scholarly journals Centered Polygonal Lacunary Sequences

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 943 ◽  
Author(s):  
Keith Sullivan ◽  
Drew Rutherford ◽  
Darin J. Ulness
Keyword(s):  
Rotational Symmetry ◽  
Natural Boundary ◽  
Self Similarity ◽  
Lacunary Sequences ◽  
Special Limit ◽  
Similarity Scaling ◽  
Scaling Features ◽  
Infinite Sum ◽  
Natural Way ◽  
P Cycle

Lacunary functions based on centered polygonal numbers have interesting features which are distinct from general lacunary functions. These features include rotational symmetry of the modulus of the functions and a notion of polished level sets. The behavior and characteristics of the natural boundary for centered polygonal lacunary sequences are discussed. These systems are complicated but, nonetheless, well organized because of their inherent rotational symmetry. This is particularly apparent at the so-called symmetry angles at which the values of the sequence at the natural boundary follow a relatively simple 4 p -cycle. This work examines special limit sequences at the natural boundary of centered polygonal lacunary sequences. These sequences arise by considering the sequence of values along integer fractions of the symmetry angle for centered polygonal lacunary functions. These sequences are referred to here as p-sequences. Several properties of the p-sequences are explored to give insight in the centered polygonal lacunary functions. Fibered spaces can organize these cycles into equivalence classes. This then provides a natural way to approach the infinite sum of the actual lacunary function. It is also seen that the inherent organization of the centered polygonal lacunary sequences gives rise to fractal-like self-similarity scaling features. These features scale in simple ways.

Experimental Evidence for Differences in the Extended Self-Similarity Scaling Laws between Fluid and Magnetohydrodynamic Turbulent Flows

1995 ◽  
Vol 75 (17) ◽  
pp. 3110-3113 ◽  
Author(s):  
Vincenzo Carbone ◽  
Pierluigi Veltri ◽  
Roberto Bruno
Keyword(s):  
Experimental Evidence ◽  
Turbulent Flows ◽  
Scaling Laws ◽  
Self Similarity ◽  
Extended Self ◽  
Similarity Scaling

scholarly journals SELF-SIMILARITY TECHNIQUES FOR CHAOTIC ATTRACTORS WITH MANY SCROLLS USING STEP SERIES SWITCHING

2021 ◽  
Vol 26 (4) ◽  
pp. 591-611
Author(s):  
Emile Franc Doungmo Goufo ◽  
Chokkalingam Ravichandran ◽  
Gunvant A. Birajdar
Keyword(s):  
Chaos Theory ◽  
Network Traffic ◽  
Chaotic Systems ◽  
Switching Process ◽  
Traffic Prediction ◽  
Chaotic Attractors ◽  
Numerical Techniques ◽  
Self Similarity ◽  
Mathematical Concepts ◽  
Natural Way

Highly applied in machining, image compressing, network traffic prediction, biological dynamics, nerve dendrite pattern and so on, self-similarity dynamic represents a part of fractal processes where an object is reproduced exactly or approximately exact to a part of itself. These reproduction processes are also very important and captivating in chaos theory. They occur naturally in our environment in the form of growth spirals, romanesco broccoli, trees and so on. Seeking alternative ways to reproduce self-similarity dynamics has called the attention of many authors working in chaos theory since the range of applications is quite wide. In this paper, three combined notions, namely the step series switching process, the Julia’s technique and the fractal-fractional dynamic are used to create various forms of self-similarity dynamics in chaotic systems of attractors, initially with two, five and seven scrolls. In each case, the solvability of the model is addressed via numerical techniques and related graphical simulations are provided. It appears that the initial systems are able to trigger a self-similarity process that generates the exact or approximately exact copy of itself or part of itself. Moreover, the dynamics of the copies are impacted by some model’s parameters involved in the process. Using mathematical concepts to re-create features that usually occur in a natural way proves to be a prowess as related applications are many for engineers.

