Certain Inequalities Involving the Fractional q-Integral Operators
Directory of Open Access Journals (Sweden)
Dumitru Baleanu
2014-01-01
Full Text Available We establish some inequalities involving Saigo fractional q-integral operator in the theory of quantum calculus by using the two parameters of deformation, q1 and q2, whose special cases are shown to yield corresponding inequalities associated with Riemann-Liouville and Kober fractional q-integral operators, respectively. Furthermore, we also consider their relevance with other related known results.
On a fractional difference operator
Directory of Open Access Journals (Sweden)
P. Baliarsingh
2016-06-01
Full Text Available In the present article, a set of new difference sequence spaces of fractional order has been introduced and subsequently, an application of these spaces, the notion of the derivatives and the integrals of a function to the case of non-integer order have been generalized. Certain results involving the unusual and non-uniform behavior of the corresponding difference operator have been investigated and also been verified by using some counter examples. We also verify these unusual and non-uniform behaviors by studying the geometry of fractional calculus.
Fractional Number Operator and Associated Fractional Diffusion Equations
Rguigui, Hafedh
2018-03-01
In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.
On the Scaled Fractional Fourier Transformation Operator
International Nuclear Information System (INIS)
Hong-Yi, Fan; Li-Yun, Hu
2008-01-01
Based on our previous study [Chin. Phys. Lett. 24 (2007) 2238] in which the Fresnel operator corresponding to classical Fresnel transform was introduced, we derive the fractional Fourier transformation operator, and the optical operator method is then enriched
Void Fraction Instrument operation and maintenance manual
International Nuclear Information System (INIS)
Borgonovi, G.; Stokes, T.I.; Pearce, K.L.; Martin, J.D.; Gimera, M.; Graves, D.B.
1994-09-01
This Operations and Maintenance Manual (O ampersand MM) addresses riser installation, equipment and personnel hazards, operating instructions, calibration, maintenance, removal, and other pertinent information necessary to safely operate and store the Void Fraction Instrument. Final decontamination and decommissioning of the Void Fraction Instrument are not covered in this document
Operator Fractional Brownian Motion and Martingale Differences
Directory of Open Access Journals (Sweden)
Hongshuai Dai
2014-01-01
Full Text Available It is well known that martingale difference sequences are very useful in applications and theory. On the other hand, the operator fractional Brownian motion as an extension of the well-known fractional Brownian motion also plays an important role in both applications and theory. In this paper, we study the relation between them. We construct an approximation sequence of operator fractional Brownian motion based on a martingale difference sequence.
Operator continued fraction and bound states
International Nuclear Information System (INIS)
Pindor, M.
1984-01-01
The effective Hamiltonian of the model space perturbation theory (multilevel Rayleigh-Schroedinger theory) is expressed as an operator continued fraction. In the case of a nondegenerate model space the expression becomes an operator branched continued fraction. The method is applied to the harmonic oscillator with the kinetic energy treated as the perturbation and to the anharmonic oscillator
Unpacking Referent Units in Fraction Operations
Philipp, Randolph A.; Hawthorne, Casey
2015-01-01
Although fraction operations are procedurally straightforward, they are complex, because they require learners to conceptualize different units and view quantities in multiple ways. Prospective secondary school teachers sometimes provide an algebraic explanation for inverting and multiplying when dividing fractions. That authors of this article…
Fractional power operation of tokamak reactors
International Nuclear Information System (INIS)
Mau, T.K.; Vold, E.L.; Conn, R.W.
1986-01-01
Methods to operate a tokamak fusion reactor at fractions of its rated power, identify the more effective control knobs and assess the impact of the requirements of fractional power operation on full power reactor design are explored. In particular, the role of burn control in maintaining the plasma at thermal equilibrium throughout these operations is studied. As a prerequisite to this task, the critical physics issues relevant to reactor performance predictions are examined and some insight into their impact on fractional power operation is offered. The basic tool of analysis consists of a zero-dimensional (0-D) time-dependent plasma power balance code which incorporates the most advanced data base and models in transport and burn plasma physics relevant to tokamaks. Because the plasma power balance is dominated by the transport loss and given the large uncertainty in the confinement model, the authors have studied the problem for a wide range of energy confinement scalings. The results of this analysis form the basis for studying the temporal behavior of the plasma under various thermal control mechanisms. Scenarios of thermally stable full and fractional power operations have been determined for a variety of transport models, with either passive or active feedback burn control. Important power control parameters, such as gas fueling rate, auxiliary power and other plasma quantities that affect transport losses, have also been identified. The results of these studies vary with the individual transport scaling used and, in particular, with respect to the effect of alpha heating power on confinement
Identities for generalized fractional integral operators associated ...
African Journals Online (AJOL)
In this present work an attempt has been made to define two generalized fractional integral operators associated with products of analogues to Dirichlet averages and special functions. Discussions on the different aspects of the obtained results have been followed by utilization in finding out the images of multivariate ...
Wu, L.; Liu, S.; Yang, Yingjie
2016-01-01
Traditional integer order buffer operator is extended to fractional order buffer operator, the corresponding relationship between the weakening buffer operator and the strengthening buffer operator is revealed. Fractional order buffer operator not only can generalize the weakening buffer operator and the strengthening buffer operator, but also realize tiny adjustment of buffer effect. The effectiveness of GM(1,1) with the fractional order buffer operator is validated by six cases.
Radial fractional Laplace operators and Hessian inequalities
Ferrari, Fausto; Verbitsky, Igor E.
In this paper we deduce a formula for the fractional Laplace operator ( on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with (, and apply it to a problem related to the Hessian inequality of Sobolev type: ∫Rn |(u| dx⩽C∫Rn -uFk[u] dx, where Fk is the k-Hessian operator on Rn, 1⩽kFerrari et al. [5] contains the extremal functions for the Hessian Sobolev inequality of X.-J. Wang (1994) [15]. This is proved using logarithmic convexity of the Gaussian ratio of hypergeometric functions which might be of independent interest.
Directory of Open Access Journals (Sweden)
Sheng-Ping Yan
2014-01-01
Full Text Available We perform a comparison between the local fractional Adomian decomposition and local fractional function decomposition methods applied to the Laplace equation. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.
Directory of Open Access Journals (Sweden)
Hossein Jafari
2016-04-01
Full Text Available In this paper, we consider the local fractional decomposition method, variational iteration method, and differential transform method for analytic treatment of linear and nonlinear local fractional differential equations, homogeneous or nonhomogeneous. The operators are taken in the local fractional sense. Some examples are given to demonstrate the simplicity and the efficiency of the presented methods.
A Quantitative Analysis of Children's Splitting Operations and Fraction Schemes
Norton, Anderson; Wilkins, Jesse L. M.
2009-01-01
Teaching experiments with pairs of children have generated several hypotheses about students' construction of fractions. For example, Steffe (2004) hypothesized that robust conceptions of improper fractions depends on the development of a splitting operation. Results from teaching experiments that rely on scheme theory and Steffe's hierarchy of…
Fractional quantum integral operator with general kernels and applications
Babakhani, Azizollah; Neamaty, Abdolali; Yadollahzadeh, Milad; Agahi, Hamzeh
In this paper, we first introduce the concept of fractional quantum integral with general kernels, which generalizes several types of fractional integrals known from the literature. Then we give more general versions of some integral inequalities for this operator, thus generalizing some previous results obtained by many researchers.2,8,25,29,30,36
The overlap Dirac operator as a continued fraction
International Nuclear Information System (INIS)
Wenger, U.; Deutsches Elektronen-Synchrotron
2004-03-01
We use a continued fraction expansion of the sign-function in order to obtain a five dimensional formulation of the overlap lattice Dirac operator. Within this formulation the inverse of the overlap operator can be calculated by a single Krylov space method and nested conjugate gradient procedures are avoided. We point out that the five dimensional linear system can be made well conditioned using equivalence transformations on the continued fractions. (orig.)
Algebraic quantization, good operators and fractional quantum numbers
International Nuclear Information System (INIS)
Aldaya, V.; Calixto, M.; Guerrero, J.
1996-01-01
The problems arising when quantizing systems with periodic boundary conditions are analysed, in an algebraic (group-) quantization scheme, and the failure of the Ehrenfest theorem is clarified in terms of the already defined notion of good (and bad) operators. The analysis of constrained Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for quantum operators without classical analogue and for new quantum (fractional) numbers extending those allowed for Chern classes in traditional Geometric Quantization. This study is illustrated with the examples of the free particle on the circumference and the charged particle in a homogeneous magnetic field on the torus, both examples featuring anomalous operators, non-equivalent quantization and the latter, fractional quantum numbers. These provide the rationale behind flux quantization in superconducting rings and Fractional Quantum Hall Effect, respectively. (orig.)
Sinc-Approximations of Fractional Operators: A Computing Approach
Directory of Open Access Journals (Sweden)
Gerd Baumann
2015-06-01
Full Text Available We discuss a new approach to represent fractional operators by Sinc approximation using convolution integrals. A spin off of the convolution representation is an effective inverse Laplace transform. Several examples demonstrate the application of the method to different practical problems.
International Nuclear Information System (INIS)
Zhang, Xiao; Wei, Chaozhen; Liu, Yingming; Luo, Maokang
2014-01-01
In this paper we use Dirac function to construct a fractional operator called fractional corresponding operator, which is the general form of momentum corresponding operator. Then we give a judging theorem for this operator and with this judging theorem we prove that R–L, G–L, Caputo, Riesz fractional derivative operator and fractional derivative operator based on generalized functions, which are the most popular ones, coincide with the fractional corresponding operator. As a typical application, we use the fractional corresponding operator to construct a new fractional quantization scheme and then derive a uniform fractional Schrödinger equation in form. Additionally, we find that the five forms of fractional Schrödinger equation belong to the particular cases. As another main result of this paper, we use fractional corresponding operator to generalize fractional quantization scheme by using Lévy path integral and use it to derive the corresponding general form of fractional Schrödinger equation, which consequently proves that these two quantization schemes are equivalent. Meanwhile, relations between the theory in fractional quantum mechanics and that in classic quantum mechanics are also discussed. As a physical example, we consider a particle in an infinite potential well. We give its wave functions and energy spectrums in two ways and find that both results are the same
A continued fraction representation of the mass operator
International Nuclear Information System (INIS)
Saraswati, D.K.
1976-01-01
We explore some further possibilities of application of the projection operator method of Zwanzig to the theory of Green's functions of quantum statistical mechanics, initiated by Ichiyanagi, and present a continued fraction representation of the mass operator involving a hierarchy of the random forces. As an application of the theory, we calculate the polarization operator of the phonon Green's function of the Frohlich Hamiltonian in the first approximation which corresponds to the assumption that the electron momenta are orthogonal to the phonon momentum. (author)
Extended Riemann-Liouville type fractional derivative operator with applications
Directory of Open Access Journals (Sweden)
Agarwal P.
2017-12-01
Full Text Available The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.
Spaces of fractional quotients, discrete operators, and their applications. II
International Nuclear Information System (INIS)
Lifanov, I K; Poltavskii, L N
1999-01-01
The theory of discrete operators in spaces of fractional quotients is developed. A theorem on the stability of discrete operators under smooth perturbations is proved. On this basis, using special quadrature formulae of rectangular kind, the convergence of approximate solutions of hypersingular integral equations to their exact solutions is demonstrated and a mathematical substantiation of the method of closed discrete vortex frameworks is obtained. The same line of argument is also applied to difference equations arising in the solution of the homogeneous Dirichlet problem for a general second-order elliptic equation with variable coefficients
TRANSFORMERLESS OPERATION OF DIII-D WITH HIGH BOOTSTRAP FRACTION
International Nuclear Information System (INIS)
POLITZER, PA; HYATT, AW; LUCE, TC; MAHDAVI, MA; MURAKAMI, M; PERKINS, FW; PRATER, R; TURNBULL, AD; CASPER, TA; FERRON, JR; JAYAKUMAR, RJ; LAHAYE, RJ; LAZARUS, EA; PETTY, CC; WADE, MR
2003-01-01
OAK-B135 The authors have initiated an experimental program to address some of the questions associated with operation of a tokamak with high bootstrap current fraction under high performance conditions, without assistance from a transformer. In these discharges they have maintained stationary (or slowly improving) conditions for > 2.2 s at β N ∼ β p ∼ 2.8. Significant current overdrive, with dI/dt > 50 kA/s and zero or negative voltage, is sustained for over 0.7 s. The overdrive condition is usually ended with the appearance of MHD activity, which alters the profiles and reduces the bootstrap current. Characteristically these plasmas have 65%-80% bootstrap current, 25%-30% NBCD, and 5%-10% ECCD. Fully noninductive operation is essential for steady-state tokamaks. For efficient operation, the bootstrap current fraction must be close to 100%, allowing for a small additional (∼ 10%) external current drive capability to be used for control. In such plasmas the current and pressure profiles are rightly coupled because J(r) is entirely determined by p(r) (or more accurately by the kinetic profiles). The pressure gradient in turn is determined by transport coefficients which depend on the poloidal field profile
Spectral results for mixed problems and fractional elliptic operators,
DEFF Research Database (Denmark)
Grubb, Gerd
2015-01-01
In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators P a of order 2a, with type and factorization index a ∈ R +, restricted to compact sets with boundary; this includes fractional powers of the Laplace operator. The domain...... and the regularity of eigenfunctions is described. In the second part, we apply this in a study of realizations A χ,Σ+ in L 2( Ω ) of mixed problems for a second-order strongly elliptic symmetric differential operator A on a bounded smooth set Ω ⊂ R n; here the boundary ∂Ω=Σ is partioned smoothly into Σ......=Σ_∪Σ+, the Dirichlet condition γ0u=0 is imposed on Σ_, and a Neumann or Robin condition χu=0 is imposed on Σ+. It is shown that the Dirichlet-to-Neumann operator Pγ,χ is principally of type 1/2 with factorization index 1/2, relative to Σ+. The above theory allows a detailed description of D (Aχ,Σ_+) with singular...
Tokamak power reactor ignition and time dependent fractional power operation
International Nuclear Information System (INIS)
Vold, E.L.; Mau, T.K.; Conn, R.W.
1986-06-01
A flexible time-dependent and zero-dimensional plasma burn code with radial profiles was developed and employed to study the fractional power operation and the thermal burn control options for an INTOR-sized tokamak reactor. The code includes alpha thermalization and a time-dependent transport loss which can be represented by any one of several currently popular scaling laws for energy confinement time. Ignition parameters were found to vary widely in density-temperature (n-T) space for the range of scaling laws examined. Critical ignition issues were found to include the extent of confinement time degradation by alpha heating, the ratio of ion to electron transport power loss, and effect of auxiliary heating on confinement. Feedback control of the auxiliary power and ion fuel sources are shown to provide thermal stability near the ignition curve
Essay on Fractional Riemann-Liouville Integral Operator versus Mikusinski’s
Directory of Open Access Journals (Sweden)
Ming Li
2013-01-01
Full Text Available This paper presents the representation of the fractional Riemann-Liouville integral by using the Mikusinski operators. The Mikusinski operators discussed in the paper may yet provide a new view to describe and study the fractional Riemann-Liouville integral operator. The present result may be useful for applying the Mikusinski operational calculus to the study of fractional calculus in mathematics and to the theory of filters of fractional order in engineering.
Directory of Open Access Journals (Sweden)
M.H.T. Alshbool
2017-01-01
Full Text Available An algorithm for approximating solutions to fractional differential equations (FDEs in a modified new Bernstein polynomial basis is introduced. Writing x→xα(0<α<1 in the operational matrices of Bernstein polynomials, the fractional Bernstein polynomials are obtained and then transformed into matrix form. Furthermore, using Caputo fractional derivative, the matrix form of the fractional derivative is constructed for the fractional Bernstein matrices. We convert each term of the problem to the matrix form by means of fractional Bernstein matrices. A basic matrix equation which corresponds to a system of fractional equations is utilized, and a new system of nonlinear algebraic equations is obtained. The method is given with some priori error estimate. By using the residual correction procedure, the absolute error can be estimated. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.
Directory of Open Access Journals (Sweden)
Dumitru Baleanu
2014-01-01
Full Text Available We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.
Directory of Open Access Journals (Sweden)
Jamal Salah
2011-01-01
Full Text Available In this article, we introduce a class of starlike functions of order α by using a fractional operator involving Caputo's fractional which was introduced recently by the authors. The coefficient inequalities and distortion theorems are determined. Further some subordination theorems are given. In addition, results involving Hadamard product are also discussed.
On certain fractional calculus operators involving generalized Mittag-Leffler function
Dinesh Kumar
2016-01-01
The object of this paper is to establish certain generalized fractional integration and differentiation involving generalized Mittag-Leffler function defined by Salim and Faraj [25]. The considered generalized fractional calculus operators contain the Appell's function $F_3$ [2, p.224] as kernel and are introduced by Saigo and Maeda [23]. The Marichev-Saigo-Maeda fractional calculus operators are the generalization of the Saigo fractional calculus operators. The established results provide ex...
Generalized Fractional Integral Operators on Generalized Local Morrey Spaces
Directory of Open Access Journals (Sweden)
V. S. Guliyev
2015-01-01
Full Text Available We study the continuity properties of the generalized fractional integral operator Iρ on the generalized local Morrey spaces LMp,φ{x0} and generalized Morrey spaces Mp,φ. We find conditions on the triple (φ1,φ2,ρ which ensure the Spanne-type boundedness of Iρ from one generalized local Morrey space LMp,φ1{x0} to another LMq,φ2{x0}, 1
Williams, P Stephen
2016-05-01
Asymmetrical flow field-flow fractionation (As-FlFFF) has become the most commonly used of the field-flow fractionation techniques. However, because of the interdependence of the channel flow and the cross flow through the accumulation wall, it is the most difficult of the techniques to optimize, particularly for programmed cross flow operation. For the analysis of polydisperse samples, the optimization should ideally be guided by the predicted fractionating power. Many experimentalists, however, neglect fractionating power and rely on light scattering detection simply to confirm apparent selectivity across the breadth of the eluted peak. The size information returned by the light scattering software is assumed to dispense with any reliance on theory to predict retention, and any departure of theoretical predictions from experimental observations is therefore considered of no importance. Separation depends on efficiency as well as selectivity, however, and efficiency can be a strong function of retention. The fractionation of a polydisperse sample by field-flow fractionation never provides a perfectly separated series of monodisperse fractions at the channel outlet. The outlet stream has some residual polydispersity, and it will be shown in this manuscript that the residual polydispersity is inversely related to the fractionating power. Due to the strong dependence of light scattering intensity and its angular distribution on the size of the scattering species, the outlet polydispersity must be minimized if reliable size data are to be obtained from the light scattering detector signal. It is shown that light scattering detection should be used with careful control of fractionating power to obtain optimized analysis of polydisperse samples. Part I is concerned with isocratic operation of As-FlFFF, and part II with programmed operation.
Directory of Open Access Journals (Sweden)
Waleed M. Abd-Elhameed
2016-09-01
Full Text Available Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. The convergence and error analysis of the suggested Fibonacci expansion were carefully investigated. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. Two accurate semi-analytic polynomial solutions for linear and nonlinear fractional differential equations are the result.
Certain fractional integral operators and the generalized multi-index ...
Indian Academy of Sciences (India)
2Department of Economics, Belarusian State University, 220030 Minsk, Belarus ... The real birth and far-reaching development of the fractional calculus ...... Particles, Fields and Media (2010) (Beijing: Springer, Heidelberg; Higher Education.
Some properties for integro-differential operator defined by a fractional formal.
Abdulnaby, Zainab E; Ibrahim, Rabha W; Kılıçman, Adem
2016-01-01
Recently, the study of the fractional formal (operators, polynomials and classes of special functions) has been increased. This study not only in mathematics but extended to another topics. In this effort, we investigate a generalized integro-differential operator [Formula: see text] defined by a fractional formal (fractional differential operator) and study some its geometric properties by employing it in new subclasses of analytic univalent functions.
Fraction Operations: An Examination of Prospective Teachers' Errors Confidence, and Bias
Young, Elaine; Zientek, Linda
2011-01-01
Fractions are important in young students' understanding of rational numbers and proportional reasoning. The teacher is fundamental in developing student understanding and competency in working with fractions. The present study spanned five years and investigated prospective teachers' competency and confidence with fraction operations as they…
DEFF Research Database (Denmark)
Gimperlein, Heiko; Grubb, Gerd
2014-01-01
The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(−t P) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbat......The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(−t P) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained...... for perturbations of fractional powers of the Laplacian. In the selfadjoint case, extensions to t∈C+ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup....
New Approaches for Solving Fokker Planck Equation on Cantor Sets within Local Fractional Operators
Directory of Open Access Journals (Sweden)
Hassan Kamil Jassim
2015-01-01
Full Text Available We discuss new approaches to handling Fokker Planck equation on Cantor sets within local fractional operators by using the local fractional Laplace decomposition and Laplace variational iteration methods based on the local fractional calculus. The new approaches maintain the efficiency and accuracy of the analytical methods for solving local fractional differential equations. Illustrative examples are given to show the accuracy and reliable results.
McAllister, Cheryl J.; Beaver, Cheryl
2012-01-01
The purpose of this research was to determine if recognizable error types exist in the work of preservice teachers required to create story problems for specific fraction operations. Students were given a particular single-operation fraction expression and asked to do the calculation and then create a story problem that would require the use of…
International Nuclear Information System (INIS)
He Qiu-Yan; Yuan Xiao; Yu Bo
2017-01-01
The performance analysis of the generalized Carlson iterating process, which can realize the rational approximation of fractional operator with arbitrary order, is presented in this paper. The reasons why the generalized Carlson iterating function possesses more excellent properties such as self-similarity and exponential symmetry are also explained. K-index, P-index, O-index, and complexity index are introduced to contribute to performance analysis. Considering nine different operational orders and choosing an appropriate rational initial impedance for a certain operational order, these rational approximation impedance functions calculated by the iterating function meet computational rationality, positive reality, and operational validity. Then they are capable of having the operational performance of fractional operators and being physical realization. The approximation performance of the impedance function to the ideal fractional operator and the circuit network complexity are also exhibited. (paper)
Preservice Elementary Teachers' Understanding of Operations for Fraction Multiplication and Division
Whitehead, Ashley; Walkowiak, Temple A.
2017-01-01
This study examined preservice elementary teachers' change in their understanding of fraction operations while taking a mathematics methods course focused on grades 3-5. The preservice teachers (n = 48) completed an assessment before and after the course's unit on the teaching and learning of fractions. Specifically, the preservice teachers were…
Directory of Open Access Journals (Sweden)
Hossein Jafari
2016-04-01
Full Text Available The non-differentiable solution of the linear and non-linear partial differential equations on Cantor sets is implemented in this article. The reduced differential transform method is considered in the local fractional operator sense. The four illustrative examples are given to show the efficiency and accuracy features of the presented technique to solve local fractional partial differential equations.
Differential-difference equations associated with the fractional Lax operators
Energy Technology Data Exchange (ETDEWEB)
Adler, V E [LD Landau Institute for Theoretical Physics, 1A Ak. Semenov, Chernogolovka 142432 (Russian Federation); Postnikov, V V, E-mail: adler@itp.ac.ru, E-mail: postnikofvv@mail.ru [Sochi Branch of Peoples' Friendship University of Russia, 32 Kuibyshev str., 354000 Sochi (Russian Federation)
2011-10-14
We study integrable hierarchies associated with spectral problems of the form P{psi} = {lambda}Q{psi}, where P and Q are difference operators. The corresponding nonlinear differential-difference equations can be viewed as inhomogeneous generalizations of the Bogoyavlensky-type lattices. While the latter turn into the Korteweg-de Vries equation under the continuous limit, the lattices under consideration provide discrete analogs of the Sawada-Kotera and Kaup-Kupershmidt equations. The r-matrix formulation and several of the simplest explicit solutions are presented. (paper)
Energy Technology Data Exchange (ETDEWEB)
Sawa, Kazuhiro; Yoshimuta, Shigeharu; Tobita, Tsutomu; Sato, Masashi [Japan Atomic Energy Research Inst., Oarai, Ibaraki (Japan). Oarai Research Establishment
1997-05-01
The High Temperature Engineering Test Reactor (HTTR) uses coated particles as fuel. During normal operation, short-lived noble gases are mainly released by diffusion from fuel particles with defects in their coating layers (i.e., failed particle). Since noble gases do not plate out on the inner surfaces of primary cooling system, their activities in primary coolant reflect fuel failure fraction in the core. An evaluation method was developed to predict failure fraction of coated fuel particles during normal operation of the HTTR. The method predicts core-average and hot plenum regionwise failure fractions based on the fractional releases, (R/B)s, of noble gases. The (R/B)s are calculated by fission gas concentration measurements in the primary cooling system of the HTTR. Recent fabrication data show that through-coatings failure fraction is extremely low. Then, fractional release from matrix contamination uranium, which is background for accurate evaluation of the fuel failure fraction, should be precisely predicted. This report describes an evaluation method of fuel failure fraction from measurements in the HTTR together with a fission gas release model from fuel compact containing failed particles and matrix contamination uranium. (author)
Ignition and time-dependent fractional power operation of tokamak reactors
International Nuclear Information System (INIS)
Vold, E.L.; Mau, T.K.; Conn, R.W.
