Calculus Of Order Research Articles

Fractional Decision Making Model for Crowds of Pedestrians in Two-Alternative Choice Evacuation

Modeling of the two-alternative decision making process for evacuation of crowds of pedestrians from bounded room has been considered in this paper under the framework of Calculus of Fractional Order. Dynamic decision making models of fractional order have been presented under symmetric interactions and asymmetric interactions, respectively. Stability analysis under asymmetric interactions has been studied using linear approximation. Final value of the collective opinion under asymmetric interactions is dependent on the initial distribution of crowd’s opinions while it is not the case for symmetric interactions. Future work based on the results of this paper are also discussed at the end of this paper.

A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus

We present a new approach based on linear integro-differential operators with logarithmic kernel related to the Hadamard fractional calculus in order to generalize, by a parameter ν ∈ (0, 1], the logarithmic creep law known in rheology as Lomnitz law (obtained for ν=1). We derive the constitutive stress-strain relation of this generalized model in a form that couples memory effects and time-varying viscosity. Then, based on the hereditary theory of linear viscoelasticity, we also derive the corresponding relaxation function by solving numerically a Volterra integral equation of the second kind. So doing we provide a full characterization of the new model both in creep and in relaxation representation, where the slow varying functions of logarithmic type play a fundamental role as required in processes of ultra slow kinetics.

A fractional model of supercapacitors for use in energy storage systems of next-generation shipboard electrical networks

The reduction of energy consumption and carbon dioxide emission has forced designers and manufacturers to develop new, more efficient ship propulsion systems. Traditional mechanical systems are replaced by hybrid or completely electric ones. In addition to the combustion, the electromechanical sources are considered. But in such small electric grids, in order to reduce the impact of load fluctuations on the power system, energy storage devices are proposed. Such devices are able to supply a peak power. As examples the batteries, flywheels, but primarily supercapacitors can be considered. Unfortunately, a very low voltage causes supercapacitors to work in stacks. This paper presents the results of analysis of the influence of changes in selected supercapacitor parameters changes on its terminal voltage during the cycles of charging and discharging. The model parameters are estimated based on the test measurements, while the differ-integral calculus of fractional order is used to derive the model output. It turns out that such an approach provides a very good fit for model responses even for its very simple structures. In addition, a very precise mathematical model allows to accurately estimate an amount of stored energy and estimate the supercapacitor reliability.

Spectral analysis of the subelliptic oblique derivative problem

This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for the usual Laplacian with a complex parameter $\lambda$. We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous oblique derivative problem when $\lvert \lambda \rvert$ tends to $\infty$. We prove the spectral properties of the closed realization of the Laplacian similar to the elliptic (non-degenerate) case. In the proof we make use of Boutet de Monvel calculus in order to study the resolvents and their adjoints in the framework of $L^2$ Sobolev spaces.

Детерминированный факторный анализ: методы интегро-дифференцирования нецелого порядка

Objective: to summarize the methods of deterministic factor economic analysis, namely the differential calculus and the integral method. Methods: mathematical methods for integro-differentiation of non-integral order, the theory of derivatives and integrals of fractional (non-integral) order.Results: the basic concepts are formulated and the new methods are developed that take into account the memory and non-locality effects in the quantitative description of the influence of individual factors on the change in the effective economic indicator. Two methods are proposed for integro-differentiation of non-integral order for the deterministic factor analysis of economic processes with memory and non-locality. It is shown that the method of integro-differentiation of non-integral order can give more accurate results compared with standard methods (method of differentiation using the first order derivatives and the integral method using the integration of the first order) for a wide class of functions describing effective economic indicators.Scientific novelty: the new methods of deterministic factor analysis are proposed: the method of differential calculus of non-integral order and the integral method of non-integral order.Practical significance: the basic concepts and formulas of the article can be used in scientific and analytical activity for factor analysis of economic processes. The proposed method for integro-differentiation of non-integral order extends the capabilities of the determined factorial economic analysis. The new quantitative method of deterministic factor analysis may become the beginning of quantitative studies of economic agents behavior with memory, hereditarity and spatial non-locality. The proposed methods of deterministic factor analysis can be used in the study of economic processes which follow the exponential law, in which the indicators (endogenous variables) are power functions of the factors (exogenous variables), including the processes described by the Cobb - Douglas production function, since these methods allow to more accurately describe the total influence of the factors in comparison with the standard method. The proposed methods can be used in the study of economic processes described by equations with a power-law non-locality in factor space and in state space.

