol 40 (1) ◽  
Author(s):  
Claudio Xavier Mendes dos Santos ◽  
Carlos Molina Mendes ◽  
Marcelo Ventura Freire
Keyword(s):  
Scale Invariance ◽  
Self Similarity ◽  
General Properties ◽  
Modern Physics ◽  
Resistive Networks ◽  
Self Similar ◽  
And Mathematics ◽  
Definition Of ◽  
The Individual

Fractals play a central role in several areas of modern physics and mathematics. In the present work we explore resistive circuits where the individual resistors are arranged in fractal-like patterns. These circuits have some of the characteristics typically found in geometric fractals, namely self-similarity and scale invariance. Considering resistive circuits as graphs, we propose a definition of self-similar circuits which mimics a self-similar fractal. General properties of the resistive circuits generated by this approach are investigated, and interesting examples are commented in detail. Specifically, we consider self-similar resistive series, tree-like resistive networks and Sierpinski’s configurations with resistors.

Sign in / Sign up

Export Citation Format

Share Document