1986-01-01
The eventual utilization of a tokamak fusion reactor for commercial power necessitates a thorough understanding of the operational requirements at full and fractional power levels and during transitions from one operating level to another. In this study we examine the role of burn control in maintaining the reactor plasma at equilibrium to avoid thermal runaway during fractional power operation. Because these requirements rely so heavily on the assumptions that govern the plasma transport, this study focuses on time-dependent analyses and a comparison of ignition requirements using a range of energy confinement
Solving Abel’s Type Integral Equation with Mikusinski’s Operator of Fractional Order
Directory of Open Access Journals (Sweden)
Ming Li
2013-01-01
Full Text Available This paper gives a novel explanation of the integral equation of Abel’s type from the point of view of Mikusinski’s operational calculus. The concept of the inverse of Mikusinski’s operator of fractional order is introduced for constructing a representation of the solution to the integral equation of Abel’s type. The proof of the existence of the inverse of the fractional Mikusinski operator is presented, providing an alternative method of treating the integral equation of Abel’s type.
The reactor plasma physics of tandem mirror startup and fractional power operation
International Nuclear Information System (INIS)
Kantrowitz, F.D.; Firestone, M.A.; Guebel, D.M.; Mau, T.K.
1984-01-01
Plasma behavior and the performance of plasma technologies are studied during the startup and fractional power operation of tandem mirrors. Five phases of machine operation are identified, some of which require plasma. The plasma phases include plasma initiation and heating, a standby phase with plasma at the density and temperature characteristics of full design performance in reactors, a deuterium-tritium fractional power operating phase in which the fusion plasma undergoes staged power increases to full power, and rated power operating phase. Plasma initiation and heating uses electron cyclotron resonance heating preionization of background gas in the plug and ion cyclotron resonance heating in the central cell. Operation of the radio-frequency systems, the neutral beams, and the direct converter are studied to determine constraints affecting plasma operation. Studies of fractional power operation, carried out using a quasi-steady-state analysis, show that the plasma Q value can be made remarkably insensitive to the level of fusion power by controlling the plasma radius. Copper insert coils used to increase the maximum choke field require considerable power and cause the recirculating power fraction to increase sharply as the fusion power is reduced. Moreover, when an efficient drift pumping scheme is used, achieved improvements in plasma Q by using high-field choke coils must be weighed against their power consumption and other technological difficulties
Ezz-Eldien, S. S.; Doha, E. H.; Bhrawy, A. H.; El-Kalaawy, A. A.; Machado, J. A. T.
2018-04-01
In this paper, we propose a new accurate and robust numerical technique to approximate the solutions of fractional variational problems (FVPs) depending on indefinite integrals with a type of fixed Riemann-Liouville fractional integral. The proposed technique is based on the shifted Chebyshev polynomials as basis functions for the fractional integral operational matrix (FIOM). Together with the Lagrange multiplier method, these problems are then reduced to a system of algebraic equations, which greatly simplifies the solution process. Numerical examples are carried out to confirm the accuracy, efficiency and applicability of the proposed algorithm
On a class of analytic functions generated by fractional integral operator
Directory of Open Access Journals (Sweden)
Ibrahim Rabha W.
2017-01-01
Full Text Available In this note, we improve the idea of the Tsallis entropy in a complex domain. This improvement is contingent on the fractional operator in a complex domain (type Alexander. We clarify some new classes of analytic functions, which are planned in view of the geometry function theory. This category of entropy is called fractional entropy; accordingly, we demand them fractional entropic geometry classes. Other geometric properties are established in the sequel. Our exhibition is supported by the Maxwell Lemma and Jack Lemma.
Directory of Open Access Journals (Sweden)
Resat Yilmazer
2016-02-01
Full Text Available In this work; we present a method for solving the second-order linear ordinary differential equation of hypergeometric type. The solutions of this equation are given by the confluent hypergeometric functions (CHFs. Unlike previous studies, we obtain some different new solutions of the equation without using the CHFs. Therefore, we obtain new discrete fractional solutions of the homogeneous and non-homogeneous confluent hypergeometric differential equation (CHE by using a discrete fractional Nabla calculus operator. Thus, we obtain four different new discrete complex fractional solutions for these equations.
On Generalized Fractional Integral Operators and the Generalized Gauss Hypergeometric Functions
Directory of Open Access Journals (Sweden)
Dumitru Baleanu
2014-01-01
Full Text Available A remarkably large number of fractional integral formulas involving the number of special functions, have been investigated by many authors. Very recently, Agarwal (National Academy Science Letters gave some integral transform and fractional integral formulas involving the Fpα,β·. In this sequel, here, we aim to establish some image formulas by applying generalized operators of the fractional integration involving Appell’s function F3(· due to Marichev-Saigo-Maeda. Some interesting special cases of our main results are also considered.
A modification of \\mathsf {WKB} method for fractional differential operators of Schrödinger's type
Sayevand, K.; Pichaghchi, K.
2017-09-01
In this paper, we were concerned with the description of the singularly perturbed differential equations within the scope of fractional calculus. However, we shall note that one of the main methods used to solve these problems is the so-called WKB method. We should mention that this was not achievable via the existing fractional derivative definitions, because they do not obey the chain rule. In order to accommodate the WKB to the scope of fractional derivative, we proposed a relatively new derivative called the local fractional derivative. By use of properties of local fractional derivative, we extend the WKB method in the scope of the fractional differential equation. By means of this extension, the WKB analysis based on the Borel resummation, for fractional differential operators of WKB type are investigated. The convergence and the Mittag-Leffler stability of the proposed approach is proven. The obtained results are in excellent agreement with the existing ones in open literature and it is shown that the present approach is very effective and accurate. Furthermore, we are mainly interested to construct the solution of fractional Schrödinger equation in the Mittag-Leffler form and how it leads naturally to this semi-classical approximation namely modified WKB.
Numerical Solution of Time-Dependent Problems with a Fractional-Power Elliptic Operator
Vabishchevich, P. N.
2018-03-01
A time-dependent problem in a bounded domain for a fractional diffusion equation is considered. The first-order evolution equation involves a fractional-power second-order elliptic operator with Robin boundary conditions. A finite-element spatial approximation with an additive approximation of the operator of the problem is used. The time approximation is based on a vector scheme. The transition to a new time level is ensured by solving a sequence of standard elliptic boundary value problems. Numerical results obtained for a two-dimensional model problem are presented.
Koornwinder, T.H.
2015-01-01
For each of the eight n-th derivative parameter changing formulas for Gauss hypergeometric functions a corresponding fractional integration formula is given. For both types of formulas the differential or integral operator is intertwining between two actions of the hypergeometric differential
Patel, J.; Mishra, A. K.
2007-08-01
In the present paper an extended fractional differintegral operator , suitable for the study of multivalent functions is introduced. Various mapping properties and inclusion relationships between certain subclasses of multivalent functions are investigated by applying the techniques of differential subordination. Relevant connections of the definitions and results presented in this paper with those obtained in several earlier works on the subject are also pointed out.
Finite element formulation of viscoelastic sandwich beams using fractional derivative operators
Galucio, A. C.; Deü, J.-F.; Ohayon, R.
This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler-Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.
Performance of combination of a Venturi and nuclear fraction meter in SAGD production operations
Energy Technology Data Exchange (ETDEWEB)
Hompoth, D.; Khun, N. [Suncor Energy, Calgary, AB (Canada); Pinguet, B.G.; Guerra, E. [Schlumberger Canada Ltd., Edmonton, AB (Canada)
2008-07-01
This paper described a multiphase flow meter (MFM) that used a Venturi and nuclear fraction meter combination for steam assisted gravity drainage (SAGD) well production testing. The device was designed by re-engineering a flow model and fluid property package to measure the steam phases. The meter was designed to improve pump monitoring processes in SAGD operations. The technology combined 2 basic measurement steps. The first was a nuclear multi Gamma-ray fraction meters which measured the fraction of each constituent at the Venturi tube's throat at high frequencies. Fractions were then determined from the solution of 3 simultaneous equations related to the Gamma ray attenuation, and a fraction balance equation. Pressure and temperature measurements were used to predict the fluid properties at line conditions. Primary outputs were based on nuclear measurements, gas fractions, water liquid ratios, and mixture densities. Secondary outputs from the meter included volumetric flow rates. Stability, dynamic responses, and reproducibility rates of the MFM were also presented. 9 refs., 6 tabs., 17 figs.
Comparison of the methods for discrete approximation of the fractional-order operator
Directory of Open Access Journals (Sweden)
Zborovjan Martin
2003-12-01
Full Text Available In this paper we will present some alternative types of discretization methods (discrete approximation for the fractional-order (FO differentiator and their application to the FO dynamical system described by the FO differential equation (FDE. With analytical solution and numerical solution by power series expansion (PSE method are compared two effective methods - the Muir expansion of the Tustin operator and continued fraction expansion method (CFE with the Tustin operator and the Al-Alaoui operator. Except detailed mathematical description presented are also simulation results. From the Bode plots of the FO differentiator and FDE and from the solution in the time domain we can see, that the CFE is a more effective method according to the PSE method, but there are some restrictions for the choice of the time step. The Muir expansion is almost unusable.
Marichev-Saigo-Maeda fractional integration operators of the Bassel functions
Directory of Open Access Journals (Sweden)
Sunil D. Purohit
2012-05-01
Full Text Available In this paper, we apply generalized operators of fractional integration involving Appell’s function F_3 (. due to Marichev-Saigo-Maeda, to the Bessel function of first kind. The results are expressed in terms of generalized Wright function and hypergeometric functions _pF_q . Special cases involving this function are mentioned. Results given recently by Kilbas and Sebastian follow as special cases of the theorems establish here.
A new fractional operator of variable order: Application in the description of anomalous diffusion
Yang, Xiao-Jun; Machado, J. A. Tenreiro
2017-09-01
In this paper, a new fractional operator of variable order with the use of the monotonic increasing function is proposed in sense of Caputo type. The properties in term of the Laplace and Fourier transforms are analyzed and the results for the anomalous diffusion equations of variable order are discussed. The new formulation is efficient in modeling a class of concentrations in the complex transport process.
Directory of Open Access Journals (Sweden)
Yaoyao Wang
2014-01-01
Full Text Available For the 4-DOF (degrees of freedom trajectory tracking control problem of underwater remotely operated vehicles (ROVs in the presence of model uncertainties and external disturbances, a novel output feedback fractional-order nonsingular terminal sliding mode control (FO-NTSMC technique is introduced in light of the equivalent output injection sliding mode observer (SMO and TSMC principle and fractional calculus technology. The equivalent output injection SMO is applied to reconstruct the full states in finite time. Meanwhile, the FO-NTSMC algorithm, based on a new proposed fractional-order switching manifold, is designed to stabilize the tracking error to equilibrium points in finite time. The corresponding stability analysis of the closed-loop system is presented using the fractional-order version of the Lyapunov stability theory. Comparative numerical simulation results are presented and analyzed to demonstrate the effectiveness of the proposed method. Finally, it is noteworthy that the proposed output feedback FO-NTSMC technique can be used to control a broad range of nonlinear second-order dynamical systems in finite time.
International Nuclear Information System (INIS)
Zhang Li-Min; Sun Ke-Hui; Liu Wen-Hao; He Shao-Bo
2017-01-01
In this paper, Adomian decomposition method (ADM) with high accuracy and fast convergence is introduced to solve the fractional-order piecewise-linear (PWL) hyperchaotic system. Based on the obtained hyperchaotic sequences, a novel color image encryption algorithm is proposed by employing a hybrid model of bidirectional circular permutation and DNA masking. In this scheme, the pixel positions of image are scrambled by circular permutation, and the pixel values are substituted by DNA sequence operations. In the DNA sequence operations, addition and substraction operations are performed according to traditional addition and subtraction in the binary, and two rounds of addition rules are used to encrypt the pixel values. The simulation results and security analysis show that the hyperchaotic map is suitable for image encryption, and the proposed encryption algorithm has good encryption effect and strong key sensitivity. It can resist brute-force attack, statistical attack, differential attack, known-plaintext, and chosen-plaintext attacks. (paper)
Diakowska, Dorota; Knast, Witold; Diakowski, Witold; Grabowski, Krzysztof; Szelachowski, Piotr; Pelczar, Piotr
2005-06-01
This study was undertaken to determine how fats digestion processes were damaged due to chronic pancreatitis, and identify, whether lipid metabolism improved after surgical treatment the patients with chronic pancreatitis. Total lipids, triglycerides, diglycerides and free fatty acids levels in serum and stool were analysed, using chemical tests, thin-layer chromatography and electrophoresis of serum lipoproteins. The patients before the operations showed higher total lipids and triglycerides concentrations, and lower concentrations of diglycerides and free fatty acids in stool. These patients had high triglycerides, chylomicrons, VLDL, LDL-CH concentrations, and low-diglycerides, free fatty acids, HDL-CH concentrations in serum. These data were statistically significant. After the operations and substitution therapy it was observed normalization of the total lipids and lipids fractions levels in stool and in serum. Concentrations of LDL-CH and HDL-CH fractions were irregular. We conclude, that these lipids parameters could be used in diagnosing and monitoring the results of chronic pancreatitis surgical treatment.
Best Approximation of the Fractional Semi-Derivative Operator by Exponential Series
Directory of Open Access Journals (Sweden)
Vladimir D. Zakharchenko
2018-01-01
Full Text Available A significant reduction in the time required to obtain an estimate of the mean frequency of the spectrum of Doppler signals when seeking to measure the instantaneous velocity of dangerous near-Earth cosmic objects (NEO is an important task being developed to counter the threat from asteroids. Spectral analysis methods have shown that the coordinate of the centroid of the Doppler signal spectrum can be found by using operations in the time domain without spectral processing. At the same time, an increase in the speed of resolving the algorithm for estimating the mean frequency of the spectrum is achieved by using fractional differentiation without spectral processing. Thus, an accurate estimate of location of the centroid for the spectrum of Doppler signals can be obtained in the time domain as the signal arrives. This paper considers the implementation of a fractional-differentiating filter of the order of ½ by a set of automation astatic transfer elements, which greatly simplifies practical implementation. Real technical devices have the ultimate time delay, albeit small in comparison with the duration of the signal. As a result, the real filter will process the signal with some error. In accordance with this, this paper introduces and uses the concept of a “pre-derivative” of ½ of magnitude. An optimal algorithm for realizing the structure of the filter is proposed based on the criterion of minimum mean square error. Relations are obtained for the quadrature coefficients that determine the structure of the filter.
Directory of Open Access Journals (Sweden)
Hua Wang
2017-01-01
Full Text Available Abstract In this paper, we first introduce some new Morrey-type spaces containing generalized Morrey space and weighted Morrey space with two weights as special cases. Then we give the weighted strong type and weak type estimates for fractional integral operators I α $I_{\\alpha}$ in these new Morrey-type spaces. Furthermore, the weighted strong type estimate and endpoint estimate of linear commutators [ b , I α ] $[b,I_{\\alpha}]$ formed by b and I α $I_{\\alpha}$ are established. Also we study related problems about two-weight, weak type inequalities for I α $I_{\\alpha}$ and [ b , I α ] $[b,I_{\\alpha}]$ in the Morrey-type spaces and give partial results.
Bonnemains, Laurent; Stos, Bertrand; Vaugrenard, Thibaud; Marie, Pierre-Yves; Odille, Freddy; Boudjemline, Younes
2012-03-01
To examine in a population of post-operative tetralogy of Fallot patients, the correlation between right ventricle (RV) ejection fractions (EF) computed from magnetic resonance imaging (MRI) and three echocardiographic indices of RV function: TAPSE, longitudinal strain and strain rate. Indeed, these patients present a pulmonary regurgitation which is responsible for progressive dilatation of the RV. An echocardiographic assessment of the RV function would be very useful in determining the timing of pulmonary revalvulation for Fallot patients. However, these indices are generally based on the ventricle contraction in the long axis direction which is impaired in this population and does not seem to correlate with the EF. Thirty-five post-operative tetralogy of Fallot patients and 20 patients with normal RVs were included. In both groups, RVEF, assessed by MRI, was compared with the three echocardiographic indices. Longitudinal strain and strain rates were computed both on the free wall and on the whole RV. No correlation was found between the echocardiographic indices and the MRI EF in our Fallot population. The accuracy of those indices as a diagnostic test of an altered RV was low with Younden's indices varying from -0.18 to 0.5 and areas under the Receiver Operating Characterictic (ROC) curves equal to 0.54 for tricuspid annulus plane systolic excursion, 0.59-0.62 for strain and 0.57-0.63 for strain rate. Three conventional echocardiographic indices based on RV longitudinal contraction failed to assess the EF in our population of post-operative tetralogy of Fallot patients.
Progress Toward Steady State Tokamak Operation Exploiting the high bootstrap current fraction regime
Ren, Q.
2015-11-01
Recent DIII-D experiments have advanced the normalized fusion performance of the high bootstrap current fraction tokamak regime toward reactor-relevant steady state operation. The experiments, conducted by a joint team of researchers from the DIII-D and EAST tokamaks, developed a fully noninductive scenario that could be extended on EAST to a demonstration of long pulse steady-state tokamak operation. Fully noninductive plasmas with extremely high values of the poloidal beta, βp >= 4 , have been sustained at βT >= 2 % for long durations with excellent energy confinement quality (H98y,2 >= 1 . 5) and internal transport barriers (ITBs) generated at large minor radius (>= 0 . 6) in all channels (Te, Ti, ne, VTf). Large bootstrap fraction (fBS ~ 80 %) has been obtained with high βp. ITBs have been shown to be compatible with steady state operation. Because of the unusually large ITB radius, normalized pressure is not limited to low βN values by internal ITB-driven modes. βN up to ~4.3 has been obtained by optimizing the plasma-wall distance. The scenario is robust against several variations, including replacing some on-axis with off-axis neutral beam injection (NBI), adding electron cyclotron (EC) heating, and reducing the NBI torque by a factor of 2. This latter observation is particularly promising for extension of the scenario to EAST, where maximum power is obtained with balanced NBI injection, and to a reactor, expected to have low rotation. However, modeling of this regime has provided new challenges to state-of-the-art modeling capabilities: quasilinear models can dramatically underpredict the electron transport, and the Sauter bootstrap current can be insufficient. The analysis shows first-principle NEO is in good agreement with experiments for the bootstrap current calculation and ETG modes with a larger saturated amplitude or EM modes may provide the missing electron transport. Work supported in part by the US DOE under DE-FC02-04ER54698, DE-AC52-07NA
Directory of Open Access Journals (Sweden)
Ya-Juan Hao
2013-01-01
Full Text Available The main object of this paper is to investigate the Helmholtz and diffusion equations on the Cantor sets involving local fractional derivative operators. The Cantor-type cylindrical-coordinate method is applied to handle the corresponding local fractional differential equations. Two illustrative examples for the Helmholtz and diffusion equations on the Cantor sets are shown by making use of the Cantorian and Cantor-type cylindrical coordinates.
Directory of Open Access Journals (Sweden)
Jieming Zhang
2013-01-01
Full Text Available We establish some sufficient conditions for the existence and uniqueness of positive solutions to a class of initial value problem for impulsive fractional differential equations involving the Caputo fractional derivative. Our analysis relies on a fixed point theorem for mixed monotone operators. Our result can not only guarantee the existence of a unique positive solution but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate our main result.
Directory of Open Access Journals (Sweden)
Yang Zhao
2013-01-01
Full Text Available The mappings for some special functions on Cantor sets are investigated. Meanwhile, we apply the local fractional Fourier series, Fourier transforms, and Laplace transforms to solve three local fractional differential equations, and the corresponding nondifferentiable solutions were presented.
A study of ∇-discrete fractional calculus operator on the radial ...
African Journals Online (AJOL)
The fractional calculus includes concepts of integrals and derivatives of any complex or real order. The fractional calculus is as old as the usual calculus. Recently, many scientists have been studying on this eld to provide the development and applicability to various areas of mathematics, physics, engineering and other ...
Povstenko, Yuriy
2015-01-01
This book is devoted to fractional thermoelasticity, i.e. thermoelasticity based on the heat conduction equation with differential operators of fractional order. Readers will discover how time-fractional differential operators describe memory effects and space-fractional differential operators deal with the long-range interaction. Fractional calculus, generalized Fourier law, axisymmetric and central symmetric problems and many relevant equations are featured in the book. The latest developments in the field are included and the reader is brought up to date with current research. The book contains a large number of figures, to show the characteristic features of temperature and stress distributions and to represent the whole spectrum of order of fractional operators. This work presents a picture of the state-of-the-art of fractional thermoelasticity and is suitable for specialists in applied mathematics, physics, geophysics, elasticity, thermoelasticity and engineering sciences. Corresponding sections of ...
Ulam stability for fractional differential equations in the sense of Caputo operator
Directory of Open Access Journals (Sweden)
Rabha W. Ibrahim
2012-12-01
Full Text Available In this paper, we consider the Hyers-Ulam stability for the following fractional differential equations, in the sense ofcomplex Caputo fractional derivative defined, in the unit disk: cDßzf(z=G(f(z, cDázf(z,zf‘(z;z 0<á<1<ß<2 . Furthermore,a generalization of the admissible functions in complex Banach spaces is imposed and applications are illustrated.
Directory of Open Access Journals (Sweden)
Sunday O. Edeki
2018-03-01
Full Text Available In this study, approximate solutions of a system of time-fractional coupled Burger equations were obtained by means of a local fractional operator (LFO in the sense of the Caputo derivative. The LFO technique was built on the basis of the standard differential transform method (DTM. Illustrative examples used in demonstrating the effectiveness and robustness of the proposed method show that the solution method is very efficient and reliable as – unlike the variational iteration method – it does not depend on any process of identifying Lagrange multipliers, even while still maintaining accuracy.
Diethelm, Kai
2010-01-01
Fractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations.
Web-Based Instruction on Preservice Teachers' Knowledge of Fraction Operations
Lin, Cheng-Yao
2010-01-01
This study determines whether web-based instruction (WBI) represents an improved method for helping preservice teachers learn procedural and conceptual knowledge of fractions.. The purpose was to compare the effectiveness of web-based instruction (WBI) with the traditional lecture in mathematics content and methods for the elementary school…
Directory of Open Access Journals (Sweden)
D. Jabari Sabeg
2016-10-01
Full Text Available In this paper, we present a new computational method for solving nonlinear singular boundary value problems of fractional order arising in biology. To this end, we apply the operational matrices of derivatives of shifted Legendre polynomials to reduce such problems to a system of nonlinear algebraic equations. To demonstrate the validity and applicability of the presented method, we present some numerical examples.
Directory of Open Access Journals (Sweden)
Zengyan Si
2012-01-01
Full Text Available We prove that b is in Lipβ(ω if and only if the commutator [b,L-α/2] of the multiplication operator by b and the general fractional integral operator L-α/2 is bounded from the weighted Morrey space Lp,k(ω to Lq,kq/p(ω1-(1-α/nq,ω, where 0(1-k/(p/(q-k, and here rω denotes the critical index of ω for the reverse Hölder condition.
Kiryakova, Virginia S.
2012-11-01
The Laplace Transform (LT) serves as a basis of the Operational Calculus (OC), widely explored by engineers and applied scientists in solving mathematical models for their practical needs. This transform is closely related to the exponential and trigonometric functions (exp, cos, sin) and to the classical differentiation and integration operators, reducing them to simple algebraic operations. Thus, the classical LT and the OC give useful tool to handle differential equations and systems with constant coefficients. Several generalizations of the LT have been introduced to allow solving, in a similar way, of differential equations with variable coefficients and of higher integer orders, as well as of fractional (arbitrary non-integer) orders. Note that fractional order mathematical models are recently widely used to describe better various systems and phenomena of the real world. This paper surveys briefly some of our results on classes of such integral transforms, that can be obtained from the LT by means of "transmutations" which are operators of the generalized fractional calculus (GFC). On the list of these Laplace-type integral transforms, we consider the Borel-Dzrbashjan, Meijer, Krätzel, Obrechkoff, generalized Obrechkoff (multi-index Borel-Dzrbashjan) transforms, etc. All of them are G- and H-integral transforms of convolutional type, having as kernels Meijer's G- or Fox's H-functions. Besides, some special functions (also being G- and H-functions), among them - the generalized Bessel-type and Mittag-Leffler (M-L) type functions, are generating Gel'fond-Leontiev (G-L) operators of generalized differentiation and integration, which happen to be also operators of GFC. Our integral transforms have operational properties analogous to those of the LT - they do algebrize the G-L generalized integrations and differentiations, and thus can serve for solving wide classes of differential equations with variable coefficients of arbitrary, including non-integer order
GHOLAMI, SAEID; BABOLIAN, ESMAIL; JAVIDI, MOHAMMAD
2016-01-01
This paper presents a new numerical approach to solve single and multiterm time fractional diffusion equations. In this work, the space dimension is discretized to the Gauss$-$Lobatto points. We use the normalized Grunwald approximation for the time dimension and a pseudospectral successive integration matrix for the space dimension. This approach shows that with fewer numbers of points, we can approximate the solution with more accuracy. Some examples with numerical results in tables and fig...
