scholarly journals Fractional Dynamics in Bioscience and Biomedicine and the Physics of Cancer

2017 ◽  
Author(s):  
Hosein Nasrolahpour
Keyword(s):  
Fractional Calculus ◽  
Fractal Theory ◽  
Memory Effects ◽  
Living Systems ◽  
Fractional Dynamics ◽  
Biological Structures ◽  
Almost All ◽  
Natural Way

AbstractAlmost all phenomena and structures in nature exhibit some degrees of fractionality or fractality. Fractional calculus and fractal theory are two interrelated concepts. In this article we study the memory effects in nature and particularly in biological structures. Based on this fact that natural way to incorporate memory effects in the modeling of various phenomena and dealing with complexities is using of fractional calculus, in this article we present different examples in various branch of science from cosmology to biology and we investigate this idea that are we able to describe all of such these phenomena using the well-know and powerful tool of fractional calculus. In particular we focus on fractional calculus approach as an effective tool for better understanding of physics of living systems and organism and especially physics of cancer.

scholarly journals Universal Qudit Hamiltonians

2021 ◽  
Author(s):  
Stephen Piddock ◽  
Ashley Montanaro
Keyword(s):  
State Energy ◽  
Entangled State ◽  
Quantum Computers ◽  
List Type ◽  
Universal Family ◽  
Finite Dimensional ◽  
Universal Quantum ◽  
Aklt Model ◽  
Almost All ◽  
Natural Way

AbstractA family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits (d-level systems): We completely characterise the k-qudit interactions which are universal, if augmented with arbitrary Hermitian 1-local terms. We find that, for all $$k \geqslant 2$$ k ⩾ 2 and all local dimensions $$d \geqslant 2$$ d ⩾ 2 , almost all such interactions are universal aside from a simple stoquastic class. We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the SU(d) and SU(2) Heisenberg interactions are universal for all local dimensions $$d \geqslant 2$$ d ⩾ 2 (spin $$\geqslant 1/2$$ ⩾ 1 / 2 ), implying that a quantum variant of the Max-d-Cut problem is QMA-complete. We also show that for $$d=3$$ d = 3 all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model. We prove universality of any interaction proportional to the projector onto a pure entangled state.

scholarly journals General Fractional Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1464
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Calculus ◽  
Exact Solutions ◽  
Arbitrary Order ◽  
Nonlinear Dynamical Systems ◽  
Fractional Integrals ◽  
Fractional Dynamics ◽  
Fractional Systems ◽  
Nonlinear Dynamical ◽  
Fractional Differential ◽  
Solutions Of Equations

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

On Fractional Hamilton Formulation Within Caputo Derivatives

2007 ◽  
Author(s):  
Dumitru Baleanu ◽  
Sami I. Muslih ◽  
Eqab M. Rabei
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Variational Principles ◽  
Hamiltonian Dynamics ◽  
Fractional Integration ◽  
Fractional Dynamics ◽  
Classical Dynamics ◽  
Lagrange Equations ◽  
Caputo Fractional Derivatives ◽  
Non Locality

The fractional Lagrangian and Hamiltonian dynamics is an important issue in fractional calculus area. The classical dynamics can be reformulated in terms of fractional derivatives. The fractional variational principles produce fractional Euler-Lagrange equations and fractional Hamiltonian equations. The fractional dynamics strongly depends of the fractional integration by parts as well as the non-locality of the fractional derivatives. In this paper we present the fractional Hamilton formulation based on Caputo fractional derivatives. One example is treated in details to show the characteristics of the fractional dynamics.

scholarly journals Memory effects on link formation in temporal networks: A fractional calculus approach

2021 ◽  
Vol 564 ◽  
pp. 125502
Author(s):  
Fereshteh Rabbani ◽  
Tamer Khraisha ◽  
Fatemeh Abbasi ◽  
Gholam Reza Jafari
Keyword(s):  
Fractional Calculus ◽  
Memory Effects ◽  
Temporal Networks ◽  
Link Formation

Formulation of a State Equation Including Fractional-Order State-Vectors

2007 ◽  
Author(s):  
Masaharu Kuroda
Keyword(s):  
Fractional Calculus ◽  
Control Theory ◽  
Fractional Order ◽  
State Equation ◽  
The State ◽  
Space Representation ◽  
Fractional Dynamics ◽  
Modern Control Theory ◽  
Order State ◽  
Modern Control

In recent years, applications of fractional calculus have flourished in various science and engineering fields. Particularly in engineering, control engineering appears to be expanding aggressively in its applications. Exemplary are the CRONE controller and the PIλDμ controller, which is categorizable into applications of fractional calculus in classical control theory. A state equation can be called the foundation of modern control theory. However, the relationship between fractional derivatives and the state equation has not been examined sufficiently. Consequently, a systematic procedure referred to by every researcher on the fractional-calculus side or control-theory side has not yet been established. For this study, therefore, involvement of fractional-order derivatives into a state equation is demonstrated here for ready comprehension by researchers. First, the procedures are explained generally; then the technique to incorporate the fractional-order state-vector into a conventional state equation is given as an example of the applications. The state-space representation in this study is useful not only for modeling a controlled system with fractional dynamics, but also for design and implementation of a controller to control fractional-order states. After we complete installation of the basic parts, we can apply the benefits of modern control theory, including robust control theories such as H-infinity and μ-analysis and synthesis in their integrities, to this fractional-order state-equation.

