SELF-SIMILARITY OF FREE STOCHASTIC PROCESSES

2006 ◽  
Vol 09 (03) ◽  
pp. 451-469 ◽  
Author(s):  
ZHAOZHI FAN
Keyword(s):  
Stochastic Processes ◽  
Self Similarity ◽  
Strict Stability ◽  
Classical Analogue ◽  
Self Similar

In this paper we study self-similarity of free stochastic processes. We establish the noncommutative counterpart of Lamperti's self-similar processes. We develop the characterization of noncommutative self-similar processes through a modification of Voiculescu transform, the free cumulant transform. We study the connection between free self-similarity, strict ⊞-stability and ⊞-self-decomposability. In particular, we derive the properties of free self-similar processes and their connection to strict ⊞-stability and ⊞-self-decomposability, that turn out to be consistent with their classical analogue.

WEAK SELF-SIMILAR SETS IN SEPARABLE COMPLETE METRIC SPACES

Fractals ◽  
2017 ◽  
Vol 25 (02) ◽  
pp. 1750021
Author(s):  
R. K. ASWATHY ◽  
SUNIL MATHEW
Keyword(s):  
Metric Spaces ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
Finite Union ◽  
Self Similarity ◽  
Complete Metric Spaces ◽  
Self Similar ◽  
Necessary And Sufficient ◽  
Physical Phenomena

Self-similarity is a common tendency in nature and physics. It is wide spread in geo-physical phenomena like diffusion and iteration. Physically, an object is self-similar if it is invariant under a set of scaling transformation. This paper gives a brief outline of the analytical and set theoretical properties of different types of weak self-similar sets. It is proved that weak sub self-similar sets are closed under finite union. Weak sub self-similar property of the topological boundary of a weak self-similar set is also discussed. The denseness of non-weak super self-similar sets in the set of all non-empty compact subsets of a separable complete metric space is established. It is proved that the power of weak self-similar sets are weak super self-similar in the product metric and weak self-similarity is preserved under isometry. A characterization of weak super self-similar sets using weak sub contractions is also presented. Exact weak sub and super self-similar sets are introduced in this paper and some necessary and sufficient conditions in terms of weak condensation IFS are presented. A condition for a set to be both exact weak super and sub self-similar is obtained and the denseness of exact weak super self similar sets in the set of all weak self-similar sets is discussed.

Some Notes on Function Spaces and Dirichlet Forms on Self-Similar Sets

2006 ◽  
Vol 6 (3) ◽  
Author(s):  
Yuan Liu ◽  
Zhi-Ying Wen
Keyword(s):  
Dirichlet Forms ◽  
Function Spaces ◽  
Metric Measure Spaces ◽  
Kernel Estimates ◽  
Self Similarity ◽  
Equivalent Characterization ◽  
Measure Spaces ◽  
Self Similar ◽  
Sierpinski Gaskets

AbstractGrigor’yan, Hu and Lau [10] introduced sub-Gaussian heat kernels on general metric measure spaces and defined a family of function spaces to characterize the domain of associated Dirichlet forms. In this paper, we will improve their results about norm equivalence. As an application, we construct self-similar Dirichlet forms on a class of self-similar sets containing the Sierpiński gaskets and carpets. Then we prove the Poincaré inequality and give effective resistance estimates by the self-similarity. Consequently, we have a new equivalent characterization of heat kernel estimates through function spaces with strong recurrent condition.

BASIC PROPERTIES AND CHARACTERIZATION OF STOCHASTICALLY SELF-SIMILAR PROCESSES IN Rd

Fractals ◽  
1999 ◽  
Vol 07 (01) ◽  
pp. 59-78 ◽  
Author(s):  
DANIELE VENEZIANO
Keyword(s):  
Random Variable ◽  
Random Processes ◽  
Affine Transformations ◽  
Spectral Measures ◽  
Self Similarity ◽  
Basic Properties ◽  
Properties And Characterization ◽  
Classical Notion ◽  
Self Similar

The classical notion of self-similarity (ss) for random X(t) as invariance under the group of positive affine transformations {X→ arX, t→rt; ar>0} is extended by allowing ar to be a random variable. The resulting property of "stochastic self-similarity" (sss) is applied to both ordinary and generalized random processes in Rd, d≥1. The class of sss processes seems to correspond to that of multifractal processes (the latter are variously defined in the literature). The spectral measures of ordinary and generalized sss processes are themselves stochastically self-similar. Two characterizations of ss processes by Lamperti are extended to the sss case and several basic properties of ordinary and generalized sss processes are derived.

