Binomial difference sequence spaces of fractional order.
Ontology highlight
ABSTRACT:
SUBMITTER: Meng J
PROVIDER: S-EPMC6182427 | biostudies-other | 2018
REPOSITORIES: biostudies-other
Publications
Binomial difference sequence spaces of fractional order.
Journal of inequalities and applications 20181010 1
In this paper, we introduce the sequence spaces b 0 r , s ( ∇ ( α ) ) , b c r , s ( ∇ ( α ) ) , and b ∞ r , s ( ∇ ( α ) ) . We investigate some functional properties, inclusion relations, and the <i>α</i>-, <i>β</i>-, <i>γ</i>-, and continuous duals of these sets. ...[more]
Similar Datasets
Project description:A high-order finite difference scheme is proposed for solving time fractional heat equations. The time fractional derivative is described in the Riemann-Liouville sense. In the proposed scheme a new second-order discretization, which is based on Crank-Nicholson method, is applied for the time fractional part and fourth-order accuracy compact approximation is applied for the second-order space derivative. The spectral stability and the Fourier stability analysis of the difference scheme are shown. Finally a detailed numerical analysis, including tables, figures, and error comparison, is given to demonstrate the theoretical results and high accuracy of the proposed scheme.
| S-EPMC3947870 | biostudies-other
Project description:Fractional derivative has a memory and non-localization features that make it very useful in modelling epidemics’ transition. The kernel of Caputo-Fabrizio fractional derivative has many features such as non-singularity, non-locality and an exponential form. Therefore, it is preferred for modeling disease spreading systems. In this work, we suggest to formulate COVID-19 epidemic transmission via
| S-EPMC7458115 | biostudies-literature
Project description:Control and synchronization of fractional-order chaotic systems have attracted wide attention due to their numerous potential applications. To get suitable control method and parameters for fractional-order chaotic systems, the stability analysis of time-varying fractional-order systems should be discussed in the first place. Therefore, this paper analyzes the stability of the time-varying fractional-order systems and presents a stability theorem for the system with the order 0<?<1. This theorem is a sufficient condition which can discriminate the stability of time-varying systems conveniently. Feedback controllers are designed for control and synchronization of the fractional-order Lü chaotic system. The simulation results demonstrate the effectiveness of the proposed theorem.
| S-EPMC5877833 | biostudies-literature
Project description:Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.
| S-EPMC7069480 | biostudies-literature
Project description:The mechanisms of atrial fibrillation (AF) are a challenging research topic. The rotor hypothesis states that the AF is sustained by a reentrant wave that propagates around an unexcited core. Cardiac tissue heterogeneities, both structural and cellular, play an important role during fibrillatory dynamics, so that the ionic characteristics of the currents, their spatial distribution and their structural heterogeneity determine the meandering of the rotor. Several studies about rotor dynamics implement the standard diffusion equation. However, this mathematical scheme carries some limitations. It assumes the myocardium as a continuous medium, ignoring, therefore, its discrete and heterogeneous aspects. A computational model integrating both, electrical and structural properties could complement experimental and clinical results. A new mathematical model of the action potential propagation, based on complex fractional order derivatives is presented. The complex derivative order appears of considering the myocardium as discrete-scale invariant fractal. The main aim is to study the role of a myocardial, with fractal characteristics, on atrial fibrillatory dynamics. For this purpose, the degree of structural heterogeneity is described through derivatives of complex order ? = ? + j?. A set of variations for ? is tested. The real part ? takes values ranging from 1.1 to 2 and the imaginary part ? from 0 to 0.28. Under this scheme, the standard diffusion is recovered when ? = 2 and ? = 0. The effect of ? on the action potential propagation over an atrial strand is investigated. Rotors are generated in a 2D model of atrial tissue under electrical remodeling due to chronic AF. The results show that the degree of structural heterogeneity, given by ?, modulates the electrophysiological properties and the dynamics of rotor-type reentrant mechanisms. The spatial stability of the rotor and the area of its unexcited core are modulated. As the real part decreases and the imaginary part increases, simulating a higher structural heterogeneity, the vulnerable window to reentrant is increased, as the total meandering of the rotor tip. This in silico study suggests that structural heterogeneity, described by means of complex order derivatives, modulates the stability of rotors and that a wide range of rotor dynamics can be generated.