Solvability of fractional multi-point boundary-value problems with p-Laplacian operator at resonance
Directory of Open Access Journals (Sweden)
Tengfei Shen
2014-02-01
Full Text Available In this article, we consider the multi-point boundary-value problem for nonlinear fractional differential equations with $p$-Laplacian operator: $$\\displaylines{ D_{0^+}^\\beta \\varphi_p (D_{0^+}^\\alpha u(t = f(t,u(t,D_{0^+}^{\\alpha - 2} u(t,D_{0^+}^{\\alpha - 1} u(t, D_{0^+}^\\alpha u(t,\\quad t \\in (0,1, \\cr u(0 = u'(0=D_{0^+}^\\alpha u(0 = 0,\\quad D_{0^+}^{\\alpha - 1} u(1 = \\sum_{i = 1}^m {\\sigma_i D_{0^+}^{\\alpha - 1} u(\\eta_i } , }$$ where $2 < \\alpha \\le 3$, $0 < \\beta \\le 1$, $3 < \\alpha + \\beta \\le 4$, $\\sum_{i = 1}^m {\\sigma_i } = 1$, $D_{0^+}^\\alpha$ is the standard Riemann-Liouville fractional derivative. $\\varphi_{p}(s=|s|^{p-2}s$ is p-Laplacians operator. The existence of solutions for above fractional boundary value problem is obtained by using the extension of Mawhin's continuation theorem due to Ge, which enrich konwn results. An example is given to illustrate the main result.
Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators
Koca, Ilknur
2018-03-01
The model of Ebola spread within a targeted population is extended to the concept of fractional differentiation and integration with non-local and non-singular fading memory introduced by Atangana and Baleanu. It is expected that the proposed model will show better approximation than the models established before. The existence and uniqueness of solutions for the spread of Ebola disease model is given via the Picard-Lindelof method. Finally, numerical solutions for the model are given by using different parameter values.
Coronel-Escamilla, A.; Gómez-Aguilar, J. F.; Torres, L.; Escobar-Jiménez, R. F.; Valtierra-Rodríguez, M.
2017-12-01
In this paper, we propose a state-observer-based approach to synchronize variable-order fractional (VOF) chaotic systems. In particular, this work is focused on complete synchronization with a so-called unidirectional master-slave topology. The master is described by a dynamical system in state-space representation whereas the slave is described by a state observer. The slave is composed of a master copy and a correction term which in turn is constituted of an estimation error and an appropriate gain that assures the synchronization. The differential equations of the VOF chaotic system are described by the Liouville-Caputo and Atangana-Baleanu-Caputo derivatives. Numerical simulations involving the synchronization of Rössler oscillators, Chua's systems and multi-scrolls are studied. The simulations show that different chaotic behaviors can be obtained if different smooths functions defined in the interval (0 , 1 ] are used as the variable order of the fractional derivatives. Furthermore, simulations show that the VOF chaotic systems can be synchronized.
Eluru, Naveen; Chakour, Vincent; Chamberlain, Morgan; Miranda-Moreno, Luis F
2013-10-01
Vehicle operating speed measured on roadways is a critical component for a host of analysis in the transportation field including transportation safety, traffic flow modeling, roadway geometric design, vehicle emissions modeling, and road user route decisions. The current research effort contributes to the literature on examining vehicle speed on urban roads methodologically and substantively. In terms of methodology, we formulate a new econometric model framework for examining speed profiles. The proposed model is an ordered response formulation of a fractional split model. The ordered nature of the speed variable allows us to propose an ordered variant of the fractional split model in the literature. The proposed formulation allows us to model the proportion of vehicles traveling in each speed interval for the entire segment of roadway. We extend the model to allow the influence of exogenous variables to vary across the population. Further, we develop a panel mixed version of the fractional split model to account for the influence of site-specific unobserved effects. The paper contributes substantively by estimating the proposed model using a unique dataset from Montreal consisting of weekly speed data (collected in hourly intervals) for about 50 local roads and 70 arterial roads. We estimate separate models for local roads and arterial roads. The model estimation exercise considers a whole host of variables including geometric design attributes, roadway attributes, traffic characteristics and environmental factors. The model results highlight the role of various street characteristics including number of lanes, presence of parking, presence of sidewalks, vertical grade, and bicycle route on vehicle speed proportions. The results also highlight the presence of site-specific unobserved effects influencing the speed distribution. The parameters from the modeling exercise are validated using a hold-out sample not considered for model estimation. The results indicate
Fractional Schroedinger equation
International Nuclear Information System (INIS)
Laskin, Nick
2002-01-01
Some properties of the fractional Schroedinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schroedinger equation we find the energy spectra of a hydrogenlike atom (fractional 'Bohr atom') and of a fractional oscillator in the semiclassical approximation. An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schroedinger equations
Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; López-López, M. G.; Alvarado-Martínez, V. M.
2018-03-01
In this paper, the two-dimensional projectile motion was studied; for this study two cases were considered, for the first one, we considered that there is no air resistance and, for the second case, we considered a resisting medium k . The study was carried out by using fractional calculus. The solution to this study was obtained by using fractional operators with power law, exponential decay and Mittag-Leffler kernel in the range of γ \\in (0,1] . These operators were considered in the Liouville-Caputo sense to use physical initial conditions with a known physical interpretation. The range and the maximum height of the projectile were obtained using these derivatives. With the aim of exploring the validity of the obtained results, we compared our results with experimental data given in the literature. A multi-objective particle swarm optimization approach was used for generating Pareto-optimal solutions for the parameters k and γ for different fixed values of velocity v0 and angle θ . The results showed some relevant qualitative differences between the use of power law, exponential decay and Mittag-Leffler law.
Directory of Open Access Journals (Sweden)
Amancio L. Cantoria, Jr.
2016-11-01
Full Text Available The study of fractions in Philippine mathematics curriculum starts as early as first grade. In spite of the regular rehearsal of this mathematical topic through secondary school level, many students reach college without showing adequate skills in fraction. This study determined the performance and analyzed the errors of preservice teachers in dealing with fractions. Findings revealed that preservice teachers’ performance in solving fractions reached an unacceptable level. Prevalent errors were demonstrated when adding dissimilar fractions, adding a mixed number and a fraction, and multiplying a mixed number by a fraction, because the dominant procedural knowledge in fraction addition interferes with their knowledge of fraction multiplication, and vice versa. Moreover, preservice teachers exhibit low level of content knowledge of fractions as shown in their inability to add common fractions and their failure to translate mixed numbers into equivalent fractions.
Structures of the fractional spaces generated by the difference neutron transport operator
International Nuclear Information System (INIS)
Ashyralyev, Allaberen; Taskin, Abdulgafur
2015-01-01
The initial boundary value problem for the neutron transport equation is considered. The first, second and third order of accuracy difference schemes for the approximate solution of this problem are presented. Highly accurate difference schemes for neutron transport equation based on Padé approximation are constructed. In applications, stability estimates for solutions of difference schemes for the approximate solution of the neutron transport equation are obtained.The positivity of the neutron transport operator in Slobodeckij spaces is proved. Numerical techniques are developed and algorithms are tested on an example in MATLAB
Operational problems related to the preparation of the seawater soluble fraction of crude oil.
Ziolli, Roberta L; Jardim, Wilson F
2002-02-01
Owing to the importance of dissolution and weathering processes following oil spills, this work focused on the operational (quantitative) aspects related to the dissolution of petroleum-derived products, as well as the influence of solar light on both dissolution and the photoproduction of hydrogen peroxide. Four Brazilian crude oil samples were used to study the transfer process of organic compounds from the crude oil film to the aqueous phase (natural seawater) over a period of up to 45 days. Dissolved organic carbon (DOC), measured by non-dispersive infrared spectroscopy followed by high temperature catalytic combustion, was used to follow the partitioning between the two phases. Aqueous DOC values increased as a function of time (up to 15 days) until equilibrium was reached at concentrations ranging from 5 to 45 mg C L(-1). The final DOC concentration as well as the rate of dissolution depends on the nature of the crude oil. When exposed to sunlight, the dissolution was enhanced by up to 67.3%, and inorganic peroxides were generated in the concentration range from 4.5 up to 8.0 micromol L(-1) after 7.3 h irradiation. These results indicate that there is a need for a standard procedure for the production of the WSF in order to generate a more reliable tool to assess the impact of oil spills on the marine environment.
DEFF Research Database (Denmark)
Angelidaki, Irini; Cui, J.; Chen, X.
2006-01-01
Three operational strategies to reduce inhibition due to ammonia during thermophilic anaerobic digestion of source-sorted organic fraction of municipal solid waste (SS-OFMSW) rich in proteins were investigated. Feed was prepared by diluting SS-OFMSW (ratio of 1:4) with tap water or reactor process...... ammonium bicarbonate additions. Dilution of SS-OFMSW with fresh water showed a stable performance with volatile fatty acids of solids (VS). Use of recirculated process water after stripping ammonia showed even better performance with a methane yield...... of 0.43 m(3) kg(-1)VS. Recirculation of process water alone on the other hand, resulted in process inhibition at both TAN levels of 3.5 and 5.5 g-N l(-1). However, after a short period, the process recovered and adapted to the tested TAN levels. Thus, use of recirculated process water after stripping...
Gembong, S.; Suwarsono, S. T.; Prabowo
2018-03-01
Schema in the current study refers to a set of action, process, object and other schemas already possessed to build an individual’s ways of thinking to solve a given problem. The current study aims to investigate the schemas built among elementary school students in solving problems related to operations of addition to fractions. The analyses of the schema building were done qualitatively on the basis of the analytical framework of the APOS theory (Action, Process, Object, and Schema). Findings show that the schemas built on students of high and middle ability indicate the following. In the action stage, students were able to add two fractions by way of drawing a picture or procedural way. In the Stage of process, they could add two and three fractions. In the stage of object, they could explain the steps of adding two fractions and change a fraction into addition of fractions. In the last stage, schema, they could add fractions by relating them to another schema they have possessed i.e. the least common multiple. Those of high and middle mathematic abilities showed that their schema building in solving problems related to operations odd addition to fractions worked in line with the framework of the APOS theory. Those of low mathematic ability, however, showed that their schema on each stage did not work properly.
FRACTIONS: CONCEPTUAL AND DIDACTIC ASPECTS
Directory of Open Access Journals (Sweden)
Sead Rešić
2016-09-01
Full Text Available Fractions represent the manner of writing parts of whole numbers (integers. Rules for operations with fractions differ from rules for operations with integers. Students face difficulties in understanding fractions, especially operations with fractions. These difficulties are well known in didactics of Mathematics throughout the world and there is a lot of research regarding problems in learning about fractions. Methods for facilitating understanding fractions have been discovered, which are essentially related to visualizing operations with fractions.
Sareen, Jitender; Belik, Shay-Lee; Afifi, Tracie O; Asmundson, Gordon J G; Cox, Brian J; Stein, Murray B
2008-12-01
We investigated mental disorders, suicidal ideation, self-perceived need for treatment, and mental health service utilization attributable to exposure to peacekeeping and combat operations among Canadian military personnel. With data from the Canadian Community Health Survey Cycle 1.2 Canadian Forces Supplement, a cross-sectional population-based survey of active Canadian military personnel (N = 8441), we estimated population attributable fractions (PAFs) of adverse mental health outcomes. Exposure to either combat or peacekeeping operations was associated with posttraumatic stress disorder (men: PAF = 46.6%; 95% confidence interval [CI] = 27.3, 62.7; women: PAF = 23.6%; 95% CI = 9.2, 40.1), 1 or more mental disorder assessed in the survey (men: PAF = 9.3%; 95% CI = 0.4, 18.1; women: PAF = 6.1%; 95% CI = 0.0, 13.4), and a perceived need for information (men: PAF = 12.3%; 95% CI = 4.1, 20.6; women: PAF = 7.9%; 95% CI = 1.3, 15.5). A substantial proportion, but not the majority, of mental health-related outcomes were attributable to combat or peacekeeping deployment. Future studies should assess traumatic events and their association with physical injury during deployment, premilitary factors, and postdeployment psychosocial factors that may influence soldiers' mental health.
Cho, Sung Bin; Lee, Ju Hee; Choi, Moon Jung; Lee, Kyu-Yeop; Oh, Sang Ho
2009-01-01
Current treatments for acne scars and enlarged facial pores have shown limited efficacy. To evaluate the efficacy and safety of the fractional photothermolysis system (FPS) with dynamic operating mode on acne scars and enlarged pores. Twelve patients with mild to moderate atrophic acne scars and enlarged pores were included in this study. Three sessions of FPS treatment were performed for acne scars and facial pores monthly. Two blinded dermatologists who compared before and after photos based on a quartile grading scale conducted objective clinical assessments of acne scar- and facial pore-treated areas. We took a biopsy immediately after one treatment with the laser from one of the authors to assess the histologic effects of the laser on facial pores. Follow-up results at 4 months after the last treatment revealed that, of the 12 patients, for acne scars, five demonstrated clinical improvements of 51% to 75% and three demonstrated improvements of 76% to 100%, and for facial pores, five demonstrated moderate clinical improvements of 26% to 50% and three demonstrated improvements of 76% to 100%. Side effects, including pain, post-treatment erythema, and edema, were resolved within 1 week. We suggest that the FPS may provide a new treatment algorithm in some cases with acne scars and enlarged pores. Considering the lack of placebo-controlled, split-face design of our study, optimized, prospective studies should be conducted to fully assess the efficacy of FPS with dynamic operating mode.
Directory of Open Access Journals (Sweden)
Jianping Liu
2016-01-01
Full Text Available An operational matrix technique is proposed to solve variable order fractional differential-integral equation based on the second kind of Chebyshev polynomials in this paper. The differential operational matrix and integral operational matrix are derived based on the second kind of Chebyshev polynomials. Using two types of operational matrixes, the original equation is transformed into the arithmetic product of several dependent matrixes, which can be viewed as an algebraic system after adopting the collocation points. Further, numerical solution of original equation is obtained by solving the algebraic system. Finally, several examples show that the numerical algorithm is computationally efficient.
Jain, Shilpi; Agarwal, Praveen; Kıymaz, I. Onur; ćetinkaya, Ayá¹£egül
2018-01-01
Authors presented some composition formulae for the Marichev-Saigo-Maeda (M-S-M) fractional integral operator with the multi-index Mittag-Leffler functions. Our results are generalizes the results obtained by Choi and Agarwal [3]. Here, we record some particular cases of our main result. Finally, we obtain Laplace transforms of the composition formulae.
FRACTIONS: CONCEPTUAL AND DIDACTIC ASPECTS
Sead Rešić; Ismet Botonjić; Maid Omerović
2016-01-01
Fractions represent the manner of writing parts of whole numbers (integers). Rules for operations with fractions differ from rules for operations with integers. Students face difficulties in understanding fractions, especially operations with fractions. These difficulties are well known in didactics of Mathematics throughout the world and there is a lot of research regarding problems in learning about fractions. Methods for facilitating understanding fractions have been discovered...
Wilde, M. V.; Sergeeva, N. V.
2018-05-01
An explicit asymptotic model extracting the contribution of a surface wave to the dynamic response of a viscoelastic half-space is derived. Fractional exponential Rabotnov's integral operators are used for describing of material properties. The model is derived by extracting the principal part of the poles corresponding to the surface waves after applying Laplace and Fourier transforms. The simplified equations for the originals are written by using power series expansions. Padè approximation is constructed to unite short-time and long-time models. The form of this approximation allows to formulate the explicit model using a fractional exponential Rabotnov's integral operator with parameters depending on the properties of surface wave. The applicability of derived models is studied by comparing with the exact solutions of a model problem. It is revealed that the model based on Padè approximation is highly effective for all the possible time domains.
Bantis, Leonidas E; Nakas, Christos T; Reiser, Benjamin; Myall, Daniel; Dalrymple-Alford, John C
2017-06-01
The three-class approach is used for progressive disorders when clinicians and researchers want to diagnose or classify subjects as members of one of three ordered categories based on a continuous diagnostic marker. The decision thresholds or optimal cut-off points required for this classification are often chosen to maximize the generalized Youden index (Nakas et al., Stat Med 2013; 32: 995-1003). The effectiveness of these chosen cut-off points can be evaluated by estimating their corresponding true class fractions and their associated confidence regions. Recently, in the two-class case, parametric and non-parametric methods were investigated for the construction of confidence regions for the pair of the Youden-index-based optimal sensitivity and specificity fractions that can take into account the correlation introduced between sensitivity and specificity when the optimal cut-off point is estimated from the data (Bantis et al., Biomet 2014; 70: 212-223). A parametric approach based on the Box-Cox transformation to normality often works well while for markers having more complex distributions a non-parametric procedure using logspline density estimation can be used instead. The true class fractions that correspond to the optimal cut-off points estimated by the generalized Youden index are correlated similarly to the two-class case. In this article, we generalize these methods to the three- and to the general k-class case which involves the classification of subjects into three or more ordered categories, where ROC surface or ROC manifold methodology, respectively, is typically employed for the evaluation of the discriminatory capacity of a diagnostic marker. We obtain three- and multi-dimensional joint confidence regions for the optimal true class fractions. We illustrate this with an application to the Trail Making Test Part A that has been used to characterize cognitive impairment in patients with Parkinson's disease.
Forss, Erik; Haupt, Dan; Stålberg, Olle; Enmark, Martin; Samuelsson, Jörgen; Fornstedt, Torgny
2017-05-26
The need to determine the actual operational conditions, instead of merely using the set operational conditions, was investigated for in packed supercritical fluid chromatography (SFC) by design of experiments (DoE) using a most important type of compounds, pharmaceutical basics, as models. The actual values of temperature, pressure, and methanol levels were recorded and calculated from external sensors, while the responses in the DoE were the retention factors and selectivity. A Kromasil CelluCoat column was used as the stationary phase, carbon dioxide containing varying methanol contents as the mobile phase, and the six racemates of alprenolol, atenolol, metoprolol, propranolol, clenbuterol, and mianserin were selected as model solutes. For the retention modeling, the most important term was the methanol fraction followed by the temperature and pressure. Significant differences (p<0.05) between most of the coefficients in the retention models were observed when comparing models from set and actual conditions. The selectivity was much less affected by operational changes, and therefore was not severely affected by difference between set and actual conditions. The temperature differences were usually small, maximum ±1.4°C, whereas the pressure differences were larger, typically approximately +10.5bar. The set and actual fractions of methanol also differed, usually by ±0.4 percentage points. A cautious conclusion is that the primary reason for the discrepancy between the models is a mismatch between the set and actual methanol fractions. This mismatch is more serious in retention models at low methanol fractions. The study demonstrates that the actual conditions should almost always be preferred. Copyright © 2017 Elsevier B.V. All rights reserved.
Kamhawi, Hani; Huang, Wensheng; Haag, Thomas
2014-01-01
The National Aeronautics and Space Administration (NASA) Science Mission Directorate In- Space Propulsion Technology office is sponsoring NASA Glenn Research Center (GRC) to develop a 4 kW-class Hall thruster propulsion system for implementation in NASA science missions. Tests were performed within NASA GRC Vacuum Facility 5 at background pressure levels that were six times lower than what has previously been attained in other vacuum facilities. A study was conducted to assess the impact of varying the cathode-to-anode flow fraction and cathode position on the performance and operational characteristics of the High Voltage Hall Accelerator (HiVHAc) thruster. In addition, the impact of injecting additional xenon propellant in the vicinity of the cathode was also assessed. Cathode-to-anode flow fraction sensitivity tests were performed for power levels between 1.0 and 3.9 kW. It was found that varying the cathode flow fraction from 5 to approximately 10% of the anode flow resulted in the cathode-to-ground voltage becoming more positive. For an operating condition of 3.8 kW and 500 V, varying the cathode position from a distance of closest approach to 600 mm away did not result in any substantial variation in thrust but resulted in the cathode-to-ground changing from -17 to -4 V. The change in the cathode-to-ground voltage along with visual observations indicated a change in how the cathode plume was coupling to the thruster discharge. Finally, the injection of secondary xenon flow in the vicinity of the cathode had an impact similar to increasing the cathode-to-anode flow fraction, where the cathode-to-ground voltage became more positive and discharge current and thrust increased slightly. Future tests of the HiVHAc thruster are planned with a centrally mounted cathode in order to further assess the impact of cathode position on thruster performance.
Wanta, Brendan T; Hanson, Kristine T; Hyder, Joseph A; Stewart, Thomas M; Curry, Timothy B; Berbari, Elie F; Habermann, Elizabeth B; Kor, Daryl J; Brown, Michael J
2018-04-02
Whether the fraction of inspired oxygen (F I O 2 ) influences the risk of surgical site infection (SSI) is controversial. The World Health Organization and the World Federation of Societies of Anesthesiologists offer conflicting recommendations. In this study, we evaluate simultaneously three different definitions of F I O 2 exposure and the risk of SSI in a large surgical population. Patients with clean (type 1) surgical incisions who developed superficial and deep organ/space SSI within 30 days after surgery from January 2003 through December 2012 in five surgical specialties were matched to specialty-specific controls. Fraction of inspired oxygen exposure was defined as (1) nadir F I O 2 , (2) percentage of operative time with F I O 2 greater than 50%, and (3) cumulative hyperoxia exposure, calculated as the area under the curve (AUC) of F I O 2 by time for the duration in which F I O 2 greater than 50%. Stratified univariable and multivariable logistic regression models tested associations between F I O 2 and SSI. One thousand two hundred fifty cases of SSI were matched to 3,248 controls. Increased oxygen exposure, by any of the three measures, was not associated with the outcome of any SSI in a multivariable logistic regression model. Elevated body mass index (BMI; 35+ vs. operative oxygen exposure was associated with higher odds of SSI in the neurosurgical and spine populations. Increased intra-operative inspired fraction of oxygen was not associated with a reduction in SSI. These findings do not support the practice of increasing F I O 2 for the purpose of SSI reduction in patients with clean surgical incisions.
Fractional vector calculus and fractional Maxwell's equations
International Nuclear Information System (INIS)
Tarasov, Vasily E.
2008-01-01
The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green's, Stokes' and Gauss's theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell's equations and the corresponding fractional wave equations are considered
Fractional gradient and its application to the fractional advection equation
D'Ovidio, M.; Garra, R.
2013-01-01
In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional fractional differential equations. A first application is discussed in relation to the d-dimensional fractional advection-dispersion equation. We also study the connection with multidimensional L\\'evy processes.
On matrix fractional differential equations
Directory of Open Access Journals (Sweden)
Adem Kılıçman
2017-01-01
Full Text Available The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann–Liouville using Laplace transform method and convolution product to the Riemann–Liouville fractional of matrices. Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objective of this article is to discuss the Laplace transform method based on operational matrices of fractional derivatives for solving several kinds of linear fractional differential equations. Moreover, we present the operational matrices of fractional derivatives with Laplace transform in many applications of various engineering systems as control system. We present the analytical technique for solving fractional-order, multi-term fractional differential equation. In other words, we propose an efficient algorithm for solving fractional matrix equation.
Maria Klimikova
2010-01-01
Understanding the reasons of the present financial problems lies In understanding the substance of fractional reserve banking. The substance of fractional banking is in lending more money than the bankers have. Banking of partial reserves is an alternative form which links deposit banking and credit banking. Fractional banking is causing many unfavorable economic impacts in the worldwide system, specifically an inflation.
-Dimensional Fractional Lagrange's Inversion Theorem
Directory of Open Access Journals (Sweden)
F. A. Abd El-Salam
2013-01-01
Full Text Available Using Riemann-Liouville fractional differential operator, a fractional extension of the Lagrange inversion theorem and related formulas are developed. The required basic definitions, lemmas, and theorems in the fractional calculus are presented. A fractional form of Lagrange's expansion for one implicitly defined independent variable is obtained. Then, a fractional version of Lagrange's expansion in more than one unknown function is generalized. For extending the treatment in higher dimensions, some relevant vectors and tensors definitions and notations are presented. A fractional Taylor expansion of a function of -dimensional polyadics is derived. A fractional -dimensional Lagrange inversion theorem is proved.
Al-Omair, Ameen; Soliman, Hany; Xu, Wei; Karotki, Aliaksandr; Mainprize, Todd; Phan, Nicolas; Das, Sunit; Keith, Julia; Yeung, Robert; Perry, James; Tsao, May; Sahgal, Arjun
2013-12-01
Our purpose was to report efficacy of hypofractionated cavity stereotactic radiotherapy (HCSRT) in patients with and without prior whole brain radiotherapy (WBRT). 32 surgical cavities in 30 patients (20 patients/21 cavities had no prior WBRT and 10 patients/11 cavities had prior WBRT) were treated with image-guided linac stereotactic radiotherapy. 7 of the 10 prior WBRT patients had "resistant" local disease given prior surgery, post-operative WBRT and a re-operation, followed by salvage HCSRT. The clinical target volume was the post-surgical cavity, and a 2-mm margin applied as planning target volume. The median total dose was 30 Gy (range: 25-37.5 Gy) in 5 fractions. In the no prior and prior WBRT cohorts, the median follow-up was 9.7 months (range: 3.0-23.6) and 15.3 months (range: 2.9-39.7), the median survival was 23.6 months and 39.7 months, and the 1-year cavity local recurrence progression- free survival (LRFS) was 79 and 100%, respectively. At 18 months the LRFS dropped to 29% in the prior WBRT cohort. Grade 3 radiation necrosis occurred in 3 prior WBRT patients. We report favorable outcomes with HCSRT, and well selected patients with prior WBRT and "resistant" disease may have an extended survival favoring aggressive salvage HCSRT at a moderate risk of radiation necrosis.