A WAVELET METHOD COUPLED WITH QUASI-SELF-SIMILAR STOCHASTIC PROCESSES FOR TIME SERIES APPROXIMATION

2011 ◽  
Vol 09 (05) ◽  
pp. 685-711 ◽  
Author(s):  
MOHAMED ESSAIED HAMRITA ◽  
NIDHAL BEN ABDALLAH ◽  
ANOUAR BEN MABROUK
Keyword(s):  
Time Series ◽  
Stochastic Processes ◽  
Scaling Laws ◽  
Financial Time Series ◽  
Wavelet Theory ◽  
Self Similarity ◽  
Wavelet Method ◽  
Financial Time ◽  
Functional Methods ◽  
Self Similar

Scaling laws and generally self-similar structures are now well known facts in financial time series. Furthermore, these signals are characterized by the presence of stochastic behavior allowing their analysis with pure functional methods being incomplete. In the present paper, some existing models are reviewed and modified, based on wavelet theory and self-similarity, to recover multi-scaling cases for approximating financial signals. The resulting models are then tested on some empirical examples and analyzed for error estimates.

Characterization of Gaussian self-similar stochastic processes using wavelet-based informational tools

2007 ◽  
Vol 75 (2) ◽  
Author(s):  
L. Zunino ◽  
D. G. Pérez ◽  
M. T. Martín ◽  
A. Plastino ◽  
M. Garavaglia ◽  
...  
Keyword(s):  
Stochastic Processes ◽  
Self Similar

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.

TUBE FORMULAS FOR GRAPH-DIRECTED FRACTALS

Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 349-361 ◽  
Author(s):  
BÜNYAMIN DEMÍR ◽  
ALI DENÍZ ◽  
ŞAHIN KOÇAK ◽  
A. ERSIN ÜREYEN
Keyword(s):  
The Self ◽  
Self Similarity ◽  
Complex Dimensions ◽  
Tubular Neighborhoods ◽  
Self Similar ◽  
Tube Formulas

Lapidus and Pearse proved recently an interesting formula about the volume of tubular neighborhoods of fractal sprays, including the self-similar fractals. We consider the graph-directed fractals in the sense of graph self-similarity of Mauldin-Williams within this framework of Lapidus-Pearse. Extending the notion of complex dimensions to the graph-directed fractals we compute the volumes of tubular neighborhoods of their associated tilings and give a simplified and pointwise proof of a version of Lapidus-Pearse formula, which can be applied to both self-similar and graph-directed fractals.

scholarly journals On the scaling of shear-driven entrainment: a DNS study

2013 ◽  
Vol 732 ◽  
pp. 150-165 ◽  
Author(s):  
Harm J. J. Jonker ◽  
Maarten van Reeuwijk ◽  
Peter P. Sullivan ◽  
Edward G. Patton
Keyword(s):  
Reynolds Number ◽  
Lower Boundary ◽  
Square Root ◽  
Simulation Design ◽  
Self Similarity ◽  
Original Experiment ◽  
Turbulent Layer ◽  
Entrainment Velocity ◽  
Constant Shear Stress ◽  
Self Similar

AbstractThe deepening of a shear-driven turbulent layer penetrating into a stably stratified quiescent layer is studied using direct numerical simulation (DNS). The simulation design mimics the classical laboratory experiments by Kato & Phillips (J. Fluid Mech., vol. 37, 1969, pp. 643–655) in that it starts with linear stratification and applies a constant shear stress at the lower boundary, but avoids sidewall and rotation effects inherent in the original experiment. It is found that the layers universally deepen as a function of the square root of time, independent of the initial stratification and the Reynolds number of the simulations, provided that the Reynolds number is large enough. Consistent with this finding, the dimensionless entrainment velocity varies with the bulk Richardson number as$R{i}^{- 1/ 2} $. In addition, it is observed that all cases evolve in a self-similar fashion. A self-similarity analysis of the conservation equations shows that only a square root growth law is consistent with self-similar behaviour.

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