| S-EPMC6066719 | biostudies-literature
Project description:Modern technological advances allow for the study of systems with additional synthetic dimensions. Higher-order topological insulators in topological states of matters have been pursued in lower physical dimensions by exploiting synthetic dimensions with phase transitions. While synthetic dimensions can be rendered in the photonics and cold atomic gases, little to no work has been succeeded in acoustics because acoustic wave-guides cannot be weakly coupled in a continuous fashion. Here, we formulate the theoretical principles and manufacture acoustic crystals composed of arrays of acoustic cavities strongly coupled through modulated channels to evidence one-dimensional (1D) and two-dimensional (2D) dynamic topological pumpings. In particular, the higher-order topological edge-bulk-edge and corner-bulk-corner transport are physically illustrated in finite-sized acoustic structures. We delineate the generated 2D and four-dimensional (4D) quantum Hall effects by calculating first and second Chern numbers and physically demonstrate robustness against the geometrical imperfections. Synthetic dimensions could provide a powerful way for acoustic topological wave steering and open up a platform to explore any continuous orbit in higher-order topological matter in dimensions four and higher.
| S-EPMC8377151 | biostudies-literature
Project description:RNA sequencing (RNA-Seq) is the current method of choice for characterizing transcriptomes and quantifying gene expression changes. This next generation sequencing-based method provides unprecedented depth and resolution. The negative binomial (NB) probability distribution has been shown to be a useful model for frequencies of mapped RNA-Seq reads and consequently provides a basis for statistical analysis of gene expression. Negative binomial exact tests are available for two-group comparisons but do not extend to negative binomial regression analysis, which is important for examining gene expression as a function of explanatory variables and for adjusted group comparisons accounting for other factors. We address the adequacy of available large-sample tests for the small sample sizes typically available from RNA-Seq studies and consider a higher-order asymptotic (HOA) adjustment to likelihood ratio tests. We demonstrate that 1) the HOA-adjusted likelihood ratio test is practically indistinguishable from the exact test in situations where the exact test is available, 2) the type I error of the HOA test matches the nominal specification in regression settings we examined via simulation, and 3) the power of the likelihood ratio test does not appear to be affected by the HOA adjustment. This work helps clarify the accuracy of the unadjusted likelihood ratio test and the degree of improvement available with the HOA adjustment. Furthermore, the HOA test may be preferable even when the exact test is available because it does not require ad hoc library size adjustments.
| S-EPMC3683603 | biostudies-literature
Project description:BackgroundIncomplete data analysis continues to be a major issue for non-inferiority clinical trials. Due to the steadily increasing use of non-inferiority study design, we believe this topic deserves an immediate attention.MethodsWe evaluated the performance of various strategies, including complete case analysis and various imputations techniques for handling incomplete non-inferiority clinical trials when outcome of interest is difference between binomial proportions. Non-inferiority of a new treatment was determined using a fixed margin approach with 95-95% confidence interval method. The methods used to construct the confidence intervals were compared as well and included: Wald, Farrington-Manning and Newcombe methods.ResultsWe found that worst-case and best-case scenario imputation methods should not be used for analysis of incomplete data in non-inferiority trial design, since such methods seriously inflate type-I error rates and produce biased estimates. In addition, we report conditions under which complete case analysis is an acceptable strategy for missing at random missingness mechanism. Importantly, we show how two-stage multiple imputation could be successfully applied for incomplete data that follow missing not at random patterns, and thus result in controlled type-I error rates and unbiased estimates.ConclusionThis thorough simulation study provides a road map for the analysis of incomplete data in non-inferiority clinical trials for different types of missingness. We believe that the results reported in this paper could serve practitioners who encounter missing data problems in their non-inferiority clinical trials.
| S-EPMC7226649 | biostudies-literature
Project description:Fractional derivative has a history as long as that of classical calculus, but it is much less popular than it should be. What is the physical meaning of fractional derivative? This is still an open problem. In modeling various memory phenomena, we observe that a memory process usually consists of two stages. One is short with permanent retention, and the other is governed by a simple model of fractional derivative. With the numerical least square method, we show that the fractional model perfectly fits the test data of memory phenomena in different disciplines, not only in mechanics, but also in biology and psychology. Based on this model, we find that a physical meaning of the fractional order is an index of memory.
| S-EPMC3852151 | biostudies-other
Project description:Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.
| S-EPMC3869418 | biostudies-literature