2. Complex physical systems (CPS)

2014 ◽  
Author(s):  
John H. Holland
Keyword(s):  
Nearest Neighbors ◽  
Power Laws ◽  
Self Similarity ◽  
Physical Systems ◽  
Self Organized ◽  
Self Organized Criticality ◽  
Newton’S Laws ◽  
Similarity Scaling ◽  
Physical Laws ◽  
Over Time

‘Complex physical systems’ considers the characteristics of complex physical systems (CPS), which are often geometric (specifically, lattice-like) arrays of elements, in which interactions typically depend only on effects propagated from nearest neighbors. The elements of a CPS follow fixed physical laws, usually expressed by differential equations—Newton’s laws of gravity and Maxwell’s laws of electromagnetism are cases in point. Neither the laws nor the elements change over time; only the positions of the elements change. CPS show several properties: self-organized criticality, self-similarity, scaling, and power laws. Examples of these properties—such as, snowflake curves, fractals, networks, dynamics, and symmetry-breaking—are discussed.

scholarly journals Helical self-similarity of tip vortex cores

2018 ◽  
Vol 859 ◽  
pp. 1084-1097 ◽  
Author(s):  
Valery L. Okulov ◽  
Ivan K. Kabardin ◽  
Robert F. Mikkelsen ◽  
Igor V. Naumov ◽  
Jens N. Sørensen
Keyword(s):  
Experimental Studies ◽  
Simple Relation ◽  
Tip Vortex ◽  
Self Similarity ◽  
Local Flow ◽  
Tip Vortices ◽  
Longitudinal Vortices ◽  
The Core ◽  
Helical Vortices ◽  
Similarity Scaling

The present work investigates local flow structures and the downstream evolution of the core of helical tip vortices generated by a three-bladed rotor. Earlier experimental studies have shown that the core of a helical tip vortex exhibits a local helical symmetry with a simple relation between the axial and azimuthal velocities. In the present study, a self-similarity scaling argument further describes the downstream development of the vortex core. Self-similarity has up to now only been investigated for longitudinal vortices and it is the first time that helical vortices have become the subject of such an analysis. Combining symmetry arguments from previous studies on helical vortices with novel experiments and knowledge regarding the self-similarity evolution of the core of longitudinal vortices, a new model describing what is referred to as ‘helical self-similarity’ is proposed. The generality of the model is verified and supported by experimental data. The proposed model is important for fundamental understanding of the behaviour of helical vortices, with a range of applications in both industry and nature. Examples of this are tip vortices behind aerodynamic devices, such as vortex generators, and fixed and rotary aircraft, and in combustion chambers and cyclone separators.

Experimental Evidence for Differences in the Extended Self-Similarity Scaling Laws between Fluid and Magnetohydrodynamic Turbulent Flows

1996 ◽  
Vol 76 (12) ◽  
pp. 2206-2206 ◽  
Author(s):  
Vincenzo Carbone ◽  
Pierluigi Veltri ◽  
Roberto Bruno
Keyword(s):  
Experimental Evidence ◽  
Turbulent Flows ◽  
Scaling Laws ◽  
Self Similarity ◽  
Extended Self ◽  
Similarity Scaling

Classical atom–diatom scattering: Self‐similarity, scaling laws, and renormalization

1993 ◽  
Vol 99 (4) ◽  
pp. 2765-2780 ◽  
Author(s):  
Ampawan Tiyapan ◽  
Charles Jaffé
Keyword(s):  
Scaling Laws ◽  
Self Similarity ◽  
Similarity Scaling

Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

1990 ◽  
Vol 48 (2) ◽  
pp. 64-65
Author(s):  
L. Reimer ◽  
R. Oelgeklaus
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Electron Energy ◽  
Rotational Symmetry ◽  
Mean Free Path ◽  
Single Scattering ◽  
Specimen Thickness ◽  
Two Dimensional ◽  
Image Position ◽  
Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

Scaling, Self-similarity, and Intermediate Asymptotics

1996 ◽  
Author(s):  
Grigory Isaakovich Barenblatt
Keyword(s):  
Self Similarity ◽  
Intermediate Asymptotics
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