Is classical mechanics a prerequisite for learning physics of the 20th century?

Physics of the 20th century has contributed significantly to modern technology, and yet many physics students are never availed the opportunity to study it as part of the curriculum. One of the possible reasons why it is not taught in high school and introductory physics courses could be because curriculum designers believe that students need a solid background in classical mechanics and calculus in order to study physics of the 20th century such as the photoelectric effect, special and general relativity, the uncertainty principle, etc. This presumption may not be justifiable or valid. The authors of this paper contend that teaching physics of the 20th century aids students in relating physics to modern technology and the real world, making studying physics exciting. In this study, the authors correlated scores for matched questions in the Mechanics Baseline Test and a physics of the 20th century test in order to examine the trend of the scores. The participants included undergraduate students attending an introductory algebra-based physics course with no intention of taking physics at a higher level. The analysis of the scores showed no significant correlation for any of the matched pairs of questions. The purpose of this article is to recommend that even without a solid background in classical mechanics, teachers can introduce physics of the 20th century to their students for increased interest.

Some Results on The Incomplete q-Gamma Function and Its First Derivative

In this paper, we use neutrix calculus in order to obtain some results on the q- analogue of the incomplete gamma function and its first derivative for all real numbers.

Analysis of Slow Modes in Ring Polymers: Threading of Rings Controls Long-Time Relaxation.

Atomistic configurations of pure, precisely monodisperse ring poly(ethylene oxide) (PEO) melts accumulated in the course of very long molecular dynamics (MD) simulations at T = 413 K and P = 1 atm have been subjected to a detailed geometric analysis involving three steps (reduction to ensembles of coarse-grained paths, triangulation of the resulting three-dimensional polygons, and analysis of interpenetrations using vector calculus) in order to locate ring-ring threading events and quantify their strength and survival times. A variety of threading situations have been identified corresponding to single and multiple penetrations. The percentage of inter-ring threadings that correspond to full penetrations has also been quantified. By repeating the analysis for several PEO melts, the dependence of the degree of inter-ring threading on molecular weight (MW) has been obtained. Simulations with MWs up to 10 times the reported entanglement molecular weight (Me) of linear PEO have revealed several multiple threading events in all systems, with their relative number increasing with increasing MW. Our analysis indicates the existence of strong ring-ring topological interactions, which can last up to several times the corresponding average orientational ring polymer relaxation time. We show that these ring-ring interactions, together with the additional ring-linear threadings due to the remaining linear impurities, can explain the appearance of slow relaxation modes observed experimentally in entangled rings.

Teoria dos registros de representação semiótica em pesquisas na Educação Matemática no Ensino Superior: uma análise de superfícies e funções de duas variáveis com intervenção do software Maple

Resumo Este artigo traz algumas reflexões motivadas pela utilização equivocada de noções da Teoria dos Registros de Representação Semiótica (TRRS), proposta por Duval, pelos estudantes e pesquisadores em suas pesquisas/palestras. Sem perda de essência e generalidade, trazemos algumas definições fundadas na proposta de Duval e utilizamos os conceitos de cálculo diferencial e integral com ênfase nas superfícies e funções de duas/três variáveis, como objetos matemáticos de referência. O objetivo é explicitar as referidas definições, na tentativa de trazer contribuições da TRRS na análise em pesquisas em Educação Matemática no Ensino Superior, além de auxiliar os estudantes na reflexão sobre o tema, discutir as técnicas instrumentais de representação do tipo de funções em questão nos diferentes registros, em especial no registro gráfico em três dimensões, utilizando o software Maple, porque o fenômeno "ver" ou "representar" objetos matemáticos em 3D é um dos entraves na aprendizagem de cálculo pela maioria dos estudantes.