Han, Chaonan; Zheng, Binghui; Qin, Yanwen; Ma, Yingqun; Yang, Chenchen; Liu, Zhichao; Cao, Wei; Chi, Minghui
2018-01-01
The impoundment of the Three Gorges Reservoir (TGR) has changed water-sand transport regime, with inevitable effects on phosphorus transport behavior in the TGR. In this study, we measured phosphorus fractions in water and suspended particles transported from upstream rivers of the TGR (the Yangtze River, the Jialing River and the Wu River) to reservoir inner region over the full operation schedule of the TGR. The aim was to determine how phosphorus fractions in water and particulate phases varied in response to natural hydrological processes and reservoir operations. The results showed that total phosphorus concentration (TP) in water in the TGR inner region was 0.17±0.05mg/L, which was lower than that in the Yangtze River (0.21±0.04mg/L) and the Wu River (0.23±0.03mg/L), but higher than that in the Jialing River (0.12±0.07mg/L). In the TGR inner region, there was no clear trend of total dissolved phosphorus (TDP), but total particulate phosphorus (TPP) showed a decreasing trend from tail area to head area because of particle deposition along the TGR mainstream. In addition, the concentrations of TPP in water and particulate phosphorus in a unit mass of suspended particles (PP) in the TGR inner region were higher in October 2014 and January 2015 (the impounding period and high water level period) than that in July 2015 (the low water level period). The temporal variations of PP and TPP concentrations in the TGR may be linked to the change of particle size distribution of suspended particles in the TGR. The particle size tended to be finer due to large-size particle deposition under stable hydrodynamic conditions in the process of TGR impoundment, resulting in high adsorption capacities of phosphorus in suspended particles. The results implied that phosphorus temporal variations in the TGR could exert different impacts on water quality in the TGR tributaries. Copyright © 2017 Elsevier B.V. All rights reserved.
Gauge invariant fractional electromagnetic fields
International Nuclear Information System (INIS)
Lazo, Matheus Jatkoske
2011-01-01
Fractional derivatives and integrations of non-integers orders was introduced more than three centuries ago but only recently gained more attention due to its application on nonlocal phenomenas. In this context, several formulations of fractional electromagnetic fields was proposed, but all these theories suffer from the absence of an effective fractional vector calculus, and in general are non-causal or spatially asymmetric. In order to deal with these difficulties, we propose a spatially symmetric and causal gauge invariant fractional electromagnetic field from a Lagrangian formulation. From our fractional Maxwell's fields arose a definition for the fractional gradient, divergent and curl operators. -- Highlights: → We propose a fractional Lagrangian formulation for fractional Maxwell's fields. → We obtain gauge invariant fractional electromagnetic fields. → Our generalized fractional Maxwell's field is spatially symmetrical. → We discuss the non-causality of the theory.
Gauge invariant fractional electromagnetic fields
Energy Technology Data Exchange (ETDEWEB)
Lazo, Matheus Jatkoske, E-mail: matheuslazo@furg.br [Instituto de Matematica, Estatistica e Fisica - FURG, Rio Grande, RS (Brazil)
2011-09-26
Fractional derivatives and integrations of non-integers orders was introduced more than three centuries ago but only recently gained more attention due to its application on nonlocal phenomenas. In this context, several formulations of fractional electromagnetic fields was proposed, but all these theories suffer from the absence of an effective fractional vector calculus, and in general are non-causal or spatially asymmetric. In order to deal with these difficulties, we propose a spatially symmetric and causal gauge invariant fractional electromagnetic field from a Lagrangian formulation. From our fractional Maxwell's fields arose a definition for the fractional gradient, divergent and curl operators. -- Highlights: → We propose a fractional Lagrangian formulation for fractional Maxwell's fields. → We obtain gauge invariant fractional electromagnetic fields. → Our generalized fractional Maxwell's field is spatially symmetrical. → We discuss the non-causality of the theory.
Srivastava, H. M.; Saxena, R. K.; Parmar, R. K.
2018-01-01
Our present investigation is inspired by the recent interesting extensions (by Srivastava et al. [35]) of a pair of the Mellin-Barnes type contour integral representations of their incomplete generalized hypergeometric functions p γ q and p Γ q by means of the incomplete gamma functions γ( s, x) and Γ( s, x). Here, in this sequel, we introduce a family of the relatively more general incomplete H-functions γ p,q m,n ( z) and Γ p,q m,n ( z) as well as their such special cases as the incomplete Fox-Wright generalized hypergeometric functions p Ψ q (γ) [ z] and p Ψ q (Γ) [ z]. The main object of this paper is to study and investigate several interesting properties of these incomplete H-functions, including (for example) decomposition and reduction formulas, derivative formulas, various integral transforms, computational representations, and so on. We apply some substantially general Riemann-Liouville and Weyl type fractional integral operators to each of these incomplete H-functions. We indicate the easilyderivable extensions of the results presented here that hold for the corresponding incomplete \\overline H -functions as well. Potential applications of many of these incomplete special functions involving (for example) probability theory are also indicated.
Discrete fractional solutions of a Legendre equation
Yılmazer, Resat
2018-01-01
One of the most popular research interests of science and engineering is the fractional calculus theory in recent times. Discrete fractional calculus has also an important position in fractional calculus. In this work, we acquire new discrete fractional solutions of the homogeneous and non homogeneous Legendre differential equation by using discrete fractional nabla operator.
Gauge invariant fractional electromagnetic fields
Lazo, Matheus Jatkoske
2011-09-01
Fractional derivatives and integrations of non-integers orders was introduced more than three centuries ago but only recently gained more attention due to its application on nonlocal phenomenas. In this context, several formulations of fractional electromagnetic fields was proposed, but all these theories suffer from the absence of an effective fractional vector calculus, and in general are non-causal or spatially asymmetric. In order to deal with these difficulties, we propose a spatially symmetric and causal gauge invariant fractional electromagnetic field from a Lagrangian formulation. From our fractional Maxwell's fields arose a definition for the fractional gradient, divergent and curl operators.
International Nuclear Information System (INIS)
Saminadayar, L.
2001-01-01
20 years ago fractional charges were imagined to explain values of conductivity in some materials. Recent experiments have proved the existence of charges whose value is the third of the electron charge. This article presents the experimental facts that have led theorists to predict the existence of fractional charges from the motion of quasi-particles in a linear chain of poly-acetylene to the quantum Hall effect. According to the latest theories, fractional charges are neither bosons nor fermions but anyons, they are submitted to an exclusive principle that is less stringent than that for fermions. (A.C.)
Punyaratabandhu, Preawphan; Wanitphakdeedecha, Rungsima; Pattanaprichakul, Penvadee; Sitthinamsuwan, Panitta; Phothong, Weeranut; Eimpunth, Sasima; Lohsiriwat, Visnu; Manuskiatti, Woraphong
2017-02-01
Topical anaesthetic cream (TAC) is commonly used as a pre-treatment of ablative fractional resurfacing (AFR) laser. Most of anaesthetic cream contains distilled water as major component. Therefore, pre-operative TAC may interfere the photothermal reaction in the skin treated with fractional carbon-dioxide (FCO 2 ) laser and fractional erbium-doped yttrium aluminium garnet (FEr:YAG) laser. The objective of the study was to compare the ablative width (AW) and coagulative depth (CD) of AFR laser with and without pre-treatment with TAC. Four Thai females who underwent abdominoplasty were included in the study. The excised skin of each subject was divided into four areas. TAC (eutectic mixture of local anaesthesia; EMLA) with 1-h occlusion was applied only on the first and second areas. The first and third areas were treated with FCO 2 at 15 mj and 5% density. The second and fourth areas were treated with FEr:YAG at 28 J/cm 2 and 5% density. Six biopsied specimens were obtained from each area. A total of 96 specimens (24 specimens from each area) were collected from four patients and examined randomly by two dermatopathologists. The ablative width and coagulative depth from each specimen were determined. In FCO 2 -treated specimens, the mean AW of the specimens that were pre-treated with TAC and control was 174.86 ± 24.57 and 188.52 ± 41.32 μm. The mean CD of the specimens that were pre-treated with TAC and control was 594.96 ± 111.72 and 520.03 ± 147.40 μm. There were no significant differences in AW and CD between both groups (p = 0.53 and p = 0.15). In FEr:YAG-treated specimens, the mean AW of the specimens that were pre-treated with TAC and control was 381.11 ± 48.02 and 423.65 ± 60.16 μm. The mean CD of the specimens that were pre-treated with TAC and control was 86.03 ± 29.44 and 71.59 ± 18.99 μm. There were no significant differences in AW and CD between both groups (p = 0.16 and p = 0.24). The pre
International Nuclear Information System (INIS)
Jackiw, R.; Massachusetts Inst. of Tech., Cambridge; Massachusetts Inst. of Tech., Cambridge
1984-01-01
The theory of fermion fractionization due to topologically generated fermion ground states is presented. Applications to one-dimensional conductors, to the MIT bag, and to the Hall effect are reviewed. (author)
Goodrich, Christopher
2015-01-01
This text provides the first comprehensive treatment of the discrete fractional calculus. Experienced researchers will find the text useful as a reference for discrete fractional calculus and topics of current interest. Students who are interested in learning about discrete fractional calculus will find this text to provide a useful starting point. Several exercises are offered at the end of each chapter and select answers have been provided at the end of the book. The presentation of the content is designed to give ample flexibility for potential use in a myriad of courses and for independent study. The novel approach taken by the authors includes a simultaneous treatment of the fractional- and integer-order difference calculus (on a variety of time scales, including both the usual forward and backwards difference operators). The reader will acquire a solid foundation in the classical topics of the discrete calculus while being introduced to exciting recent developments, bringing them to the frontiers of the...
Bhattacharyya, Sonalee; Namakshi, Nama; Zunker, Christina; Warshauer, Hiroko K.; Warshauer, Max
2016-01-01
Making math more engaging for students is a challenge that every teacher faces on a daily basis. These authors write that they are constantly searching for rich problem-solving tasks that cover the necessary content, develop critical-thinking skills, and engage student interest. The Mystery Fraction activity provided here focuses on a key number…
Fractional Differential Equation
Directory of Open Access Journals (Sweden)
Moustafa El-Shahed
2007-01-01
where 2<α<3 is a real number and D0+α is the standard Riemann-Liouville fractional derivative. Our analysis relies on Krasnoselskiis fixed point theorem of cone preserving operators. An example is also given to illustrate the main results.
Owolabi, Kolade M.
2018-03-01
In this work, we are concerned with the solution of non-integer space-fractional reaction-diffusion equations with the Riemann-Liouville space-fractional derivative in high dimensions. We approximate the Riemann-Liouville derivative with the Fourier transform method and advance the resulting system in time with any time-stepping solver. In the numerical experiments, we expect the travelling wave to arise from the given initial condition on the computational domain (-∞, ∞), which we terminate in the numerical experiments with a large but truncated value of L. It is necessary to choose L large enough to allow the waves to have enough space to distribute. Experimental results in high dimensions on the space-fractional reaction-diffusion models with applications to biological models (Fisher and Allen-Cahn equations) are considered. Simulation results reveal that fractional reaction-diffusion equations can give rise to a range of physical phenomena when compared to non-integer-order cases. As a result, most meaningful and practical situations are found to be modelled with the concept of fractional calculus.
Fraction Reduction through Continued Fractions
Carley, Holly
2011-01-01
This article presents a method of reducing fractions without factoring. The ideas presented may be useful as a project for motivated students in an undergraduate number theory course. The discussion is related to the Euclidean Algorithm and its variations may lead to projects or early examples involving efficiency of an algorithm.
Fractional dynamic calculus and fractional dynamic equations on time scales
Georgiev, Svetlin G
2018-01-01
Pedagogically organized, this monograph introduces fractional calculus and fractional dynamic equations on time scales in relation to mathematical physics applications and problems. Beginning with the definitions of forward and backward jump operators, the book builds from Stefan Hilger’s basic theories on time scales and examines recent developments within the field of fractional calculus and fractional equations. Useful tools are provided for solving differential and integral equations as well as various problems involving special functions of mathematical physics and their extensions and generalizations in one and more variables. Much discussion is devoted to Riemann-Liouville fractional dynamic equations and Caputo fractional dynamic equations. Intended for use in the field and designed for students without an extensive mathematical background, this book is suitable for graduate courses and researchers looking for an introduction to fractional dynamic calculus and equations on time scales. .
Toward lattice fractional vector calculus
Tarasov, Vasily E.
2014-09-01
An analog of fractional vector calculus for physical lattice models is suggested. We use an approach based on the models of three-dimensional lattices with long-range inter-particle interactions. The lattice analogs of fractional partial derivatives are represented by kernels of lattice long-range interactions, where the Fourier series transformations of these kernels have a power-law form with respect to wave vector components. In the continuum limit, these lattice partial derivatives give derivatives of non-integer order with respect to coordinates. In the three-dimensional description of the non-local continuum, the fractional differential operators have the form of fractional partial derivatives of the Riesz type. As examples of the applications of the suggested lattice fractional vector calculus, we give lattice models with long-range interactions for the fractional Maxwell equations of non-local continuous media and for the fractional generalization of the Mindlin and Aifantis continuum models of gradient elasticity.
Fermion fractionization and index theorem
International Nuclear Information System (INIS)
Hirayama, Minoru; Torii, Tatsuo
1982-01-01
The relation between the fermion fractionization and the Callias-Bott-Seeley index theorem for the Dirac operator in the open space of odd dimension is clarified. Only the case of one spatial dimension is discussed in detail. Sum rules for the expectation values of various quantities in fermion-fractionized configurations are derived. (author)
Fractional supersymmetry through generalized anyonic algebra
International Nuclear Information System (INIS)
Douari, Jamila; Abdus Salam International Centre for Theoretical Physics, Trieste; Hassouni, Yassine
2001-01-01
The construction of anyonic operators and algebra is generalized by using quons operators. Therefore, the particular version of fractional supersymmetry is constructed on the two-dimensional lattice by associating two generalized anyons of different kinds. The fractional supersymmetry Hamiltonian operator is obtained on the two-dimensional lattice and the quantum algebra U q (sl 2 ) is realized. (author)
International Nuclear Information System (INIS)
Shpakauskas, V.V.; Kychkin, I.S.; Rudzikas, Z.B.
1976-01-01
Certain symmetry properties of standard quantities of the atomic shell theory for LS coupling are studied, namely, the commutation of quantum numbers of spin and quasispin in genealogical coefficients and in submatrix elements of irreducible tensor operators. The method of second quantization and quasispin has been used for obtaining new relations between genealogical coefficients. The similar relations have been also found for the submatrix elements of the irreducible tensor operators, as well as for genealogical coefficients with two and more split-off electrons. For the first time in special cases for the quantities under study the explicit algebraic expressions are obtained
Fractional vector calculus for fractional advection dispersion
Meerschaert, Mark M.; Mortensen, Jeff; Wheatcraft, Stephen W.
2006-07-01
We develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient, divergence, and curl, and a fractional divergence theorem and Stokes theorem. These basic tools are then applied to provide a physical explanation for the fractional advection-dispersion equation for flow in heterogeneous porous media.
Pagnanelli, Francesca; Moscardini, Emanuela; Altimari, Pietro; Abo Atia, Thomas; Toro, Luigi
2017-02-01
Experimental results of leaching tests using waste fractions obtained by mechanical pretreatment of lithium ion batteries (LIB) were reported. Two physical pretreatments were performed at pilot scale in order to recover electrodic powders: the first including crushing, milling, and sieving and the second granulation, and sieving. Recovery yield of electrodic powder was significantly influenced by the type of pretreatment. About 50% of initial LIB wastes was recovered by the first treatment (as electrodic powder with size extraction. Solid/liquid ratios and sulfuric acid concentrations were changed according to factorial designs at constant temperature (80°C). Optimized conditions for quantitative extraction (>99%) of Co and Li from Sample 1 are 1/10g/mL as solid/liquid ratio and +50% stoichiometric excess of acid (1.1M). Using the same solid/liquid ratio, +100% acid excess (1.2M) is necessary to extract 96% of Co and 86% of Li from Sample 2. Best conditions for leaching of Sample 2 using glucose are +200% acid excess (1.7M) and 0.05M glucose concentration. Optimized conditions found in this work are among the most effective reported in the literature in term of Co extraction and reagent consumption. Copyright © 2016 Elsevier Ltd. All rights reserved.
Bergstra, Jan A.
2015-01-01
In the context of an involutive meadow a precise definition of fractions is formulated and on that basis formal definitions of various classes of fractions are given. The definitions follow the fractions as terms paradigm. That paradigm is compared with two competing paradigms for storytelling on fractions: fractions as values and fractions as pairs.
Yang, W; Liu, Y; Zhang, Y; Zhao, Q-H; He, S-F
2016-08-01
Surgical site infection (SSI) causes significant mortality and morbidity. Administration of a high inspired oxygen fraction (FiO2) to patients undergoing surgery may represent a potential preventive strategy. To conduct a meta-analysis of randomized controlled trials in which high FiO2 was compared with normal FiO2 in patients undergoing surgery to estimate the effect on the development of SSI. A comprehensive search was undertaken for randomized controlled trials (until December 2015) that compared high FiO2 with normal FiO2 in adults undergoing surgery with general anaesthesia and reported on SSI. This study included 17 randomized controlled trials with 8093 patients. Infection rates were 13.11% in the control group and 11.53% in the hyperoxic group, while the overall risk ratio was 0.893 [95% confidence interval (CI) 0.794-1.003; P = 0.057]. Subgroup analyses stratified by country, definition of SSI, and type of surgery were also performed, and showed similar results. However, high FiO2 was found to be of significant benefit in patients undergoing colorectal surgery, with a risk ratio of 0.735 (95% CI 0.573-0.944; P=0.016). There is moderate evidence to suggest that administration of high FiO2 to patients undergoing surgery, especially colorectal surgery, reduces the risk of SSI. Further studies with better adherence to the intervention may affect the results of this meta-analysis. Copyright © 2016 The Healthcare Infection Society. Published by Elsevier Ltd. All rights reserved.
On Generalized Fractional Differentiator Signals
Directory of Open Access Journals (Sweden)
Hamid A. Jalab
2013-01-01
Full Text Available By employing the generalized fractional differential operator, we introduce a system of fractional order derivative for a uniformly sampled polynomial signal. The calculation of the bring in signal depends on the additive combination of the weighted bring-in of N cascaded digital differentiators. The weights are imposed in a closed formula containing the Stirling numbers of the first kind. The approach taken in this work is to consider that signal function in terms of Newton series. The convergence of the system to a fractional time differentiator is discussed.
International Nuclear Information System (INIS)
Kaminska, G.; Kawczynska, M.
1984-01-01
The ''afterloading'' technique with use of Curietron (41 cases) or Selectron (43 cases) apparatus was applied in 83 cervix uteri carcinoma patients in phase 1b (79 cases) and 2a (4 cases). The Wertheim-Magis operation was performed after several weeks. Post-operative histologic investigation showed complete destruction of the tumour in 72% cases; persistent cancer cells in cervix uteri were stated in 11 patients (13%), while lymphatic node metastasis was seen in 14 patients (17%). Supplementary teleradiotherapy was performed in those cases. 39 patients (95%) of the 41 observed during 3 years survived without any cancer symptoms. No explicit relationship between dose administered to vaginal disc and cancer persistence frequency in the cervix uteri was stated. However, such relationship was stated for the dose in point A. In the group of 19 patients who received point A dose of the least 6000 rads, with average TDF value of 135, presence of cancer cells in cervix uteri was not stated in any of the cases. In 7 patients (17%) observed for over 3 years light and medium heavy postradiation complications were stated; in that 5 rectum side complications were stated in patients additionally irradiated from external fields, where the dose absorbed by rectum wall exceeded 8000 rads, with TDF - 150. No complications were observed when the dose was less than 5000 rads and TDF below 100. 16 refs., 1 fig. (author)
Multivariate fractional Poisson processes and compound sums
Beghin, Luisa; Macci, Claudio
2015-01-01
In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (non-fractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes.
Fractional Vector Calculus and Fractional Special Function
Li, Ming-Fan; Ren, Ji-Rong; Zhu, Tao
2010-01-01
Fractional vector calculus is discussed in the spherical coordinate framework. A variation of the Legendre equation and fractional Bessel equation are solved by series expansion and numerically. Finally, we generalize the hypergeometric functions.
The Whole Story: Understanding Fraction Computation
Dixon, Juli K.; Tobias, Jennifer M.
2013-01-01
What does it look like to "understand" operations with fractions? The Common Core State Standards for Mathematics (CCSSM) uses the term "understand" when describing expectations for students' knowledge related to each of the fraction operations within grades 4 through 6 (CCSSI 2010). Furthermore, CCSSM elaborates that…
Toward lattice fractional vector calculus
International Nuclear Information System (INIS)
Tarasov, Vasily E
2014-01-01
An analog of fractional vector calculus for physical lattice models is suggested. We use an approach based on the models of three-dimensional lattices with long-range inter-particle interactions. The lattice analogs of fractional partial derivatives are represented by kernels of lattice long-range interactions, where the Fourier series transformations of these kernels have a power-law form with respect to wave vector components. In the continuum limit, these lattice partial derivatives give derivatives of non-integer order with respect to coordinates. In the three-dimensional description of the non-local continuum, the fractional differential operators have the form of fractional partial derivatives of the Riesz type. As examples of the applications of the suggested lattice fractional vector calculus, we give lattice models with long-range interactions for the fractional Maxwell equations of non-local continuous media and for the fractional generalization of the Mindlin and Aifantis continuum models of gradient elasticity. (papers)
Laskin, Nick
2018-01-01
Fractional quantum mechanics is a recently emerged and rapidly developing field of quantum physics. This is the first monograph on fundamentals and physical applications of fractional quantum mechanics, written by its founder. The fractional Schrödinger equation and the fractional path integral are new fundamental physical concepts introduced and elaborated in the book. The fractional Schrödinger equation is a manifestation of fractional quantum mechanics. The fractional path integral is a new mathematical tool based on integration over Lévy flights. The fractional path integral method enhances the well-known Feynman path integral framework. Related topics covered in the text include time fractional quantum mechanics, fractional statistical mechanics, fractional classical mechanics and the α-stable Lévy random process. The book is well-suited for theorists, pure and applied mathematicians, solid-state physicists, chemists, and others working with the Schrödinger equation, the path integral technique...
Error analysis of pupils in calculating with fractions
Uranič, Petra
2016-01-01
In this thesis I examine the correlation between the frequency of errors that seventh grade pupils make in their calculations with fractions and their level of understanding of fractions. Fractions are a relevant and demanding theme in the mathematics curriculum. Although we use fractions on a daily basis, pupils find learning fractions to be very difficult. They generally do not struggle with the concept of fractions itself, but they frequently have problems with mathematical operations ...
Fractional statistics and fractional quantized Hall effect
International Nuclear Information System (INIS)
Tao, R.; Wu, Y.S.
1985-01-01
The authors suggest that the origin of the odd-denominator rule observed in the fractional quantized Hall effect (FQHE) may lie in fractional statistics which govern quasiparticles in FQHE. A theorem concerning statistics of clusters of quasiparticles implies that fractional statistics do not allow coexistence of a large number of quasiparticles at fillings with an even denominator. Thus, no Hall plateau can be formed at these fillings, regardless of the presence of an energy gap. 15 references
A fast fractional difference algorithm
DEFF Research Database (Denmark)
Jensen, Andreas Noack; Nielsen, Morten Ørregaard
2014-01-01
We provide a fast algorithm for calculating the fractional difference of a time series. In standard implementations, the calculation speed (number of arithmetic operations) is of order T 2, where T is the length of the time series. Our algorithm allows calculation speed of order T log...
A Fast Fractional Difference Algorithm
DEFF Research Database (Denmark)
Jensen, Andreas Noack; Nielsen, Morten Ørregaard
We provide a fast algorithm for calculating the fractional difference of a time series. In standard implementations, the calculation speed (number of arithmetic operations) is of order T 2, where T is the length of the time series. Our algorithm allows calculation speed of order T log...
Initialized Fractional Calculus
Lorenzo, Carl F.; Hartley, Tom T.
2000-01-01
This paper demonstrates the need for a nonconstant initialization for the fractional calculus and establishes a basic definition set for the initialized fractional differintegral. This definition set allows the formalization of an initialized fractional calculus. Two basis calculi are considered; the Riemann-Liouville and the Grunwald fractional calculi. Two forms of initialization, terminal and side are developed.