scholarly journals Fractional Dynamics in Soccer Leagues

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 356 ◽  
Author(s):  
António M. Lopes ◽  
Jose A. Tenreiro Machado
Keyword(s):  
Mathematical Modeling ◽  
Fractional Calculus ◽  
Real World ◽  
Complex Dynamics ◽  
Statistical Information ◽  
Fractional Dynamics ◽  
Entropy Analysis ◽  
Real World Data ◽  
Proposed Model ◽  
Soccer Teams

This paper addresses the dynamics of four European soccer teams over the season 2018–2019. The modeling perspective adopts the concepts of fractional calculus and power law. The proposed model embeds implicitly details such as the behavior of players and coaches, strategical and tactical maneuvers during the matches, errors of referees and a multitude of other effects. The scale of observation focuses the teams’ behavior at each round. Two approaches are considered, namely the evaluation of the team progress along the league by a variety of heuristic models fitting real-world data, and the analysis of statistical information by means of entropy. The best models are also adopted for predicting the future results and their performance compared with the real outcome. The computational and mathematical modeling lead to results that are analyzed and interpreted in the light of fractional dynamics. The emergence of patterns both with the heuristic modeling and the entropy analysis highlight similarities in different national leagues and point towards some underlying complex dynamics.

scholarly journals Fractional dynamics in liquid manipulation

2010 ◽  
Vol 58 (4) ◽  
pp. 555-560
Author(s):  
M. Lima ◽  
J. Machado
Keyword(s):  
Experimental Study ◽  
Control System ◽  
Fractional Calculus ◽  
Trajectory Planning ◽  
Fourier Spectrum ◽  
Fractional Dynamics ◽  
Second Phase ◽  
Experimental Setup ◽  
Recorded Data ◽  
Analysis Package

Fractional dynamics in liquid manipulationThis paper presents a fractional calculus perspective in the study of signals captured during the movement of a mechanical manipulator carrying a liquid container. In order to study the signals an experimental setup is implemented. The system acquires data from the sensors, in real time, and, in a second phase, processes them through an analysis package. The analysis package runs off-line and handles the recorded data. The results show that the Fourier spectrum of several signals presents a fractional behavior. The experimental study provides useful information that can assist in the design of a control system and the trajectory planning to be used in reducing or eliminating the effect of vibrations.

INTRODUCTION

2012 ◽  
Vol 22 (04) ◽  
pp. 1202002 ◽  
Author(s):  
CHANGPIN LI ◽  
YANG QUAN CHEN ◽  
BLAS M. VINAGRE ◽  
IGOR PODLUBNY
Keyword(s):  
Fractional Calculus ◽  
Scaling Laws ◽  
Power Laws ◽  
Fractional Dynamics ◽  
Long Range Interactions ◽  
Potential Applications ◽  
Dynamics And Control ◽  
Long Memory Processes ◽  
Fractional Differential ◽  
And Control

Fractional Dynamics and Control is emerging as a new hot topic of research which draws tremendous attention and great interest. Although the fractional calculus appeared almost in the same era when the classical (or integer-order) calculus was born, it has recently been found that it can better characterize long-memory processes and materials, anomalous diffusion, long-range interactions, long-term behaviors, power laws, allometric scaling laws, and so on. Complex dynamical evolutions of these fractional differential equation models, as well as their controls, are becoming more and more important due to their potential applications in the real world. This special issue includes one review article and twenty-three regular papers, covering fundamental theories of fractional calculus, dynamics and control of fractional differential systems, and numerical calculation of fractional differential equations.

scholarly journals On Aggregation and Computation on Domain Values

1992 ◽  
Vol 21 (414) ◽  
Author(s):  
Kim Skak Larsen
Keyword(s):  
Query Languages ◽  
Relational Algebra ◽  
Way Of Thinking ◽  
Definition Of ◽  
Almost All ◽  
Natural Way

<p>Query languages often allow a limited amount of anthmetic and string operations on domain values, and sometimes sets of values can be dealt with through aggregation and sometimes even set comparisons. We address the question of how these facilities can be added to a relational language in a natural way. Our discussions lead us to reconsider the definition of the standard operators, and we introduce a new way of thinking about relational algebra computations.</p><p>We define a language FC, which has an iteration mechanism as its basis. A tuple language is used to carry out almost all computations. We prove equivalence results relating FC to relational algebra under various circumstances.</p>

COMPLEXITY-BASED CLASSIFICATION OF THE CORONAVIRUS GENOME VERSUS GENOMES OF THE HUMAN IMMUNODEFICIENCY VIRUS (HIV) AND DENGUE VIRUS

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050129
Author(s):  
HAMIDREZA NAMAZI
Keyword(s):  
Dengue Virus ◽  
Human Life ◽  
Fractal Theory ◽  
Complex Structure ◽  
Approximate Entropy ◽  
Virus Genome ◽  
The Other ◽  
Complexity Measures ◽  
Significant Difference ◽  
Almost All

Coronavirus disease (COVID-19) is a pandemic disease that has affected almost all around the world. The most crucial step in the treatment of patients with COVID-19 is to investigate about the coronavirus itself. In this research, for the first time, we analyze the complex structure of the coronavirus genome and compare it with the other two dangerous viruses, namely, dengue and HIV. For this purpose, we employ fractal theory, sample entropy, and approximate entropy to analyze the genome walk of coronavirus, dengue virus, and HIV. Based on the obtained results, the genome walk of coronavirus has greater complexity than the other two deadly viruses. The result of statistical analysis also showed the significant difference between the complexity of genome walks in case of all complexity measures. The result of this analysis opens new doors to scientists to consider the complexity of a virus genome as an index to investigate its danger for human life.

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