Existence and regularity of the density for solutions of stochastic differential equations with boundary noise

We study the existence and regularity of densities for the solution of a nonlinear heat diffusion with stochastic perturbation of Brownian and fractional Brownian motion type: we use the Malliavin calculus in order to prove that, if the nonlinear term is suitably regular, then the law of the solution has a smooth density with respect to the Lebesgue measure.

A pseudo-extractor approach to hidden boundary regularity for the wave equation with mixed boundary conditions

In this paper we introduce a new approach to “hidden” boundary regularity for the linear wave equation with mixed Dirichlet–Neumann boundary conditions, where the Neumann data is non-smooth. First, we obtain existence and uniqueness of solution by Galerkin estimates. Then we use a new, pseudo-extractor technique (based on the Fourier transform and shape and tangential calculus) in order to provide sharp regularity for the solution at the boundary.

A conformable fractional calculus on arbitrary time scales

A conformable time-scale fractional calculus of order α∈]0,1] is introduced. The basic tools for fractional differentiation and fractional integration are then developed. The Hilger time-scale calculus is obtained as a particular case, by choosing α=1.

On the use of high-order cumulant and bispectrum for muscular-activity detection

The electromyographic (EMG) signals are extensively used on feature extraction methods for movement classification purposes. High-order statistics (HOS) is being employed increasingly in myoelectric research. HOS techniques could be represented in the frequency domain (high-order spectra, e.g., bispectrum, trispectrum) or in the time domain (higher-order cumulants). More calculus is required for computing the HOS in the frequency domain. On the one hand, classical bispectrum-based features were applied to EMG signals. We propose novel third-order cumulant-based features for EMG signals. Three different classifiers are implemented for muscular-activity detection. Different analysis and evaluations were applied to both HOS-based features in order to qualify and quantify similarities. Based on these results, it is possible to conclude that cumulant-based features and bispectrum-based features had comparable behavior and allowed similar classification rates. Hence, extra calculus in order to convert time- to frequency-domain should be avoided.

Theoretical and Experimental Model for the Analysis of a Cardan Cross

The work presents the aproach of initial moddeling and calculus in order to prepare an finite element investigation of the cardan transmission joint. For this, it is necessary to develop a flexible informational system that would connect the constructive design of cardan transmissions to the technological design within a unitary structure. Due to the typodimensional diversity of cardan transmissions, the design and manufacturing of the various components for a certain application domain (medical aparature, dental and agricultural machines, lifting and transportation machines etc.) has been typified. In order to study the problems that occur in the practice of using cardan transmissions, there has been realised a questionnaire-based descriptive research. We insisted on calculation of the cardan transmission for Dacia 1304/1307.

Confluent Hypergeometric Equation via Fractional Calculus Approach

In this paper, using the theory of the so-called fractional calculus we show that it is possible to easily obtain the solutions for the confluent hypergeometric equation. Our approach is to be compared with the standard one (Frobenius) which is based on the ordinary calculus of integer order.

Electrochemical impedance spectroscopy in understanding the influence of ultrasonic dental scaling on the dental structure-dental filling interface.

The purpose of this study was to investigate the effect of ultrasonic scaling on teeth restored with a light-cured resin. Ultrasonic scaling is a very popular periodontal therapy among dentists, and used for the removal of dental plaque and calculus in order to reduce and eliminate inflammation. Given the fact that most ultrasonic devices are used at high frequencies to perform scaling, undesirable consequences, such as loss of adhesion and increase in surface roughness, may occur in teeth that have been restored with light-cured resins. Electrochemical impedance spectroscopy (EIS) and scanning electron microscopy were used to investigate the effects of ultrasonic treatments at the dental material-hard dental tissue interface. After ultrasonic scaling, EIS measurements were performed on a human tooth that had been restored with a light-cured resin filling. The data were analyzed and the influence of ultrasound was shown after visualization of the hard dental tissues and the dental material as equivalent electrical circuits. The study revealed, through EIS measurements, that ultrasonic scaling affected the resistance of the light-cured resin filling and dentin, whereas the enamel was affected only slightly. Scanning electron microscopy revealed an increase in roughness of the dental material.