Energy Technology Data Exchange (ETDEWEB)
Sabzikar, Farzad, E-mail: sabzika2@stt.msu.edu [Department of Statistics and Probability, Michigan State University, East Lansing, MI 48823 (United States); Meerschaert, Mark M., E-mail: mcubed@stt.msu.edu [Department of Statistics and Probability, Michigan State University, East Lansing, MI 48823 (United States); Chen, Jinghua, E-mail: cjhdzdz@163.com [School of Sciences, Jimei University, Xiamen, Fujian, 361021 (China)
2015-07-15
Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series.
Sabzikar, Farzad; Meerschaert, Mark M.; Chen, Jinghua
2015-07-01
Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series.
International Nuclear Information System (INIS)
Sabzikar, Farzad; Meerschaert, Mark M.; Chen, Jinghua
2015-01-01
Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series
Higher fractions theory of fractional hall effect
International Nuclear Information System (INIS)
Kostadinov, I.Z.; Popov, V.N.
1985-07-01
A theory of fractional quantum Hall effect is generalized to higher fractions. N-particle model interaction is used and the gap is expressed through n-particles wave function. The excitation spectrum in general and the mean field critical behaviour are determined. The Hall conductivity is calculated from first principles. (author)
Paper Plate Fractions: The Counting Connection
McCoy, Ann; Barnett, Joann; Stine, Tammy
2016-01-01
Without a doubt, fractions prove to be a stumbling block for many children. Researchers have suggested a variety of explanations for why this is the case. The introduction of symbolization and operations before the development of conceptual understanding of fractions, a lack of understanding of the role of the numerator and denominator, and an…
Estimation's Role in Calculations with Fractions
Johanning, Debra I.
2011-01-01
Estimation is more than a skill or an isolated topic. It is a thinking tool that needs to be emphasized during instruction so that students will learn to develop algorithmic procedures and meaning for fraction operations. For students to realize when fractions should be added, subtracted, multiplied, or divided, they need to develop a sense of…
Asphalt chemical fractionation
International Nuclear Information System (INIS)
Obando P, Klever N.
1998-01-01
Asphalt fractionation were carried out in the Esmeraldas Oil Refinery using n-pentane, SiO 2 and different mixture of benzene- methane. The fractions obtained were analyzed by Fourier's Transformed Infrared Spectrophotometry (FTIR)
Early Fractions Learning of 3rd Grade Students in SD Laboratorium Unesa
Sari, Elisabet Ayunika Permata; Juniati, Dwi; Patahudin, Sitti Maesuri
2012-01-01
Fractions varied meanings is one of the causes of difficulties in learning fractions. These students should be given greater opportunities to explore the meaning of fractions before they learn the relationship between fractions and operations on fractions. Although students shading an area represents a fraction, it does not mean they really…
Smarandache Continued Fractions
Ibstedt, H.
2001-01-01
The theory of general continued fractions is developed to the extent required in order to calculate Smarandache continued fractions to a given number of decimal places. Proof is given for the fact that Smarandache general continued fractions built with positive integer Smarandache sequences baving only a finite number of terms equal to 1 is convergent. A few numerical results are given.
Shamim, Atif
2011-03-01
For the first time, a generalized Smith chart is introduced here to represent fractional order circuit elements. It is shown that the standard Smith chart is a special case of the generalized fractional order Smith chart. With illustrations drawn for both the conventional integer based lumped elements and the fractional elements, a graphical technique supported by the analytical method is presented to plot impedances on the fractional Smith chart. The concept is then applied towards impedance matching networks, where the fractional approach proves to be much more versatile and results in a single element matching network for a complex load as compared to the two elements in the conventional approach. © 2010 IEEE.
Directory of Open Access Journals (Sweden)
Hamid A. Jalab
2014-01-01
Full Text Available The interest in using fractional mask operators based on fractional calculus operators has grown for image denoising. Denoising is one of the most fundamental image restoration problems in computer vision and image processing. This paper proposes an image denoising algorithm based on convex solution of fractional heat equation with regularized fractional power parameters. The performances of the proposed algorithms were evaluated by computing the PSNR, using different types of images. Experiments according to visual perception and the peak signal to noise ratio values show that the improvements in the denoising process are competent with the standard Gaussian filter and Wiener filter.
Fractional equivalent Lagrangian densities for a fractional higher-order equation
International Nuclear Information System (INIS)
Fujioka, J
2014-01-01
In this communication we show that the equivalent Lagrangian densities (ELDs) of a fractional higher-order nonlinear Schrödinger equation with stable soliton-like solutions can be related in a hitherto unknown way. This new relationship is described in terms of a new fractional operator that includes both left- and right-sided fractional derivatives. Using this operator it is possible to generate new ELDs that contain different fractional parts, in addition to the already known ELDs, which only differ by a sum of first-order partial derivatives of two arbitrary functions. (fast track communications)
Early Fractions Learning of 3rd Grade Students in SD Laboratorium Unesa
Directory of Open Access Journals (Sweden)
Elisabet Ayunika Permata Sari
2012-01-01
Full Text Available Fractions varied meanings is one of the causes of difficulties in learning fractions. These students should be given greater opportunities to explore the meaning of fractions before they learn the relationship between fractions and operations on fractions. Although students can shading area represents a fraction, does not mean they really understand the meaning of fractions as a whole. With a realistic approach to mathematics, students are given the contextual issues of equitable distribution and measurements that involve fractions
Dey, Aloke
2009-01-01
A one-stop reference to fractional factorials and related orthogonal arrays.Presenting one of the most dynamic areas of statistical research, this book offers a systematic, rigorous, and up-to-date treatment of fractional factorial designs and related combinatorial mathematics. Leading statisticians Aloke Dey and Rahul Mukerjee consolidate vast amounts of material from the professional literature--expertly weaving fractional replication, orthogonal arrays, and optimality aspects. They develop the basic theory of fractional factorials using the calculus of factorial arrangements, thereby providing a unified approach to the study of fractional factorial plans. An indispensable guide for statisticians in research and industry as well as for graduate students, Fractional Factorial Plans features: * Construction procedures of symmetric and asymmetric orthogonal arrays. * Many up-to-date research results on nonexistence. * A chapter on optimal fractional factorials not based on orthogonal arrays. * Trend-free plans...
Fractional Dynamics and Control
Machado, José; Luo, Albert
2012-01-01
Fractional Dynamics and Control provides a comprehensive overview of recent advances in the areas of nonlinear dynamics, vibration and control with analytical, numerical, and experimental results. This book provides an overview of recent discoveries in fractional control, delves into fractional variational principles and differential equations, and applies advanced techniques in fractional calculus to solving complicated mathematical and physical problems.Finally, this book also discusses the role that fractional order modeling can play in complex systems for engineering and science. Discusses how fractional dynamics and control can be used to solve nonlinear science and complexity issues Shows how fractional differential equations and models can be used to solve turbulence and wave equations in mechanics and gravity theories and Schrodinger’s equation Presents factional relaxation modeling of dielectric materials and wave equations for dielectrics Develops new methods for control and synchronization of...
Dividing Fractions: A Pedagogical Technique
Lewis, Robert
2016-01-01
When dividing one fraction by a second fraction, invert, that is, flip the second fraction, then multiply it by the first fraction. To multiply fractions, simply multiply across the denominators, and multiply across the numerators to get the resultant fraction. So by inverting the division of fractions it is turned into an easy multiplication of…
Fractional distillation of oil
Energy Technology Data Exchange (ETDEWEB)
Jones, L D
1931-10-31
A method of dividing oil into lubricating oil fractions without substantial cracking by introducing the oil in a heated state into a fractionating column from which oil fractions having different boiling points are withdrawn at different levels, while reflux liquid is supplied to the top of the column, and additional heat is introduced into the column by contacting with the oil therein a heated fluid of higher monlecular weight than water and less susceptible to thermal decomposition than is the highest boiling oil fraction resulting from the distillation, or of which any products produced by thermal decomposition will not occur in the highest boiling distillate withdrawn from the column.
Fractional Poisson process (II)
International Nuclear Information System (INIS)
Wang Xiaotian; Wen Zhixiong; Zhang Shiying
2006-01-01
In this paper, we propose a stochastic process W H (t)(H-bar (12,1)) which we call fractional Poisson process. The process W H (t) is self-similar in wide sense, displays long range dependence, and has more fatter tail than Gaussian process. In addition, it converges to fractional Brownian motion in distribution
Wilkerson, Trena L.; Bryan, Tommy; Curry, Jane
2012-01-01
This article describes how using candy bars as models gives sixth-grade students a taste for learning to represent fractions whose denominators are factors of twelve. Using paper models of the candy bars, students explored and compared fractions. They noticed fewer different representations for one-third than for one-half. The authors conclude…
Can Kindergartners Do Fractions?
Cwikla, Julie
2014-01-01
Mathematics professor Julie Cwikla decided that she needed to investigate young children's understandings and see what precurricular partitioning notions young minds bring to the fraction table. Cwikla realized that only a handful of studies have examined how preschool-age and early elementary school-age students solve fraction problems (Empson…
Diaz, Victor Alfonzo; Giusti, Andrea
2018-03-01
The aim of this paper is to present a simple generalization of bosonic string theory in the framework of the theory of fractional variational problems. Specifically, we present a fractional extension of the Polyakov action, for which we compute the general form of the equations of motion and discuss the connection between the new fractional action and a generalization the Nambu-Goto action. Consequently, we analyze the symmetries of the modified Polyakov action and try to fix the gauge, following the classical procedures. Then we solve the equations of motion in a simplified setting. Finally, we present a Hamiltonian description of the classical fractional bosonic string and introduce the fractional light-cone gauge. It is important to remark that, throughout the whole paper, we thoroughly discuss how to recover the known results as an "integer" limit of the presented model.
Early Fractions Learning of 3rd Grade Students in SD Laboratorium Unesa
Elisabet Ayunika Permata Sari; Dwi Juniati; Sitti Maesuri Patahudin
2012-01-01
Fractions varied meanings is one of the causes of difficulties in learning fractions. These students should be given greater opportunities to explore the meaning of fractions before they learn the relationship between fractions and operations on fractions. Although students can shading area represents a fraction, does not mean they really understand the meaning of fractions as a whole. With a realistic approach to mathematics, students are given the contextual issues of equitable distributio...
A fractional spline collocation-Galerkin method for the time-fractional diffusion equation
Directory of Open Access Journals (Sweden)
Pezza L.
2018-03-01
Full Text Available The aim of this paper is to numerically solve a diffusion differential problem having time derivative of fractional order. To this end we propose a collocation-Galerkin method that uses the fractional splines as approximating functions. The main advantage is in that the derivatives of integer and fractional order of the fractional splines can be expressed in a closed form that involves just the generalized finite difference operator. This allows us to construct an accurate and efficient numerical method. Several numerical tests showing the effectiveness of the proposed method are presented.
Fractional Nottale's Scale Relativity and emergence of complexified gravity
International Nuclear Information System (INIS)
EL-Nabulsi, Ahmad Rami
2009-01-01
Fractional calculus of variations has recently gained significance in studying weak dissipative and nonconservative dynamical systems ranging from classical mechanics to quantum field theories. In this paper, fractional Nottale's Scale Relativity (NSR) for an arbitrary fractal dimension is introduced within the framework of fractional action-like variational approach recently introduced by the author. The formalism is based on fractional differential operators that generalize the differential operators of conventional NSR but that reduces to the standard formalism in the integer limit. Our main aim is to build the fractional setting for the NSR dynamical equations. Many interesting consequences arise, in particular the emergence of complexified gravity and complex time.
Fractional Order Generalized Information
Directory of Open Access Journals (Sweden)
José Tenreiro Machado
2014-04-01
Full Text Available This paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow an high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The concepts are also extended to relative distances and tested with several sets of data, confirming the goodness of the generalization.
Fractional finite Fourier transform.
Khare, Kedar; George, Nicholas
2004-07-01
We show that a fractional version of the finite Fourier transform may be defined by using prolate spheroidal wave functions of order zero. The transform is linear and additive in its index and asymptotically goes over to Namias's definition of the fractional Fourier transform. As a special case of this definition, it is shown that the finite Fourier transform may be inverted by using information over a finite range of frequencies in Fourier space, the inversion being sensitive to noise. Numerical illustrations for both forward (fractional) and inverse finite transforms are provided.
Social Trust and Fractionalization:
DEFF Research Database (Denmark)
Bjørnskov, Christian
2008-01-01
This paper takes a closer look at the importance of fractionalization for the creation of social trust. It first argues that the determinants of trust can be divided into two categories: those affecting individuals' trust radii and those affecting social polarization. A series of estimates using...... a much larger country sample than in previous literature confirms that fractionalization in the form of income inequality and political diversity adversely affects social trust while ethnic diversity does not. However, these effects differ systematically across countries, questioning standard...... interpretations of the influence of fractionalization on trust....
Fractional calculus in bioengineering, part 3.
Magin, Richard L
2004-01-01
Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub
Almost Periodic Solutions for Impulsive Fractional Stochastic Evolution Equations
Directory of Open Access Journals (Sweden)
Toufik Guendouzi
2014-08-01
Full Text Available In this paper, we consider the existence of square-mean piecewise almost periodic solutions for impulsive fractional stochastic evolution equations involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, fractional calculus, fixed point technique and stochastic analysis theory and methods adopted directly from deterministic fractional equations. Some known results are improved and generalized.
Fractional separation of hydrocarbon vapours
Energy Technology Data Exchange (ETDEWEB)
1937-07-10
A process is described for converting higher boiling hydrocarbons to lower boiling hydrocarbons by subjecting them at elevated temperatures to a conversion operation, then separating the higher and lower boiling fractions. The separation takes place while the reaction products are maintained in the vapor phase by contact with a mass of solid porous material which has little or no catalytic activity but does have a preferential absorption property for higher boiling hydrocarbons so that the lower boiling part of the reaction products pass through the separation zone while the heavier hydrocarbons are retained. The separation is accomplished without substantial loss of heat of these reaction products.
Fractional Stochastic Field Theory
Honkonen, Juha
2018-02-01
Models describing evolution of physical, chemical, biological, social and financial processes are often formulated as differential equations with the understanding that they are large-scale equations for averages of quantities describing intrinsically random processes. Explicit account of randomness may lead to significant changes in the asymptotic behaviour (anomalous scaling) in such models especially in low spatial dimensions, which in many cases may be captured with the use of the renormalization group. Anomalous scaling and memory effects may also be introduced with the use of fractional derivatives and fractional noise. Construction of renormalized stochastic field theory with fractional derivatives and fractional noise in the underlying stochastic differential equations and master equations and the interplay between fluctuation-induced and built-in anomalous scaling behaviour is reviewed and discussed.
Shamim, Atif; Radwan, Ahmed Gomaa; Salama, Khaled N.
2011-01-01
matching networks, where the fractional approach proves to be much more versatile and results in a single element matching network for a complex load as compared to the two elements in the conventional approach. © 2010 IEEE.
Intracellular Cadmium Isotope Fractionation
Horner, T. J.; Lee, R. B.; Henderson, G. M.; Rickaby, R. E.
2011-12-01
Recent stable isotope studies into the biological utilization of transition metals (e.g. Cu, Fe, Zn, Cd) suggest several stepwise cellular processes can fractionate isotopes in both culture and nature. However, the determination of fractionation factors is often unsatisfactory, as significant variability can exist - even between different organisms with the same cellular functions. Thus, it has not been possible to adequately understand the source and mechanisms of metal isotopic fractionation. In order to address this problem, we investigated the biological fractionation of Cd isotopes within genetically-modified bacteria (E. coli). There is currently only one known biological use or requirement of Cd, a Cd/Zn carbonic anhydrase (CdCA, from the marine diatom T. weissfloggii), which we introduce into the E. coli genome. We have also developed a cleaning procedure that allows for the treating of bacteria so as to study the isotopic composition of different cellular components. We find that whole cells always exhibit a preference for uptake of the lighter isotopes of Cd. Notably, whole cells appear to have a similar Cd isotopic composition regardless of the expression of CdCA within the E. coli. However, isotopic fractionation can occur within the genetically modified E. coli during Cd use, such that Cd bound in CdCA can display a distinct isotopic composition compared to the cell as a whole. Thus, the externally observed fractionation is independent of the internal uses of Cd, with the largest Cd isotope fractionation occurring during cross-membrane transport. A general implication of these experiments is that trace metal isotopic fractionation most likely reflects metal transport into biological cells (either actively or passively), rather than relating to expression of specific physiological function and genetic expression of different metalloenzymes.
Fractional laser skin resurfacing.
Alexiades-Armenakas, Macrene R; Dover, Jeffrey S; Arndt, Kenneth A
2012-11-01
Laser skin resurfacing (LSR) has evolved over the past 2 decades from traditional ablative to fractional nonablative and fractional ablative resurfacing. Traditional ablative LSR was highly effective in reducing rhytides, photoaging, and acne scarring but was associated with significant side effects and complications. In contrast, nonablative LSR was very safe but failed to deliver consistent clinical improvement. Fractional LSR has achieved the middle ground; it combined the efficacy of traditional LSR with the safety of nonablative modalities. The first fractional laser was a nonablative erbium-doped yttrium aluminum garnet (Er:YAG) laser that produced microscopic columns of thermal injury in the epidermis and upper dermis. Heralding an entirely new concept of laser energy delivery, it delivered the laser beam in microarrays. It resulted in microscopic columns of treated tissue and intervening areas of untreated skin, which yielded rapid reepithelialization. Fractional delivery was quickly applied to ablative wavelengths such as carbon dioxide, Er:YAG, and yttrium scandium gallium garnet (2,790 nm), providing more significant clinical outcomes. Adjustable laser parameters, including power, pitch, dwell time, and spot density, allowed for precise determination of percent surface area, affected penetration depth, and clinical recovery time and efficacy. Fractional LSR has been a significant advance to the laser field, striking the balance between safety and efficacy.
International Nuclear Information System (INIS)
El-Nabulsi, Ahmad Rami
2009-01-01
Multidimensional fractional actionlike variational problem with time-dependent dynamical fractional exponents is constructed. Fractional Euler-Lagrange equations are derived and discussed in some details. The results obtained are used to explore some novel aspects of fractional quantum field theory where many interesting consequences are revealed, in particular the complexification of quantum field theory, in particular Dirac operators and the novel notion of 'mass without mass'.
Existence of a coupled system of fractional differential equations
International Nuclear Information System (INIS)
Ibrahim, Rabha W.; Siri, Zailan
2015-01-01
We manage the existence and uniqueness of a fractional coupled system containing Schrödinger equations. Such a system appears in quantum mechanics. We confirm that the fractional system under consideration admits a global solution in appropriate functional spaces. The solution is shown to be unique. The method is based on analytic technique of the fixed point theory. The fractional differential operator is considered from the virtue of the Riemann-Liouville differential operator
Existence of a coupled system of fractional differential equations
Energy Technology Data Exchange (ETDEWEB)
Ibrahim, Rabha W. [Multimedia unit, Department of Computer System and Technology Faculty of Computer Science & IT, University of Malaya, 50603 Kuala Lumpur (Malaysia); Siri, Zailan [Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur (Malaysia)
2015-10-22
We manage the existence and uniqueness of a fractional coupled system containing Schrödinger equations. Such a system appears in quantum mechanics. We confirm that the fractional system under consideration admits a global solution in appropriate functional spaces. The solution is shown to be unique. The method is based on analytic technique of the fixed point theory. The fractional differential operator is considered from the virtue of the Riemann-Liouville differential operator.
Preservice Teachers’ Conceptual and Procedural Knowledge of Fraction
Lin, Cheng-Yao; Becker, Jerry; Byun, Mi-Ran; Yang, Der-Ching; Huang, Tsai-Wei
2013-01-01
This study examined (a) the differences in preservice teachers’ procedural knowledge in four areas of fraction operations in Taiwan and the United States, (b) the differences in preservice teachers’ conceptual knowledge in four areas of fraction operations in Taiwan and the United States, and (c) correlation in preservice teachers’ conceptual…
Series expansion in fractional calculus and fractional differential equations
Li, Ming-Fan; Ren, Ji-Rong; Zhu, Tao
2009-01-01
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this theorem, in this paper we introduce fractional series expansion method to fractional calculus. We define a kind of fractional Taylor series of an infinitely fractionally-differentiable function. Further, based on our definition we generalize hypergeometric functio...
Fractional quantization and the quantum hall effect
International Nuclear Information System (INIS)
Guerrero, J.; Calixto, M.; Aldaya, V.
1998-01-01
Quantization with constrains is considered in a group-theoretical framework, providing a precise characterization of the set of good operators, i.e., those preserving the constrained Hilbert space, in terms of the representation of the subgroup of constraints. This machinery is applied to the quantization of the torus as symplectic manifold, obtaining that fractional quantum numbers are permitted, provided that we allow for vector valued representations. The good operators turn out to be the Wilson loops and, for certain representations of the subgroup of constraints, the modular transformations. These results are applied to the Fractional Quantum Hall Effect, where interesting implications are derived
2010-07-01
... 40 Protection of Environment 9 2010-07-01 2010-07-01 false Fraction Measured (Fm) and Fraction... Vessels, Transfer Operations, and Wastewater Pt. 63, Subpt. G, Table 34 Table 34 to Subpart G of Part 63—Fraction Measured (Fm) and Fraction Emitted (Fe) For HAP Compounds in Wastewater Streams Chemical name CAS...
Fractional Calculus in Hydrologic Modeling: A Numerical Perspective
Energy Technology Data Exchange (ETDEWEB)
David A. Benson; Mark M. Meerschaert; Jordan Revielle
2012-01-01
Fractional derivatives can be viewed either as a handy extension of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Levy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Levy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus.
Biswas, Karabi; Caponetto, Riccardo; Mendes Lopes, António; Tenreiro Machado, José António
2017-01-01
This book focuses on two specific areas related to fractional order systems – the realization of physical devices characterized by non-integer order impedance, usually called fractional-order elements (FOEs); and the characterization of vegetable tissues via electrical impedance spectroscopy (EIS) – and provides readers with new tools for designing new types of integrated circuits. The majority of the book addresses FOEs. The interest in these topics is related to the need to produce “analogue” electronic devices characterized by non-integer order impedance, and to the characterization of natural phenomena, which are systems with memory or aftereffects and for which the fractional-order calculus tool is the ideal choice for analysis. FOEs represent the building blocks for designing and realizing analogue integrated electronic circuits, which the authors believe hold the potential for a wealth of mass-market applications. The freedom to choose either an integer- or non-integer-order analogue integrator...
Fractional Diffusion in Gaussian Noisy Environment
Directory of Open Access Journals (Sweden)
Guannan Hu
2015-03-01
Full Text Available We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: \\(D_t^{(\\alpha} u(t, x=\\textit{B}u+u\\cdot \\dot W^H\\, where \\(D_t^{(\\alpha}\\ is the Caputo fractional derivative of order \\(\\alpha\\in (0,1\\ with respect to the time variable \\(t\\, \\(\\textit{B}\\ is a second order elliptic operator with respect to the space variable \\(x\\in\\mathbb{R}^d\\ and \\(\\dot W^H\\ a time homogeneous fractional Gaussian noise of Hurst parameter \\(H=(H_1, \\cdots, H_d\\. We obtain conditions satisfied by \\(\\alpha\\ and \\(H\\, so that the square integrable solution \\(u\\ exists uniquely.
Fractional diffusion equations and anomalous diffusion
Evangelista, Luiz Roberto
2018-01-01
Anomalous diffusion has been detected in a wide variety of scenarios, from fractal media, systems with memory, transport processes in porous media, to fluctuations of financial markets, tumour growth, and complex fluids. Providing a contemporary treatment of this process, this book examines the recent literature on anomalous diffusion and covers a rich class of problems in which surface effects are important, offering detailed mathematical tools of usual and fractional calculus for a wide audience of scientists and graduate students in physics, mathematics, chemistry and engineering. Including the basic mathematical tools needed to understand the rules for operating with the fractional derivatives and fractional differential equations, this self-contained text presents the possibility of using fractional diffusion equations with anomalous diffusion phenomena to propose powerful mathematical models for a large variety of fundamental and practical problems in a fast-growing field of research.
Generalized variational formulations for extended exponentially fractional integral
Directory of Open Access Journals (Sweden)
Zuo-Jun Wang
2016-01-01
Full Text Available Recently, the fractional variational principles as well as their applications yield a special attention. For a fractional variational problem based on different types of fractional integral and derivatives operators, corresponding fractional Lagrangian and Hamiltonian formulation and relevant Euler–Lagrange type equations are already presented by scholars. The formulations of fractional variational principles still can be developed more. We make an attempt to generalize the formulations for fractional variational principles. As a result we obtain generalized and complementary fractional variational formulations for extended exponentially fractional integral for example and corresponding Euler–Lagrange equations. Two illustrative examples are presented. It is observed that the formulations are in exact agreement with the Euler–Lagrange equations.
Fractional diffusion models of nonlocal transport
International Nuclear Information System (INIS)
Castillo-Negrete, D. del
2006-01-01
A class of nonlocal models based on the use of fractional derivatives (FDs) is proposed to describe nondiffusive transport in magnetically confined plasmas. FDs are integro-differential operators that incorporate in a unified framework asymmetric non-Fickian transport, non-Markovian ('memory') effects, and nondiffusive scaling. To overcome the limitations of fractional models in unbounded domains, we use regularized FDs that allow the incorporation of finite-size domain effects, boundary conditions, and variable diffusivities. We present an α-weighted explicit/implicit numerical integration scheme based on the Grunwald-Letnikov representation of the regularized fractional diffusion operator in flux conserving form. In sharp contrast with the standard diffusive model, the strong nonlocality of fractional diffusion leads to a linear in time response for a decaying pulse at short times. In addition, an anomalous fractional pinch is observed, accompanied by the development of an uphill transport region where the 'effective' diffusivity becomes negative. The fractional flux is in general asymmetric and, for steady states, it has a negative (toward the core) component that enhances confinement and a positive component that increases toward the edge and leads to poor confinement. The model exhibits the characteristic anomalous scaling of the confinement time, τ, with the system's size, L, τ∼L α , of low-confinement mode plasma where 1<α<2 is the order of the FD operator. Numerical solutions of the model with an off-axis source show that the fractional inward transport gives rise to profile peaking reminiscent of what is observed in tokamak discharges with auxiliary off-axis heating. Also, cold-pulse perturbations to steady sates in the model exhibit fast, nondiffusive propagation phenomena that resemble perturbative experiments
Multidimensional fractional Schrödinger equation
Rodrigues, M. M.; Vieira, N.
2012-11-01
This work is intended to investigate the multi-dimensional space-time fractional Schrödinger equation of the form (CDt0+αu)(t,x) = iħ/2m(C∇βu)(t,x), with ħ the Planck's constant divided by 2π, m is the mass and u(t,x) is a wave function of the particle. Here (CDt0+α,C∇β are operators of the Caputo fractional derivatives, where α ∈]0,1] and β ∈]1,2]. The wave function is obtained using Laplace and Fourier transforms methods and a symbolic operational form of solutions in terms of the Mittag-Leffler functions is exhibited. It is presented an expression for the wave function and for the quantum mechanical probability density. Using Banach fixed point theorem, the existence and uniqueness of solutions is studied for this kind of fractional differential equations.
Fractional virus epidemic model on financial networks
Directory of Open Access Journals (Sweden)
Balci Mehmet Ali
2016-01-01
Full Text Available In this study, we present an epidemic model that characterizes the behavior of a financial network of globally operating stock markets. Since the long time series have a global memory effect, we represent our model by using the fractional calculus. This model operates on a network, where vertices are the stock markets and edges are constructed by the correlation distances. Thereafter, we find an analytical solution to commensurate system and use the well-known differential transform method to obtain the solution of incommensurate system of fractional differential equations. Our findings are confirmed and complemented by the data set of the relevant stock markets between 2006 and 2016. Rather than the hypothetical values, we use the Hurst Exponent of each time series to approximate the fraction size and graph theoretical concepts to obtain the variables.
Vinogradova, Natalya; Blaine, Larry
2013-01-01
Almost everyone loves chocolate. However, the same cannot be said about fractions, which are loved by markedly fewer. Middle school students tend to view them with wary respect, but little affection. The authors attempt to sweeten the subject by describing a type of game involving division of chocolate bars. The activity they describe provides a…
Fermion Number Fractionization
Indian Academy of Sciences (India)
Srimath
1 . In tro d u ctio n. T he N obel P rize in C hem istry for the year 2000 w as aw arded to A lan J H ... soliton, the ground state of the ferm ion-soliton system can have ..... probability density,in a heuristic w ay that a fractional ferm ion num ber m ay ...
Momentum fractionation on superstrata
International Nuclear Information System (INIS)
Bena, Iosif; Martinec, Emil; Turton, David; Warner, Nicholas P.
2016-01-01
Superstrata are bound states in string theory that carry D1, D5, and momentum charges, and whose supergravity descriptions are parameterized by arbitrary functions of (at least) two variables. In the D1-D5 CFT, typical three-charge states reside in high-degree twisted sectors, and their momentum charge is carried by modes that individually have fractional momentum. Understanding this momentum fractionation holographically is crucial for understanding typical black-hole microstates in this system. We use solution-generating techniques to add momentum to a multi-wound supertube and thereby construct the first examples of asymptotically-flat superstrata. The resulting supergravity solutions are horizonless and smooth up to well-understood orbifold singularities. Upon taking the AdS_3 decoupling limit, our solutions are dual to CFT states with momentum fractionation. We give a precise proposal for these dual CFT states. Our construction establishes the very nontrivial fact that large classes of CFT states with momentum fractionation can be realized in the bulk as smooth horizonless supergravity solutions.
Vapor liquid fraction determination
International Nuclear Information System (INIS)
1980-01-01
This invention describes a method of measuring liquid and vapor fractions in a non-homogeneous fluid flowing through an elongate conduit, such as may be required with boiling water, non-boiling turbulent flows, fluidized bed experiments, water-gas mixing analysis, and nuclear plant cooling. (UK)
Brewing with fractionated barley
Donkelaar, van L.H.G.
2016-01-01
Brewing with fractionated barley
Beer is a globally consumed beverage, which is produced from malted barley, water, hops and yeast. In recent years, the use of unmalted barley and exogenous enzymes have become more popular because they enable simpler processing and reduced environmental
Fractionation and rectification apparatus
Energy Technology Data Exchange (ETDEWEB)
Sauerwald, A
1932-05-25
Fractionation and rectifying apparatus with a distillation vessel and a stirring tube, drainage tubes leading from its coils to a central collecting tube, the drainage tubes being somewhat parallel and attached to the outer half of the stirring tube and partly on the inner half of the central collecting tube, whereby distillation and rectification can be effected in a single apparatus.
International Nuclear Information System (INIS)
Innes, W.; Klein, S.; Perl, M.; Price, J.C.
1982-06-01
A device to search for fractional charge in matter is described. The sample is coupled to a low-noise amplifier by a periodically varying capacitor and the resulting signal is synchronously detected. The varying capacitor is constructed as a rapidly spinning wheel. Samples of any material in volumes of up to 0.05 ml may be searched in less than an hour
On solutions of variable-order fractional differential equations
Directory of Open Access Journals (Sweden)
Ali Akgül
2017-01-01
solutions to fractional differential equations are compelling to get in real applications, due to the nonlocality and complexity of the fractional differential operators, especially for variable-order fractional differential equations. Therefore, it is significant to enhanced numerical methods for fractional differential equations. In this work, we consider variable-order fractional differential equations by reproducing kernel method. There has been much attention in the use of reproducing kernels for the solutions to many problems in the recent years. We give two examples to demonstrate how efficiently our theory can be implemented in practice.
Fractional Quantum Field Theory: From Lattice to Continuum
Directory of Open Access Journals (Sweden)
Vasily E. Tarasov
2014-01-01
Full Text Available An approach to formulate fractional field theories on unbounded lattice space-time is suggested. A fractional-order analog of the lattice quantum field theories is considered. Lattice analogs of the fractional-order 4-dimensional differential operators are proposed. We prove that continuum limit of the suggested lattice field theory gives a fractional field theory for the continuum 4-dimensional space-time. The fractional field equations, which are derived from equations for lattice space-time with long-range properties of power-law type, contain the Riesz type derivatives on noninteger orders with respect to space-time coordinates.
Table-sized matrix model in fractional learning
Soebagyo, J.; Wahyudin; Mulyaning, E. C.
2018-05-01
This article provides an explanation of the fractional learning model i.e. a Table-Sized Matrix model in which fractional representation and its operations are symbolized by the matrix. The Table-Sized Matrix are employed to develop problem solving capabilities as well as the area model. The Table-Sized Matrix model referred to in this article is used to develop an understanding of the fractional concept to elementary school students which can then be generalized into procedural fluency (algorithm) in solving the fractional problem and its operation.
Subordination principle for fractional evolution equations
Bazhlekova, E.G.
2000-01-01
The abstract Cauchy problem for the fractional evolution equation Daa = Au, a > 0, (1) where A is a closed densely de??ned operator in a Banach space, is investigated. The subordination principle, presented earlier in [J. P r ??u s s, Evolutionary In- tegral Equations and Applications. Birkh??auser,
Higher Order and Fractional Diffusive Equations
Directory of Open Access Journals (Sweden)
D. Assante
2015-07-01
Full Text Available We discuss the solution of various generalized forms of the Heat Equation, by means of different tools ranging from the use of Hermite-Kampé de Fériet polynomials of higher and fractional order to operational techniques. We show that these methods are useful to obtain either numerical or analytical solutions.
Content knowledge of prospective elementary school teacher for fractional concepts
Pattimukay, N.; Juniati, D.; Budiarto, M. T.
2018-03-01
The aim of this study was to describe the content knowledge especially the concept of fraction of prospective elementary school teacher. The purpose of this study is to describe the content knowledge, especially the concept of fraction of prospective elementary school teacher. The subject of the study was one of prospective elementary school teacher of Pattimura University. This research is qualitative research. Data were collected through the provision of tests to explore the knowledge content of primary school teacher candidates about fractional concepts. Then continued with qualitative data analysis. The results of this study are as follows: that the prospective primary school teacher defines fractions as part of the whole if an object is divided into equal parts, so that the part that has been divided is part of the whole. Furthermore, the prospective elementary school teacher understood the fractions as division shown in two ways, namely the prospective elementary school teacher understood the fraction as a division operation, the primary school teacher candidate interpreted the fraction as a division when an object is divided be part of the same. Meanwhile, the fraction as a ratio is interpreted as the relationship between a pair of numbers. Then, the denominations are interpreted as a ratio between the numerator and the denominator of the same value. The prospective elementary school teacher also understands fractions of value when simplifying fractions. Primary school teacher candidates understand the concept of fractional operations.
Operator theory, operator algebras and applications
Lebre, Amarino; Samko, Stefan; Spitkovsky, Ilya
2014-01-01
This book consists of research papers that cover the scientific areas of the International Workshop on Operator Theory, Operator Algebras and Applications, held in Lisbon in September 2012. The volume particularly focuses on (i) operator theory and harmonic analysis (singular integral operators with shifts; pseudodifferential operators, factorization of almost periodic matrix functions; inequalities; Cauchy type integrals; maximal and singular operators on generalized Orlicz-Morrey spaces; the Riesz potential operator; modification of Hadamard fractional integro-differentiation), (ii) operator algebras (invertibility in groupoid C*-algebras; inner endomorphisms of some semi group, crossed products; C*-algebras generated by mappings which have finite orbits; Folner sequences in operator algebras; arithmetic aspect of C*_r SL(2); C*-algebras of singular integral operators; algebras of operator sequences) and (iii) mathematical physics (operator approach to diffraction from polygonal-conical screens; Poisson geo...
The Local Fractional Bootstrap
DEFF Research Database (Denmark)
Bennedsen, Mikkel; Hounyo, Ulrich; Lunde, Asger
We introduce a bootstrap procedure for high-frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our...... new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first order validity of the bootstrap method...... and in simulations we observe that the bootstrap-based hypothesis test provides considerable finite-sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data; we illustrate this by applying the bootstrap method...
Fractionalization and Entrepreneurial Activities
Awaworyi Churchill, Sefa
2015-01-01
The vast majority of the literature on ethnicity and entrepreneurship focuses on the construct of ethnic entrepreneurship. However, very little is known about how ethnic heterogeneity affects entrepreneurship. This study attempts to fill the gap, and thus examines the effect of ethnic heterogeneity on entrepreneurial activities in a cross-section of 90 countries. Using indices of ethnic and linguistic fractionalization, we show that ethnic heterogeneity negatively influences entrepreneurship....
Schmithorst, Vincent J; Brown, Rhonda Douglas
2004-07-01
The suitability of a previously hypothesized triple-code model of numerical processing, involving analog magnitude, auditory verbal, and visual Arabic codes of representation, was investigated for the complex mathematical task of the mental addition and subtraction of fractions. Functional magnetic resonance imaging (fMRI) data from 15 normal adult subjects were processed using exploratory group Independent Component Analysis (ICA). Separate task-related components were found with activation in bilateral inferior parietal, left perisylvian, and ventral occipitotemporal areas. These results support the hypothesized triple-code model corresponding to the activated regions found in the individual components and indicate that the triple-code model may be a suitable framework for analyzing the neuropsychological bases of the performance of complex mathematical tasks. Copyright 2004 Elsevier Inc.
Generalized fractional integration of the \\overline{H}-function
Directory of Open Access Journals (Sweden)
Praveen Agarwal
2012-11-01
Full Text Available A significantly large number of earlier works on the subject of fractional calculus give interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, summation of series, et cetera. In the present paper, we study and develop the generalized fractional integral operators given by Saigo. First, we establish two Theorems that give the images of the product of H-function and a general class of polynomials inSaigo operators. On account of the general nature of the Saigo operators, H-function and a general class of polynomials a large number of new and known Images involving Riemann-Liouville and Erdélyi-Kober fractional integral operators and several special functions notably generalized Wright hypergeometric function, generalized Wright-Bessel function, the polylogarithm and Mittag-Leffler functions follow as special cases of our main findings.
An extended integrable fractional-order KP soliton hierarchy
International Nuclear Information System (INIS)
Li Li
2011-01-01
In this Letter, we consider the modified derivatives and integrals of fractional-order pseudo-differential operators. A sequence of Lax KP equations hierarchy and extended fractional KP (fKP) hierarchy are introduced, and the fKP hierarchy has Lax presentations with the extended Lax operators. In the case of the extension with the half-order pseudo-differential operators, a new integrable fKP hierarchy is obtained. A few particular examples of fractional order will be listed, together with their Lax pairs.
An extended integrable fractional-order KP soliton hierarchy
Energy Technology Data Exchange (ETDEWEB)
Li Li, E-mail: li07099@163.co [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China)
2011-01-17
In this Letter, we consider the modified derivatives and integrals of fractional-order pseudo-differential operators. A sequence of Lax KP equations hierarchy and extended fractional KP (fKP) hierarchy are introduced, and the fKP hierarchy has Lax presentations with the extended Lax operators. In the case of the extension with the half-order pseudo-differential operators, a new integrable fKP hierarchy is obtained. A few particular examples of fractional order will be listed, together with their Lax pairs.
The Extended Fractional Subequation Method for Nonlinear Fractional Differential Equations
Zhao, Jianping; Tang, Bo; Kumar, Sunil; Hou, Yanren
2012-01-01
An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powe...
Functional Fractional Calculus
Das, Shantanu
2011-01-01
When a new extraordinary and outstanding theory is stated, it has to face criticism and skeptism, because it is beyond the usual concept. The fractional calculus though not new, was not discussed or developed for a long time, particularly for lack of its application to real life problems. It is extraordinary because it does not deal with 'ordinary' differential calculus. It is outstanding because it can now be applied to situations where existing theories fail to give satisfactory results. In this book not only mathematical abstractions are discussed in a lucid manner, with physical mathematic
Andreasen, Niels; Bjerregaard, Mads; Lund, Jonas; Olsen, Ove Bitsch; Rasmussen, Andreas Dalgas
2012-01-01
Projektet er bygget op omkring kritisk realisme, som er det gennemgående videnskabelige fundament til undersøgelsen af hvilke strukturelle grunde der er til finansiel ustabilitet i Danmark. Projektet går i dybden med Fractional Reserve Banking og incitamentsstrukturen i banksystemet. Vi bevæger os både på det makro- og mikroøkonomiske niveau i analysen. På makro niveau bruger vi den østrigske skole om konjunktur teori (The Positive Theory of the Cycle). På mikro niveau arbejder vi med princip...
Farrugia, Albert; Evers, Theo; Falcou, Pierre-Francois; Burnouf, Thierry; Amorim, Luiz; Thomas, Sylvia
2009-04-01
Procurement and processing of human plasma for fractionation of therapeutic proteins or biological medicines used in clinical practice is a multi-billion dollar international trade. Together the private sector and public sector (non-profit) provide large amounts of safe and effective therapeutic plasma proteins needed worldwide. The principal therapeutic proteins produced by the dichotomous industry include gamma globulins or immunoglobulins (including pathogen-specific hyperimmune globulins, such as hepatitis B immune globulins) albumin, factor VIII and Factor IX concentrates. Viral inactivation, principally by solvent detergent and other processes, has proven highly effective in preventing transmission of enveloped viruses, viz. HBV, HIV, and HCV.
An improved search for elementary particles with fractional electric charge
International Nuclear Information System (INIS)
Lee, E.R.
1996-08-01
The SLAC Quark Search Group has demonstrated successful operation of a low cost, high mass throughput Millikan apparatus designed to search for fractionally charged particles. About six million silicone oil drops were measured with no evidence of fractional charges. A second experiment is under construction with 100 times greater throughput which will utilize optimized search fluids
Concept of fractional parentage for arbitrary molecular point groups
International Nuclear Information System (INIS)
Koenig, E.; Kremer, S.
1977-01-01
The method of fractional parentage is extended to the general case of mixed configurations in arbitrary nonsimply reducible groups, G is contained in SO(3). Particular attention is devoted to the calculation of coefficients of fractional parentage (CFP) and expressions are provided for the matrix elements of F and G type operators between N electron functions. 29 references
The fundamental solutions for fractional evolution equations of parabolic type
Directory of Open Access Journals (Sweden)
Mahmoud M. El-Borai
2004-01-01
Full Text Available The fundamental solutions for linear fractional evolution equations are obtained. The coefficients of these equations are a family of linear closed operators in the Banach space. Also, the continuous dependence of solutions on the initial conditions is studied. A mixed problem of general parabolic partial differential equations with fractional order is given as an application.
Some Consequences of Learning Theory Applied to Division of Fractions
Bidwell, James K.
1971-01-01
Reviews the learning theories of Robert Gagne and David Ausubel, and applies these theories to the three most common approaches to teaching division of fractions: common denominator, complex fraction, and inverse operation methods. Such analysis indicates the inverse approach should be most effective for meaningful teaching, as is verified by…
Fractional Branes and Dynamical Supersymmetry Breaking
Franco, S; Saad, F; Uranga, Angel M; Franco, Sebastian; Hanany, Amihay; Saad, Fouad; Uranga, Angel M.
2006-01-01
We study the dynamics of fractional branes at toric singularities, including cones over del Pezzo surfaces and the recently constructed Y^{p,q} theories. We find that generically the field theories on such fractional branes show dynamical supersymmetry breaking, due to the appearance of non-perturbative superpotentials. In special cases, one recovers the known cases of supersymmetric infrared behaviors, associated to SYM confinement (mapped to complex deformations of the dual geometries, in the gauge/string correspondence sense) or N=2 fractional branes. In the supersymmetry breaking cases, when the dynamics of closed string moduli at the singularity is included, the theories show a runaway behavior (involving moduli such as FI terms or equivalently dibaryonic operators), rather than stable non-supersymmetric minima. We comment on the implications of this gauge theory behavior for the infrared smoothing of the dual warped throat solutions with 3-form fluxes, describing duality cascades ending in such field th...
Advances in robust fractional control
Padula, Fabrizio
2015-01-01
This monograph presents design methodologies for (robust) fractional control systems. It shows the reader how to take advantage of the superior flexibility of fractional control systems compared with integer-order systems in achieving more challenging control requirements. There is a high degree of current interest in fractional systems and fractional control arising from both academia and industry and readers from both milieux are catered to in the text. Different design approaches having in common a trade-off between robustness and performance of the control system are considered explicitly. The text generalizes methodologies, techniques and theoretical results that have been successfully applied in classical (integer) control to the fractional case. The first part of Advances in Robust Fractional Control is the more industrially-oriented. It focuses on the design of fractional controllers for integer processes. In particular, it considers fractional-order proportional-integral-derivative controllers, becau...
International Nuclear Information System (INIS)
Turner, R.E.
1984-01-01
A search was made for fractional charges of the form Z plus two-thirds e, where Z is an integer. It was assumed that the charges exist in natural form bound with other fractional charges in neutral molecules. It was further assumed that these neutral molecules are present in air. Two concentration schemes were employed. One sample was derived from the waste gases from a xenon distillation plant. This assumes that high mass, low vapor pressure components of air are concentrated along with the xenon. The second sample involved ionizing air, allowing a brief recombination period, and then collecting residual ions on the surface of titanium discs. Both samples were analyzed at the University of Rochester in a system using a tandem Van de Graff to accelerate particles through an essentially electrostatic beam handling system. The detector system employed both a Time of Flight and an energy-sensitive gas ionization detector. In the most sensitive mode of analysis, a gas absorber was inserted in the beam path to block the intense background. The presence of an absorber limited the search to highly penetrating particles. Effectively, this limited the search to particles with low Z and masses greater than roughly fifty GeV. The final sensitivities attained were on the order of 1 x 10 -20 for the ionized air sample and 1 x 10 -21 for the gas sample. A discussion of the caveats that could reduce the actual level of sensitivity is included
Fractional Reserve in Banking System
Valkonen, Maria
2016-01-01
This thesis is aimed to provide understanding of the role of the fractional reserve in the mod-ern banking system worldwide and particularly in Finland. The fractional reserve banking is used worldwide, but the benefits of this system are very disputable. On the one hand, experts say that the fractional reserve is a necessary instrument for the normal business and profit making. On the other hand, sceptics openly criticize the fractional reserve system and blame it for fiat money (money n...
On matrix fractional differential equations
Adem Kılıçman; Wasan Ajeel Ahmood
2017-01-01
The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann–Liouville using Laplace transform method and convolution product to the Riemann–Liouville fractional of matrices. Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objec...
ALMOST AUTOMORPHIC MILD SOLUTIONS TO SOME FRACTIONAL DELAY DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In this paper,a new and general existence and uniqueness theorem of almost automorphic mild solutions is obtained for some fractional delay differential equations,using sectorial operators and the Banach contraction principle.
Fractional Hopfield Neural Networks: Fractional Dynamic Associative Recurrent Neural Networks.
Pu, Yi-Fei; Yi, Zhang; Zhou, Ji-Liu
2017-10-01
This paper mainly discusses a novel conceptual framework: fractional Hopfield neural networks (FHNN). As is commonly known, fractional calculus has been incorporated into artificial neural networks, mainly because of its long-term memory and nonlocality. Some researchers have made interesting attempts at fractional neural networks and gained competitive advantages over integer-order neural networks. Therefore, it is naturally makes one ponder how to generalize the first-order Hopfield neural networks to the fractional-order ones, and how to implement FHNN by means of fractional calculus. We propose to introduce a novel mathematical method: fractional calculus to implement FHNN. First, we implement fractor in the form of an analog circuit. Second, we implement FHNN by utilizing fractor and the fractional steepest descent approach, construct its Lyapunov function, and further analyze its attractors. Third, we perform experiments to analyze the stability and convergence of FHNN, and further discuss its applications to the defense against chip cloning attacks for anticounterfeiting. The main contribution of our work is to propose FHNN in the form of an analog circuit by utilizing a fractor and the fractional steepest descent approach, construct its Lyapunov function, prove its Lyapunov stability, analyze its attractors, and apply FHNN to the defense against chip cloning attacks for anticounterfeiting. A significant advantage of FHNN is that its attractors essentially relate to the neuron's fractional order. FHNN possesses the fractional-order-stability and fractional-order-sensitivity characteristics.
The random continued fraction transformation
Kalle, Charlene; Kempton, Tom; Verbitskiy, Evgeny
2017-03-01
We introduce a random dynamical system related to continued fraction expansions. It uses random combinations of the Gauss map and the Rényi (or backwards) continued fraction map. We explore the continued fraction expansions that this system produces, as well as the dynamical properties of the system.
How Weird Are Weird Fractions?
Stuffelbeam, Ryan
2013-01-01
A positive rational is a weird fraction if its value is unchanged by an illegitimate, digit-based reduction. In this article, we prove that each weird fraction is uniquely weird and initiate a discussion of the prevalence of weird fractions.
Do Children Understand Fraction Addition?
Braithwaite, David W.; Tian, Jing; Siegler, Robert S.
2017-01-01
Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, in press) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments…
On fractional Fourier transform moments
Alieva, T.; Bastiaans, M.J.