Spectral Convergence Bounds for Classical and Quantum Markov Processes

We introduce a new framework that yields spectral bounds on norms of functions of transition maps for finite, homogeneous Markov chains. The techniques employed work for bounded semigroups, in particular for classical as well as for quantum Markov chains and they do not require additional assumptions like detailed balance, irreducibility or aperiodicity. We use the method in order to derive convergence bounds that improve significantly upon known spectral bounds. The core technical observation is that power-boundedness of transition maps of Markov chains enables a Wiener algebra functional calculus in order to upper bound any norm of any holomorphic function of the transition map. Finally, we discuss how general detailed balance conditions for quantum Markov processes lead to spectral convergence bounds.

A revised FMEA application to the quality control management

Purpose – The purpose of this paper is to propose a cost-effective, time-saving and easy-to-use failure modes and effects analysis (FMEA) system applied on the quality control of supplied products. The traditional FMEA has been modified and adapted to fit the quality control features and requirements. The paper introduces a new and revised FMEA approach, where the “failure concept” has been modified with “defect concept.” Design/methodology/approach – The typical FMEA parameters have been modified, and a non-linear scale has been introduced to better evaluate the FMEA parameters. In addition, two weight functions have been introduced in the risk priority number (RPN) calculus in order to consider different critical situations previously ignored and the RPN is assigned to several similar products in order to reduce the problem of complexity. Findings – A complete procedure is provided in order to assist managers in deciding on the critical suppliers, the creation of homogeneous families overcome the complexity of single product code approach, in RPN definition the relative importance of factors is evaluated. Originality/value – This different approach facilitates the quality control managers acting as a structured and “friendly” decision support system: the quality control manager can easily evaluate the critical situations and simulate different scenarios of corrective actions in order to choose the best one. This FMEA technique is a dynamic tool and the performed process is an iterative one. The method has been applied in a small medium enterprise producing hydro massage bathtub, shower, spas and that commercializes bathroom furniture. The firm application has been carried out involving a cross-functional and multidisciplinary team.

An approach to a model disordered birefringent medium for light depolarization applied to a liquid crystal device

A general model consisting of micro-wave plates is presented as a straightforward extension of Jones–Mueller calculus in order to describe the effect that undergoes a perfectly polarized quasi-monochromatic light beam transmitted through a disordered birefringent medium. The model treats the birefringent medium as an array of micro-wave plates lying side by side. The transmitted light beam results spatially divided in micro-beams which inherit from micro-wave plates different polarization states and hence, spatial depolarization is achieved. The model is applied to a liquid crystal sandwich-like cell subject to a splay-bend perturbation in the bulk. The model output is cast against the experimental data with a set of parameters throughout simulations. A good agreement is obtained between experimental data and model results.

Differential and integral calculus for logical operations. A matrix-vector approach

A variety of problems emerged investigating electronic circuits, computer devices and cellular automata motivated a number of attempts to create a differential and integral calculus for Boolean functions. In the present article, we extend this kind of calculus in order to include the semantic of classical logical operations. We show that this extension to logics is strongly helped if we submerge the elementary logical calculus in a matrix-vector formalism that naturally includes a kind of fuzzy-logic. In this way, guided by the laws of matrix algebra, we can construct compact representations for the derivatives and the integrals of logical functions. Inside this semantic-algebraic calculus, we obtain expressions for the derivatives of some of the basic logical operations and show the general way to obtain the derivatives of any well-formed formula of propositional calculus. We show that some of the basic tautologies (Excluded middle, Modus ponens, Hypothetical syllogism) are members of a kind of hierarchical system linked by the differentiation algorithm. In addition using the logical derivatives we show that relatively complex formulas can collapse in simple expressions that reveal clearly their hidden logical meaning. The search for the antiderivatives produces naturally an integral calculus. Within this logical formalism an indefinite integral can always be found for any logical expression. Moreover, particular integrals can be constructed based on detachment properties that lead to logical expressions of growing complexity. We show that these particular integrals have some similarities with the 'generalizing deduction' procedures investigated by Lukasiewicz.