2000-01-01
Based on the relation between the ambiguity function represented in a quasi-polar coordinate system and the fractional power spectra, the fractional Fourier transform moments are introduced. Important equalities for the global second-order fractional Fourier transform moments are derived and their
Particle Simulation of Fractional Diffusion Equations
Allouch, Samer
2017-07-12
This work explores different particle-based approaches to the simulation of one-dimensional fractional subdiffusion equations in unbounded domains. We rely on smooth particle approximations, and consider four methods for estimating the fractional diffusion term. The first method is based on direct differentiation of the particle representation, it follows the Riesz definition of the fractional derivative and results in a non-conservative scheme. The other three methods follow the particle strength exchange (PSE) methodology and are by construction conservative, in the sense that the total particle strength is time invariant. The first PSE algorithm is based on using direct differentiation to estimate the fractional diffusion flux, and exploiting the resulting estimates in an integral representation of the divergence operator. Meanwhile, the second one relies on the regularized Riesz representation of the fractional diffusion term to derive a suitable interaction formula acting directly on the particle representation of the diffusing field. A third PSE construction is considered that exploits the Green\\'s function of the fractional diffusion equation. The performance of all four approaches is assessed for the case of a one-dimensional diffusion equation with constant diffusivity. This enables us to take advantage of known analytical solutions, and consequently conduct a detailed analysis of the performance of the methods. This includes a quantitative study of the various sources of error, namely filtering, quadrature, domain truncation, and time integration, as well as a space and time self-convergence analysis. These analyses are conducted for different values of the order of the fractional derivatives, and computational experiences are used to gain insight that can be used for generalization of the present constructions.
Particle Simulation of Fractional Diffusion Equations
Allouch, Samer; Lucchesi, Marco; Maî tre, O. P. Le; Mustapha, K. A.; Knio, Omar
2017-01-01
This work explores different particle-based approaches to the simulation of one-dimensional fractional subdiffusion equations in unbounded domains. We rely on smooth particle approximations, and consider four methods for estimating the fractional diffusion term. The first method is based on direct differentiation of the particle representation, it follows the Riesz definition of the fractional derivative and results in a non-conservative scheme. The other three methods follow the particle strength exchange (PSE) methodology and are by construction conservative, in the sense that the total particle strength is time invariant. The first PSE algorithm is based on using direct differentiation to estimate the fractional diffusion flux, and exploiting the resulting estimates in an integral representation of the divergence operator. Meanwhile, the second one relies on the regularized Riesz representation of the fractional diffusion term to derive a suitable interaction formula acting directly on the particle representation of the diffusing field. A third PSE construction is considered that exploits the Green's function of the fractional diffusion equation. The performance of all four approaches is assessed for the case of a one-dimensional diffusion equation with constant diffusivity. This enables us to take advantage of known analytical solutions, and consequently conduct a detailed analysis of the performance of the methods. This includes a quantitative study of the various sources of error, namely filtering, quadrature, domain truncation, and time integration, as well as a space and time self-convergence analysis. These analyses are conducted for different values of the order of the fractional derivatives, and computational experiences are used to gain insight that can be used for generalization of the present constructions.
An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations
Burrage, Kevin; Hale, Nicholas; Kay, David
2012-01-01
Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time
Modeling electron fractionalization with unconventional Fock spaces.
Cobanera, Emilio
2017-08-02
It is shown that certain fractionally-charged quasiparticles can be modeled on D-dimensional lattices in terms of unconventional yet simple Fock algebras of creation and annihilation operators. These unconventional Fock algebras are derived from the usual fermionic algebra by taking roots (the square root, cubic root, etc) of the usual fermionic creation and annihilation operators. If the fermions carry non-Abelian charges, then this approach fractionalizes the Abelian charges only. In particular, the mth-root of a spinful fermion carries charge e/m and spin 1/2. Just like taking a root of a complex number, taking a root of a fermion yields a mildly non-unique result. As a consequence, there are several possible choices of quantum exchange statistics for fermion-root quasiparticles. These choices are tied to the dimensionality [Formula: see text] of the lattice by basic physical considerations. One particular family of fermion-root quasiparticles is directly connected to the parafermion zero-energy modes expected to emerge in certain mesoscopic devices involving fractional quantum Hall states. Hence, as an application of potential mesoscopic interest, I investigate numerically the hybridization of Majorana and parafermion zero-energy edge modes caused by fractionalizing but charge-conserving tunneling.
On the singular perturbations for fractional differential equation.
Atangana, Abdon
2014-01-01
The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.
On the Singular Perturbations for Fractional Differential Equation
Directory of Open Access Journals (Sweden)
Abdon Atangana
2014-01-01
Full Text Available The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.
Electronically Tunable Fully Integrated Fractional-Order Resonator
Tsirimokou, Georgia
2017-03-20
A fully integrated implementation of a parallel fractional-order resonator which employs together a fractional order capacitor and a fractional-order inductor is proposed in this paper. The design utilizes current-controlled Operational Transconductance Amplifiers as building blocks, designed and fabricated in AMS 0:35m CMOS process, and based on a second-order approximation of a fractional-order differentiator/ integrator magnitude optimized in the range 10Hz–700Hz. An attractive benefit of the proposed scheme is its electronic tuning capability.
Finite element method for time-space-fractional Schrodinger equation
Directory of Open Access Journals (Sweden)
Xiaogang Zhu
2017-07-01
Full Text Available In this article, we develop a fully discrete finite element method for the nonlinear Schrodinger equation (NLS with time- and space-fractional derivatives. The time-fractional derivative is described in Caputo's sense and the space-fractional derivative in Riesz's sense. Its stability is well derived; the convergent estimate is discussed by an orthogonal operator. We also extend the method to the two-dimensional time-space-fractional NLS and to avoid the iterative solvers at each time step, a linearized scheme is further conducted. Several numerical examples are implemented finally, which confirm the theoretical results as well as illustrate the accuracy of our methods.
Electronically Tunable Fully Integrated Fractional-Order Resonator
Tsirimokou, Georgia; Psychalinos, Costas; Elwakil, Ahmed S.; Salama, Khaled N.
2017-01-01
A fully integrated implementation of a parallel fractional-order resonator which employs together a fractional order capacitor and a fractional-order inductor is proposed in this paper. The design utilizes current-controlled Operational Transconductance Amplifiers as building blocks, designed and fabricated in AMS 0:35m CMOS process, and based on a second-order approximation of a fractional-order differentiator/ integrator magnitude optimized in the range 10Hz–700Hz. An attractive benefit of the proposed scheme is its electronic tuning capability.
Nonhomogeneous fractional Poisson processes
Energy Technology Data Exchange (ETDEWEB)
Wang Xiaotian [School of Management, Tianjin University, Tianjin 300072 (China)]. E-mail: swa001@126.com; Zhang Shiying [School of Management, Tianjin University, Tianjin 300072 (China); Fan Shen [Computer and Information School, Zhejiang Wanli University, Ningbo 315100 (China)
2007-01-15
In this paper, we propose a class of non-Gaussian stationary increment processes, named nonhomogeneous fractional Poisson processes W{sub H}{sup (j)}(t), which permit the study of the effects of long-range dependance in a large number of fields including quantum physics and finance. The processes W{sub H}{sup (j)}(t) are self-similar in a wide sense, exhibit more fatter tail than Gaussian processes, and converge to the Gaussian processes in distribution in some cases. In addition, we also show that the intensity function {lambda}(t) strongly influences the existence of the highest finite moment of W{sub H}{sup (j)}(t) and the behaviour of the tail probability of W{sub H}{sup (j)}(t)
Nonhomogeneous fractional Poisson processes
International Nuclear Information System (INIS)
Wang Xiaotian; Zhang Shiying; Fan Shen
2007-01-01
In this paper, we propose a class of non-Gaussian stationary increment processes, named nonhomogeneous fractional Poisson processes W H (j) (t), which permit the study of the effects of long-range dependance in a large number of fields including quantum physics and finance. The processes W H (j) (t) are self-similar in a wide sense, exhibit more fatter tail than Gaussian processes, and converge to the Gaussian processes in distribution in some cases. In addition, we also show that the intensity function λ(t) strongly influences the existence of the highest finite moment of W H (j) (t) and the behaviour of the tail probability of W H (j) (t)
Membrane Assisted Enzyme Fractionation
DEFF Research Database (Denmark)
Yuan, Linfeng
to the variation in size of the proteins and a reasonable separation factor can be observed only when the size difference is in the order of 10 or more. This is partly caused by concentration polarization and membrane fouling which hinders an effective separation of the proteins. Application of an electric field...... across the porous membrane has been demonstrated to be an effective way to reduce concentration polarization and membrane fouling. In addition, this technique can also be used to separate the proteins based on difference in charge, which to some extent overcome the limitations of size difference...... of proteins on the basis of their charge, degree of hydrophobicity, affinity or size. Adequate purity is often not achieved unless several purification steps are combined thereby increasing cost and reducing product yield. Conventional fractionation of proteins using ultrafiltration membranes is limited...
Fraction Reduction in Membrane Systems
Directory of Open Access Journals (Sweden)
Ping Guo
2014-01-01
Full Text Available Fraction reduction is a basic computation for rational numbers. P system is a new computing model, while the current methods for fraction reductions are not available in these systems. In this paper, we propose a method of fraction reduction and discuss how to carry it out in cell-like P systems with the membrane structure and the rules with priority designed. During the application of fraction reduction rules, synchronization is guaranteed by arranging some special objects in these rules. Our work contributes to performing the rational computation in P systems since the rational operands can be given in the form of fraction.
Thermochemical transformations of anthracite fractions
Energy Technology Data Exchange (ETDEWEB)
Belkina, T.V.; Privalov, V.E.; Stepanenko, atM.A.
1979-08-01
Research on the nature of thermochemical transformations of anthracite fractions and the possibility of increasing their activity and identifying conditions for their use in the electrode pitch process is described. From research done on different anthracite fractions processed at varying temperatures it was concluded that accumulations of condensates from heating anthracite fractions occur significantly slower in comparison with pitch. As a result the electrode pitch process is prolonged. Thermal treatment of an anthracite fraction causes the formation and accumulation of condensates and promotes thermochemical transformations. Lastly, the use of thermally treated anthracite fractions apparently intensifies the electrode pitch process and improves its quality. (16 refs.) (In Russian)
On the numerical solution of the neutron fractional diffusion equation
International Nuclear Information System (INIS)
Maleki Moghaddam, Nader; Afarideh, Hossein; Espinosa-Paredes, Gilberto
2014-01-01
Highlights: • The new version of neutron diffusion equation which established on the fractional derivatives is presented. • The Neutron Fractional Diffusion Equation (NFDE) is solved in the finite differences frame. • NFDE is solved using shifted Grünwald-Letnikov definition of fractional operators. • The results show that “K eff ” strongly depends on the order of fractional derivative. - Abstract: In order to core calculation in the nuclear reactors there is a new version of neutron diffusion equation which is established on the fractional partial derivatives, named Neutron Fractional Diffusion Equation (NFDE). In the NFDE model, neutron flux in each zone depends directly on the all previous zones (not only on the nearest neighbors). Under this circumstance, it can be said that the NFDE has the space history. We have developed a one-dimension code, NFDE-1D, which can simulate the reactor core using arbitrary exponent of differential operators. In this work a numerical solution of the NFDE is presented using shifted Grünwald-Letnikov definition of fractional derivative in finite differences frame. The model is validated with some numerical experiments where different orders of fractional derivative are considered (e.g. 0.999, 0.98, 0.96, and 0.94). The results show that the effective multiplication factor (K eff ) depends strongly on the order of fractional derivative
A Caputo fractional derivative of a function with respect to another function
Almeida, Ricardo
2017-03-01
In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor's Theorem, Fermat's Theorem, etc., are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.
Chiral anomaly, bosonization and fractional charge
International Nuclear Information System (INIS)
Mignaco, J.A.; Rego Monteiro, M.A. do.
1984-01-01
A method to evaluate the Jacobian of chiral rotations, regulating determinants through the proper time method and using Seeley's asymptotic expansion is presented. With this method the chiral anomaly ofr ν=4,6 dimensions is computed easily, bosonization of some massless two-dimensional models is discussed and the problem of charge fractionization is handled. Besides, the general validity of Fujikawa's approach to regulate the Jacobian of chiral rotations with non-hermitean operators is commented. (Author) [pt
Chiral anomaly, bosonization, and fractional charge
International Nuclear Information System (INIS)
Mignaco, J.A.; Monteiro, M.A.R.
1985-01-01
We present a method to evaluate the Jacobian of chiral rotations, regulating determinants through the proper-time method and using Seeley's asymptotic expansion. With this method we compute easily the chiral anomaly for ν = 4,6 dimensions, discuss bosonization of some massless two-dimensional models, and handle the problem of charge fractionization. In addition, we comment on the general validity of Fujikawa's approach to regulate the Jacobian of chiral rotations with non-Hermitian operators
Implementation of quantum and classical discrete fractional Fourier transforms
Weimann, Steffen; Perez-Leija, Armando; Lebugle, Maxime; Keil, Robert; Tichy, Malte; Gräfe, Markus; Heilmann, René; Nolte, Stefan; Moya-Cessa, Hector; Weihs, Gregor; Christodoulides, Demetrios N.; Szameit, Alexander
2016-01-01
Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools. PMID:27006089
Implementation of quantum and classical discrete fractional Fourier transforms.
Weimann, Steffen; Perez-Leija, Armando; Lebugle, Maxime; Keil, Robert; Tichy, Malte; Gräfe, Markus; Heilmann, René; Nolte, Stefan; Moya-Cessa, Hector; Weihs, Gregor; Christodoulides, Demetrios N; Szameit, Alexander
2016-03-23
Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.
Request for approval, vented container annual release fraction
International Nuclear Information System (INIS)
HILL, J.S.
1999-01-01
In accordance with the approval conditions for Modification to the Central Waste Complex (CWC) Radioactive Air Emissions Notice of Construction (NOC). dated August 24,1998, a new release fraction has been developed for submittal to the Washington State Department of Health (WDOH). The proposed annual release fraction of 2.50 E-14 is proposed for use in future NOCs involving the storage and handling operations associated with vented containers on the Hanford Site. The proposed annual release fraction was the largest release fraction calculated from alpha measurements of the NucFil filters from 10 vented containers consisting of nine 55-gallon drums and one burial box with dimensions of 9.3 x 5.7 x 6.4 feet. An annual release fraction of 2.0 E-09 was used in the modification to the CWC radioactive air emissions NOC. This study confirmed that the release fraction used in the CWC radioactive air emissions NOC was conservative
Request for approval, vented container annual release fraction; FINAL
International Nuclear Information System (INIS)
HILL, J.S.
1999-01-01
In accordance with the approval conditions for Modification to the Central Waste Complex (CWC) Radioactive Air Emissions Notice of Construction (NOC). dated August 24,1998, a new release fraction has been developed for submittal to the Washington State Department of Health (WDOH). The proposed annual release fraction of 2.50 E-14 is proposed for use in future NOCs involving the storage and handling operations associated with vented containers on the Hanford Site. The proposed annual release fraction was the largest release fraction calculated from alpha measurements of the NucFil filters from 10 vented containers consisting of nine 55-gallon drums and one burial box with dimensions of 9.3 x 5.7 x 6.4 feet. An annual release fraction of 2.0 E-09 was used in the modification to the CWC radioactive air emissions NOC. This study confirmed that the release fraction used in the CWC radioactive air emissions NOC was conservative
Dependence of stability of metastable superconductors on copper fraction
International Nuclear Information System (INIS)
Elrod, S.A.; Lue, J.W.; Miller, J.R.; Dresner, L.
1980-12-01
The stability of composite superconductors operating in the metastable regime depends upon such factors as matrix resistivity, cooled surface dimensions, fraction of critical current, and volume fraction of stabilizer. By assuming constant thermophysical properties, we developed analytic expressions for the energy and voltage of the minimum propagating zone (MPZ). With other factors held constant, these expressions have been used to predict composite superconductor stability as a function of copper fraction: lower copper fractions lead to higher MPZ energies. MPZ voltages have been measured for three NbTi/Cu composites having different copper fractions and different critical current densities for several magnetic fields and transport currents. Experimental MPZ voltages have been used to calculate an effective heat transfer coefficient, which is subsequently used to calculate the MPZ energy. The experimental MPZ energies support the theoretical expectation that lower copper fractions lead to higher stability in the metastable regime
Misonidazole in fractionated radiotherapy: are many small fractions best
International Nuclear Information System (INIS)
Denekamp, J.; McNally, N.J.; Fowler, J.F.; Joiner, M.C.
1980-01-01
The largest sensitizing effect is always demonstrated with six fractions, each given with 2 g/m 2 of misonidazole. In the absence of reoxygenation a sensitizer enhancement ratio of 1.7 is predicted, but this falls to 1.1-1.2 if extensive reoxygenation occurs. Less sensitization is observed with 30 fractions, each with 0.4 g/m 2 of drug. However, for clinical use, the important question is which treatment kills the maximum number of tumour cells. Many of the simulations predict a marked disadvantage of reducing the fraction number for X rays alone. The circumstances in which this disadvantage is offset by the large Sensitizer enhancement ratio values with a six-fraction schedule are few. The model calculations suggest that many small fractions, each with a low drug dose, are safest unless the clinician has some prior knowledge that a change in fraction number is not disadvantageous. (author)
Fractional statistics and fractional quantized Hall effect. Revision
International Nuclear Information System (INIS)
Tao, R.; Wu, Y.S.
1984-01-01
We suggest that the origin of the odd denominator rule observed in the fractional quantized Hall effect (FQHE) may lie in fractional statistics which governs quasiparticles in FQHE. A theorem concerning statistics of clusters of quasiparticles implies that fractional statistics does not allow coexistence of a large number of quasiparticles at fillings with an even denominator. Thus no Hall plateau can be formed at these fillings, regardless of the presence of an energy gap. 15 references
Fractionation of Pb and Cu in the fine fraction (landfill.
Kaczala, Fabio; Orupõld, Kaja; Augustsson, Anna; Burlakovs, Juris; Hogland, Marika; Bhatnagar, Amit; Hogland, William
2017-11-01
The fractionation of metals in the fine fraction (landfill was carried out to evaluate the metal (Pb and Cu) contents and their potential towards not only mobility but also possibilities of recovery/extraction. The fractionation followed the BCR (Community Bureau of Reference) sequential extraction, and the exchangeable (F1), reducible (F2), oxidizable (F3) and residual fractions were determined. The results showed that Pb was highly associated with the reducible (F2) and oxidizable (F3) fractions, suggesting the potential mobility of this metal mainly when in contact with oxygen, despite the low association with the exchangeable fraction (F1). Cu has also shown the potential for mobility when in contact with oxygen, since high associations with the oxidizable fraction (F3) were observed. On the other hand, the mobility of metals in excavated waste can be seen as beneficial considering the circular economy and recovery of such valuables back into the economy. To conclude, not only the total concentration of metals but also a better understanding of fractionation and in which form metals are bound is very important to bring information on how to manage the fine fraction from excavated waste both in terms of environmental impacts and also recovery of such valuables in the economy.
Fractional variational calculus in terms of Riesz fractional derivatives
International Nuclear Information System (INIS)
Agrawal, O P
2007-01-01
This paper presents extensions of traditional calculus of variations for systems containing Riesz fractional derivatives (RFDs). Specifically, we present generalized Euler-Lagrange equations and the transversality conditions for fractional variational problems (FVPs) defined in terms of RFDs. We consider two problems, a simple FVP and an FVP of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives, functions and parameters, and to unspecified boundary conditions. For the second problem, we present Lagrange-type multiplier rules. For both problems, we develop the Euler-Lagrange-type necessary conditions which must be satisfied for the given functional to be extremum. Problems are considered to demonstrate applications of the formulations. Explicitly, we introduce fractional momenta, fractional Hamiltonian, fractional Hamilton equations of motion, fractional field theory and fractional optimal control. The formulations presented and the resulting equations are similar to the formulations for FVPs given in Agrawal (2002 J. Math. Anal. Appl. 272 368, 2006 J. Phys. A: Math. Gen. 39 10375) and to those that appear in the field of classical calculus of variations. These formulations are simple and can be extended to other problems in the field of fractional calculus of variations
Modelling altered fractionation schedules
International Nuclear Information System (INIS)
Fowler, J.F.
1993-01-01
The author discusses the conflicting requirements of hyperfractionation and accelerated fractionation used in radiotherapy, and the development of computer modelling to predict how to obtain an optimum of tumour cell kill without exceeding normal-tissue tolerance. The present trend is to shorten hyperfractionated schedules from 6 or 7 weeks to give overall times of 4 or 5 weeks as in new schedules by Herskovic et al (1992) and Harari (1992). Very high doses are given, much higher than can be given when ultrashort schedules such as CHART (12 days) are used. Computer modelling has suggested that optimum overall times, to yield maximum cell kill in tumours ((α/β = 10 Gy) for a constant level of late complications (α/β = 3 Gy) would be X or X-1 weeks, where X is the doubling time of the tumour cells in days (Fowler 1990). For median doubling times of about 5 days, overall times of 4 or 5 weeks should be ideal. (U.K.)
An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations
Burrage, Kevin
2012-01-01
Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator. © 2012 Society for Industrial and Applied Mathematics.
Accessible solitons of fractional dimension
Energy Technology Data Exchange (ETDEWEB)
Zhong, Wei-Ping, E-mail: zhongwp6@126.com [Department of Electronic and Information Engineering, Shunde Polytechnic, Guangdong Province, Shunde 528300 (China); Texas A& M University at Qatar, P.O. Box 23874, Doha (Qatar); Belić, Milivoj [Texas A& M University at Qatar, P.O. Box 23874, Doha (Qatar); Zhang, Yiqi [Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Shaanxi Key Lab of Information Photonic Technique, Xi’an Jiaotong University, Xi’an 710049 (China)
2016-05-15
We demonstrate that accessible solitons described by an extended Schrödinger equation with the Laplacian of fractional dimension can exist in strongly nonlocal nonlinear media. The soliton solutions of the model are constructed by two special functions, the associated Legendre polynomials and the Laguerre polynomials in the fraction-dimensional space. Our results show that these fractional accessible solitons form a soliton family which includes crescent solitons, and asymmetric single-layer and multi-layer necklace solitons. -- Highlights: •Analytic solutions of a fractional Schrödinger equation are obtained. •The solutions are produced by means of self-similar method applied to the fractional Schrödinger equation with parabolic potential. •The fractional accessible solitons form crescent, asymmetric single-layer and multilayer necklace profiles. •The model applies to the propagation of optical pulses in strongly nonlocal nonlinear media.
Fractional Calculus and Shannon Wavelet
Directory of Open Access Journals (Sweden)
Carlo Cattani
2012-01-01
Full Text Available An explicit analytical formula for the any order fractional derivative of Shannon wavelet is given as wavelet series based on connection coefficients. So that for any 2(ℝ function, reconstructed by Shannon wavelets, we can easily define its fractional derivative. The approximation error is explicitly computed, and the wavelet series is compared with Grünwald fractional derivative by focusing on the many advantages of the wavelet method, in terms of rate of convergence.
Fractional variational principles in action
Energy Technology Data Exchange (ETDEWEB)
Baleanu, Dumitru [Department of Mathematics and Computer Science, Faculty of Art and Sciences, Cankaya University, 06530 Ankara (Turkey); Institute of Space Sciences, PO Box MG-23, R 76900, Magurele-Bucharest (Romania)], E-mail: dumitru@cankaya.edu.tr
2009-10-15
The fractional calculus has gained considerable importance in various fields of science and engineering, especially during the last few decades. An open issue in this emerging field is represented by the fractional variational principles area. Therefore, the fractional Euler-Lagrange and Hamilton equations started to be examined intensely during the last decade. In this paper, we review some new trends in this field and we discuss some of their potential applications.
Kimura, Taro; Pestun, Vasily
2018-04-01
We introduce quiver gauge theory associated with the non-simply laced type fractional quiver and define fractional quiver W-algebras by using construction of Kimura and Pestun (Lett Math Phys, 2018. https://doi.org/10.1007/s11005-018-1072-1; Lett Math Phys, 2018. https://doi.org/10.1007/s11005-018-1073-0) with representation of fractional quivers.
Hosseinabadi, Abdolali Neamaty; Nategh, Mehdi
2014-01-01
This work, dealt with the classical mean value theorem and took advantage of it in the fractional calculus. The concept of a fractional critical point is introduced. Some sufficient conditions for the existence of a critical point is studied and an illustrative example rele- vant to the concept of the time dilation effect is given. The present paper also includes, some connections between convexity (and monotonicity) with fractional derivative in the Riemann-Liouville sense.
Boundary value problemfor multidimensional fractional advection-dispersion equation
Directory of Open Access Journals (Sweden)
Khasambiev Mokhammad Vakhaevich
2015-05-01
Full Text Available In recent time there is a very great interest in the study of differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. It is due to the development of the theory of fractional integro-differential theory and application of it in different fields.The fractional integrals and derivatives of fractional integro-differential equations are widely used in modern investigations of theoretical physics, mechanics, and applied mathematics. The fractional calculus is a very powerful tool for describing physical systems, which have a memory and are non-local. Many processes in complex systems have nonlocality and long-time memory. Fractional integral operators and fractional differential operators allow describing some of these properties. The use of the fractional calculus will be helpful for obtaining the dynamical models, in which integro-differential operators describe power long-time memory by time and coordinates, and three-dimensional nonlocality for complex medium and processes.Differential equations of fractional order appear when we use fractal conception in physics of the condensed medium. The transfer, described by the operator with fractional derivatives at a long distance from the sources, leads to other behavior of relatively small concentrations as compared with classic diffusion. This fact redefines the existing ideas about safety, based on the ideas on exponential velocity of damping. Fractional calculus in the fractal theory and the systems with memory have the same importance as the classic analysis in mechanics of continuous medium.In recent years, the application of fractional derivatives for describing and studying the physical processes of stochastic transfer is very popular too. Many problems of filtration of liquids in fractal (high porous medium lead to the need to study boundary value problems for partial differential equations in fractional order.In this paper the
Fractionated Spacecraft Architectures Seeding Study
National Research Council Canada - National Science Library
Mathieu, Charlotte; Weigel, Annalisa
2006-01-01
.... Models were developed from a customer-centric perspective to assess different fractionated spacecraft architectures relative to traditional spacecraft architectures using multi-attribute analysis...
q-fractional calculus and equations
Annaby, Mahmoud H
2012-01-01
This nine-chapter monograph introduces a rigorous investigation of q-difference operators in standard and fractional settings. It starts with elementary calculus of q-differences and integration of Jackson’s type before turning to q-difference equations. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Regular q-Sturm–Liouville theory is also introduced; Green’s function is constructed and the eigenfunction expansion theorem is given. The monograph also discusses some integral equations of Volterra and Abel type, as introductory material for the study of fractional q-calculi. Hence fractional q-calculi of the types Riemann–Liouville; Grünwald–Letnikov; Caputo; Erdélyi–Kober and Weyl are defined analytically. Fractional q-Leibniz rules with applications in q-series are also obtained with rigorous proofs of the formal results of Al-Salam-Verma, which remained unproved for decades. In working ...
Early Fractions Learning of 3rd Grade Students in SD Laboratorium Unesa
Directory of Open Access Journals (Sweden)
Elisabet Ayunika Permata Sari
2012-01-01
Full Text Available Fractions varied meanings is one of the causes of difficulties in learning fractions. These students should be given greater opportunities to explore the meaning of fractions before they learn the relationship between fractions and operations on fractions. Although students can shading area represents a fraction, does not mean they really understand the meaning of fractions as a whole. With a realistic approach to mathematics, students are given the contextual issues of equitable distribution and measurements that involve fractions. Keyword: fraction meaning, relation of fraction, design research,realistic mathematics education, equitable distribution, measurement DOI: http://dx.doi.org/10.22342/jme.3.1.617.17-28
COMMERCIAL SNF ACCIDENT RELEASE FRACTIONS
Energy Technology Data Exchange (ETDEWEB)
S.O. Bader
1999-10-18
The purpose of this design analysis is to specify and document the total and respirable fractions for radioactive materials that are released from an accident event at the Monitored Geologic Repository (MGR) involving commercial spent nuclear fuel (CSNF) in a dry environment. The total and respirable release fractions will be used to support the preclosure licensing basis for the MGR. The total release fraction is defined as the fraction of total CSNF assembly inventory, typically expressed as an activity inventory (e.g., curies), of a given radionuclide that is released to the environment from a waste form. The radionuclides are released from the inside of breached fuel rods (or pins) and from the detachment of radioactive material (crud) from the outside surfaces of fuel rods and other components of fuel assemblies. The total release fraction accounts for several mechanisms that tend to retain, retard, or diminish the amount of radionuclides that are available for transport to dose receptors or otherwise can be shown to reduce exposure of receptors to radiological releases. The total release fraction includes a fraction of airborne material that is respirable and could result in inhalation doses. This subset of the total release fraction is referred to as the respirable release fraction. Potential accidents may involve waste forms that are characterized as either bare (unconfined) fuel assemblies or confined fuel assemblies. The confined CSNF assemblies at the MGR are contained in shipping casks, canisters, or disposal containers (waste packages). In contrast to the bare fuel assemblies, the container that confines the fuel assemblies has the potential of providing an additional barrier for diminishing the total release fraction should the fuel rod cladding breach during an accident. However, this analysis will not take credit for this additional bamer and will establish only the total release fractions for bare unconfined CSNF assemblies, which may however be
COMMERCIAL SNF ACCIDENT RELEASE FRACTIONS
International Nuclear Information System (INIS)
S.O. Bader
1999-01-01
The purpose of this design analysis is to specify and document the total and respirable fractions for radioactive materials that are released from an accident event at the Monitored Geologic Repository (MGR) involving commercial spent nuclear fuel (CSNF) in a dry environment. The total and respirable release fractions will be used to support the preclosure licensing basis for the MGR. The total release fraction is defined as the fraction of total CSNF assembly inventory, typically expressed as an activity inventory (e.g., curies), of a given radionuclide that is released to the environment from a waste form. The radionuclides are released from the inside of breached fuel rods (or pins) and from the detachment of radioactive material (crud) from the outside surfaces of fuel rods and other components of fuel assemblies. The total release fraction accounts for several mechanisms that tend to retain, retard, or diminish the amount of radionuclides that are available for transport to dose receptors or otherwise can be shown to reduce exposure of receptors to radiological releases. The total release fraction includes a fraction of airborne material that is respirable and could result in inhalation doses. This subset of the total release fraction is referred to as the respirable release fraction. Potential accidents may involve waste forms that are characterized as either bare (unconfined) fuel assemblies or confined fuel assemblies. The confined CSNF assemblies at the MGR are contained in shipping casks, canisters, or disposal containers (waste packages). In contrast to the bare fuel assemblies, the container that confines the fuel assemblies has the potential of providing an additional barrier for diminishing the total release fraction should the fuel rod cladding breach during an accident. However, this analysis will not take credit for this additional bamer and will establish only the total release fractions for bare unconfined CSNF assemblies, which may however be
Power filtering of nth order in the fractional Fourier domain
International Nuclear Information System (INIS)
Alieva, Tatiana; Calvo, Maria Luisa; Bastiaans, Martin J.
2002-01-01
The main properties of the power filtering operation in the fractional Fourier domain and its relationship to the differentiation operation are considered. The application of linear power filtering for solving the phase retrieval problem from intensity distributions only is proposed. The optical configuration for the experimental realization of the method is discussed. (author)
Validating a Written Instrument for Assessing Students' Fractions Schemes and
Wilkins, Jesse L. M.; Norton, Anderson; Boyce, Steven J.
2013-01-01
Previous research has documented schemes and operations that undergird students' understanding of fractions. This prior research was based, in large part, on small-group teaching experiments. However, written assessments are needed in order for teachers and researchers to assess students' ways of operating on a whole-class scale. In this study,…
Fractions, Number Lines, Third Graders
Cramer, Kathleen; Ahrendt, Sue; Monson, Debra; Wyberg, Terry; Colum, Karen
2017-01-01
The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) outlines ambitious goals for fraction learning, starting in third grade, that include the use of the number line model. Understanding and constructing fractions on a number line are particularly complex tasks. The current work of the authors centers on ways to successfully…
Unwrapping Students' Ideas about Fractions
Lewis, Rebecca M.; Gibbons, Lynsey K.; Kazemi, Elham; Lind, Teresa
2015-01-01
Supporting students to develop an understanding of the meaning of fractions is an important goal of elementary school mathematics. This involves developing partitioning strategies, creating representations, naming fractional quantities, and using symbolic notation. This article describes how teachers can use a formative assessment problem to…
Understanding Magnitudes to Understand Fractions
Gabriel, Florence
2016-01-01
Fractions are known to be difficult to learn and difficult to teach, yet they are vital for students to have access to further mathematical concepts. This article uses evidence to support teachers employing teaching methods that focus on the conceptual understanding of the magnitude of fractions.
Financial Planning with Fractional Goals
Goedhart, Marc; Spronk, Jaap
1995-01-01
textabstractWhen solving financial planning problems with multiple goals by means of multiple objective programming, the presence of fractional goals leads to technical difficulties. In this paper we present a straightforward interactive approach for solving such linear fractional programs with multiple goal variables. The approach is illustrated by means of an example in financial planning.
Deterministic ratchets for suspension fractionation
Kulrattanarak, T.
2010-01-01
Driven by the current insights in sustainability and technological development in
biorefining natural renewable resources, the food industry has taken an interest in
fractionation of agrofood materials, like milk and cereal crops. The purpose of fractionation
is to split the raw
Resolvent estimates in homogenisation of periodic problems of fractional elasticity
Cherednichenko, Kirill; Waurick, Marcus
2018-03-01
We provide operator-norm convergence estimates for solutions to a time-dependent equation of fractional elasticity in one spatial dimension, with rapidly oscillating coefficients that represent the material properties of a viscoelastic composite medium. Assuming periodicity in the coefficients, we prove operator-norm convergence estimates for an operator fibre decomposition obtained by applying to the original fractional elasticity problem the Fourier-Laplace transform in time and Gelfand transform in space. We obtain estimates on each fibre that are uniform in the quasimomentum of the decomposition and in the period of oscillations of the coefficients as well as quadratic with respect to the spectral variable. On the basis of these uniform estimates we derive operator-norm-type convergence estimates for the original fractional elasticity problem, for a class of sufficiently smooth densities of applied forces.
The usage of Maxwell fractional equations for the investigation of the waveguide processes
International Nuclear Information System (INIS)
Maksyuta, M.V.; Slinchenko, Yu.A.; Grygoruk, V.I.
2016-01-01
By means of nabla operator written down with using both of some differential operators with integer orders and fractional differential Caputo operators, gradient, divergence and rotor operators are determined, it is checked up the fulfillment of vector relations in fractional vector analysis, fractional Green, Stocks and Ostrogradsky-Gauss formulas. For a specific expression of nabla operator (nabla components along x and y axes have a unit order and along z axis, correspondingly, a fractional value in the interval from zero till unit) Maxwell fractional equations are written down. Based on the following from them some fractional wave equations, dissipative and polarization processes at electromagnetic waves distribution both in rectangular (planar) and in cylindrical waveguide structures are analyzed.
A new fractional wavelet transform
Dai, Hongzhe; Zheng, Zhibao; Wang, Wei
2017-03-01
The fractional Fourier transform (FRFT) is a potent tool to analyze the time-varying signal. However, it fails in locating the fractional Fourier domain (FRFD)-frequency contents which is required in some applications. A novel fractional wavelet transform (FRWT) is proposed to solve this problem. It displays the time and FRFD-frequency information jointly in the time-FRFD-frequency plane. The definition, basic properties, inverse transform and reproducing kernel of the proposed FRWT are considered. It has been shown that an FRWT with proper order corresponds to the classical wavelet transform (WT). The multiresolution analysis (MRA) associated with the developed FRWT, together with the construction of the orthogonal fractional wavelets are also presented. Three applications are discussed: the analysis of signal with time-varying frequency content, the FRFD spectrum estimation of signals that involving noise, and the construction of fractional Harr wavelet. Simulations verify the validity of the proposed FRWT.
Intelligent numerical methods applications to fractional calculus
Anastassiou, George A
2016-01-01
In this monograph the authors present Newton-type, Newton-like and other numerical methods, which involve fractional derivatives and fractional integral operators, for the first time studied in the literature. All for the purpose to solve numerically equations whose associated functions can be also non-differentiable in the ordinary sense. That is among others extending the classical Newton method theory which requires usual differentiability of function. Chapters are self-contained and can be read independently and several advanced courses can be taught out of this book. An extensive list of references is given per chapter. The book’s results are expected to find applications in many areas of applied mathematics, stochastics, computer science and engineering. As such this monograph is suitable for researchers, graduate students, and seminars of the above subjects, also to be in all science and engineering libraries.
Generalized fractional Schroedinger equation with space-time fractional derivatives
International Nuclear Information System (INIS)
Wang Shaowei; Xu Mingyu
2007-01-01
In this paper the generalized fractional Schroedinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by the method of integral transforms, Fourier transform and Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function. The Green function for free particle is also presented in this paper. Finally, we discuss the relationship between the cases of the generalized fractional Schroedinger equation and the ones in standard quantum
Permutation entropy of fractional Brownian motion and fractional Gaussian noise
International Nuclear Information System (INIS)
Zunino, L.; Perez, D.G.; Martin, M.T.; Garavaglia, M.; Plastino, A.; Rosso, O.A.
2008-01-01
We have worked out theoretical curves for the permutation entropy of the fractional Brownian motion and fractional Gaussian noise by using the Bandt and Shiha [C. Bandt, F. Shiha, J. Time Ser. Anal. 28 (2007) 646] theoretical predictions for their corresponding relative frequencies. Comparisons with numerical simulations show an excellent agreement. Furthermore, the entropy-gap in the transition between these processes, observed previously via numerical results, has been here theoretically validated. Also, we have analyzed the behaviour of the permutation entropy of the fractional Gaussian noise for different time delays
Meanings for Fraction as Number-Measure by Exploring the Number Line
Psycharis, Giorgos; Latsi, Maria; Kynigos, Chronis
2009-01-01
This paper reports on a case-study design experiment in the domain of fraction as number-measure. We designed and implemented a set of exploratory tasks concerning comparison and ordering of fractions as well as operations with fractions. Two groups of 12-year-old students worked collaboratively using paper and pencil as well as a specially…
A remark on fractional differential equation involving I-function
Mishra, Jyoti
2018-02-01
The present paper deals with the solution of the fractional differential equation using the Laplace transform operator and its corresponding properties in the fractional calculus; we derive an exact solution of a complex fractional differential equation involving a special function known as I-function. The analysis of the some fractional integral with two parameters is presented using the suggested Theorem 1. In addition, some very useful corollaries are established and their proofs presented in detail. Some obtained exact solutions are depicted to see the effect of each fractional order. Owing to the wider applicability of the I-function, we can conclude that, the obtained results in our work generalize numerous well-known results obtained by specializing the parameters.
Fractional Nottale's Scale Relativity and emergence of complexified gravity
Energy Technology Data Exchange (ETDEWEB)
EL-Nabulsi, Ahmad Rami [Department of Nuclear and Energy Engineering, Cheju National University, Ara-dong 1, Jeju 690-756 (Korea, Republic of)], E-mail: nabulsiahmadrami@yahoo.fr
2009-12-15
Fractional calculus of variations has recently gained significance in studying weak dissipative and nonconservative dynamical systems ranging from classical mechanics to quantum field theories. In this paper, fractional Nottale's Scale Relativity (NSR) for an arbitrary fractal dimension is introduced within the framework of fractional action-like variational approach recently introduced by the author. The formalism is based on fractional differential operators that generalize the differential operators of conventional NSR but that reduces to the standard formalism in the integer limit. Our main aim is to build the fractional setting for the NSR dynamical equations. Many interesting consequences arise, in particular the emergence of complexified gravity and complex time.
New Approach for the Analysis of Damped Vibrations of Fractional Oscillators
Directory of Open Access Journals (Sweden)
Yuriy A. Rossikhin
2009-01-01
Full Text Available The dynamic behavior of linear and nonlinear mechanical oscillators with constitutive equations involving fractional derivatives defined as a fractional power of the operator of conventional time-derivative is considered. Such a definition of the fractional derivative enables one to analyse approximately vibratory regimes of the oscillator without considering the drift of its position of equilibrium. The assumption of small fractional derivative terms allows one to use the method of multiple time scales whereby a comparative analysis of the solutions obtained for different orders of low-level fractional derivatives and nonlinear elastic terms is possible to be carried out. The interrelationship of the fractional parameter (order of the fractional operator and nonlinearity manifests itself in full measure when orders of the small fractional derivative term and of the cubic nonlinearity entering in the oscillator's constitutive equation coincide.
Symmetric duality for left and right Riemann–Liouville and Caputo fractional differences
Directory of Open Access Journals (Sweden)
Thabet Abdeljawad
2017-07-01
Full Text Available A discrete version of the symmetric duality of Caputo–Torres, to relate left and right Riemann–Liouville and Caputo fractional differences, is considered. As a corollary, we provide an evidence to the fact that in case of right fractional differences, one has to mix between nabla and delta operators. As an application, we derive right fractional summation by parts formulas and left fractional difference Euler–Lagrange equations for discrete fractional variational problems whose Lagrangians depend on right fractional differences.
Measurement of unattached fractions in open-pit uranium mines
International Nuclear Information System (INIS)
Solomon, S.B.; Wise, K.N.
1983-01-01
A preliminary set of measurements of the unattached fraction of potential alpha energy was made at the Ranger open pit uranium uranium mine and the Nabarlek uranium mill. The measurement system, which incorporated a parallel plate diffusion battery and diffuse junction detectors, is described. Results for RaA show a wide variation in the unattached fraction. They range up to 0.76 and are higher than corresponding values for underground mining operations
Ferroelectric Fractional-Order Capacitors
Agambayev, Agamyrat
2017-07-25
Poly(vinylidene fluoride)-based polymers and their blends are used to fabricate electrostatic fractional-order capacitors. This simple but effective method allows us to precisely tune the constant phase angle of the resulting fractional-order capacitor by changing the blend composition. Additionally, we have derived an empirical relation between the ratio of the blend constituents and the constant phase angle to facilitate the design of a fractional order capacitor with a desired constant phase angle. The structural composition of the fabricated blends is investigated using Fourier transform infrared spectroscopy and X-ray diffraction techniques.
Ferroelectric Fractional-Order Capacitors
Agambayev, Agamyrat; Patole, Shashikant P.; Farhat, Mohamed; Elwakil, Ahmed; Bagci, Hakan; Salama, Khaled N.
2017-01-01
Poly(vinylidene fluoride)-based polymers and their blends are used to fabricate electrostatic fractional-order capacitors. This simple but effective method allows us to precisely tune the constant phase angle of the resulting fractional-order capacitor by changing the blend composition. Additionally, we have derived an empirical relation between the ratio of the blend constituents and the constant phase angle to facilitate the design of a fractional order capacitor with a desired constant phase angle. The structural composition of the fabricated blends is investigated using Fourier transform infrared spectroscopy and X-ray diffraction techniques.
Regularization by fractional filter methods and data smoothing
International Nuclear Information System (INIS)
Klann, E; Ramlau, R
2008-01-01
This paper is concerned with the regularization of linear ill-posed problems by a combination of data smoothing and fractional filter methods. For the data smoothing, a wavelet shrinkage denoising is applied to the noisy data with known error level δ. For the reconstruction, an approximation to the solution of the operator equation is computed from the data estimate by fractional filter methods. These fractional methods are based on the classical Tikhonov and Landweber method, but avoid, at least partially, the well-known drawback of oversmoothing. Convergence rates as well as numerical examples are presented
Use of Angle Model to Understand Addition and Subtraction of Fractions
Directory of Open Access Journals (Sweden)
Muzwangowenyu Mukwambo
2018-02-01
Full Text Available Learners in lower primary and even some in upper primary grades grapple to perform mathematical operations which involve fractions. Failure to solve these mathematical operations creates a gap in the teaching and learning processes of mathematics. We opine that this is attributed to use of traditional mathematical approaches of teaching and learning (TMATL of operations of fraction. With the hope of engaging the reformed mathematical approach of teaching and learning (RMATL this study investigated the following: How can trainee teachers use the angle model in RMATL operations of fractions? What are the perceptions of trainee teachers in the use of the angle model which engages RMATL to teach the operations of fractions? With the goal to fill the mentioned gap in which learners struggle to perform operations involving fractions, we observed and analysed worksheets on operation with fractions students wrote. Observations and interviews with trainee teachers of lower primary revealed poor performance in problems related to operations with fractions. Observed patterns supported by cognitivism revealed that invented methods or strategies on which RMATL is anchored are suitable enough to engage learner–centred teaching and learning which can prevent the abstractness of the concept of operations with fractions.
A new operational matrix of fractional order integration for the ...
Indian Academy of Sciences (India)
M H HEYDARI
1Department of Mathematics, Shiraz University of Technology, Shiraz, Iran. 2Center of ... published online 24 April 2018. Abstract. ... a realistic modeling of a physical phenomenon having dependence not only at the time instant, but also on ...
A new operational matrix of fractional order integration for the ...
Indian Academy of Sciences (India)
49
1,2The Laboratory of Quantum Information Processing and Cryptography. ... the orthogonal functions may be widely classified into four families [41]. ... segmentations, time-frequency analysis and fast algorithms for easy implementation.
Conservation laws for certain time fractional nonlinear systems of partial differential equations
Singla, Komal; Gupta, R. K.
2017-12-01
In this study, an extension of the concept of nonlinear self-adjointness and Noether operators is proposed for calculating conserved vectors of the time fractional nonlinear systems of partial differential equations. In our recent work (J Math Phys 2016; 57: 101504), by proposing the symmetry approach for time fractional systems, the Lie symmetries for some fractional nonlinear systems have been derived. In this paper, the obtained infinitesimal generators are used to find conservation laws for the corresponding fractional systems.
Simulation and optimization of fractional crystallization processes
DEFF Research Database (Denmark)
Thomsen, Kaj; Rasmussen, Peter; Gani, Rafiqul
1998-01-01
A general method for the calculation of various types of phase diagrams for aqueous electrolyte mixtures is outlined. It is shown how the thermodynamic equilibrium precipitation process can be used to satisfy the operational needs of industrial crystallizer/centrifuge units. Examples of simulation...... and optimization of fractional crystallization processes are shown. In one of these examples, a process with multiple steady states is analyzed. The thermodynamic model applied for describing the highly non-ideal aqueous electrolyte systems is the Extended UNIQUAC model. (C) 1998 Published by Elsevier Science Ltd...
Contextual Fraction as a Measure of Contextuality
Abramsky, Samson; Barbosa, Rui Soares; Mansfield, Shane
2017-08-01
We consider the contextual fraction as a quantitative measure of contextuality of empirical models, i.e., tables of probabilities of measurement outcomes in an experimental scenario. It provides a general way to compare the degree of contextuality across measurement scenarios; it bears a precise relationship to violations of Bell inequalities; its value, and a witnessing inequality, can be computed using linear programing; it is monotonic with respect to the "free" operations of a resource theory for contextuality; and it measures quantifiable advantages in informatic tasks, such as games and a form of measurement-based quantum computing.
Energy Technology Data Exchange (ETDEWEB)
Vilppunen, P.; Aaltonen, H.; Sohlo, J. [Oulu Univ. (Finland). Dept. of Process Engineering
1997-12-01
Separation processes for energy and fibre fractions, predominantly those for seed flax, using traditional pulp classifiers and the new pressure classifier process were studied in the wet-separation part of the project. A combined plant fibre further-refining process, based on mechanical and biotechnical separation, operating on the basis of fibre length, was developed on the basis of dry and wet fraction tests. (orig.)
On generalized fractional vibration equation
International Nuclear Information System (INIS)
Dai, Hongzhe; Zheng, Zhibao; Wang, Wei
2017-01-01
Highlights: • The paper presents a generalized fractional vibration equation for arbitrary viscoelastically damped system. • Some classical vibration equations can be derived from the developed equation. • The analytic solution of developed equation is derived under some special cases. • The generalized equation is particularly useful for developing new fractional equivalent linearization method. - Abstract: In this paper, a generalized fractional vibration equation with multi-terms of fractional dissipation is developed to describe the dynamical response of an arbitrary viscoelastically damped system. It is shown that many classical equations of motion, e.g., the Bagley–Torvik equation, can be derived from the developed equation. The Laplace transform is utilized to solve the generalized equation and the analytic solution under some special cases is derived. Example demonstrates the generalized transfer function of an arbitrary viscoelastic system.
Physcicists rewarded for 'fractional electrons'
Ball, P
1998-01-01
The 1998 Nobel prize for physics has been awarded to Horst Stormer, Daniel Tsui and Robert Laughlin.Stormer and Tsui were the first to observe the fractional quantum Hall effect and Laughlin provided the theory shortly afterwards (1 page).
Ultracentrifugation for ultrafine nanodiamond fractionation
Koniakhin, S. V.; Besedina, N. A.; Kirilenko, D. A.; Shvidchenko, A. V.; Eidelman, E. D.
2018-01-01
In this paper we propose a method for ultrafine fractionation of nanodiamonds using the differential centrifugation in the fields up to 215000g. The developed protocols yield 4-6 nm fraction giving main contribution to the light scattering intensity. The desired 4-6 nm fraction can be obtained from various types of initial nanodiamonds: three types of detonation nanodiamonds differing in purifying methods, laser synthesis nanodiamonds and nanodiamonds made by milling. The characterization of the obtained hydrosols was conducted with Dynamic Light Scattering, Zeta potential measurements, powder XRD and TEM. According to powder XRD and TEM data ultracentrifugation also leads to a further fractionation of the primary diamond nanocrystallites in the hydrosols from 4 to 2 nm.
Commercial SNF Accident Release Fractions
Energy Technology Data Exchange (ETDEWEB)
J. Schulz
2004-11-05
The purpose of this analysis is to specify and document the total and respirable fractions for radioactive materials that could be potentially released from an accident at the repository involving commercial spent nuclear fuel (SNF) in a dry environment. The total and respirable release fractions are used to support the preclosure licensing basis for the repository. The total release fraction is defined as the fraction of total commercial SNF assembly inventory, typically expressed as an activity inventory (e.g., curies), of a given radionuclide that is released to the environment from a waste form. Radionuclides are released from the inside of breached fuel rods (or pins) and from the detachment of radioactive material (crud) from the outside surfaces of fuel rods and other components of fuel assemblies. The total release fraction accounts for several mechanisms that tend to retain, retard, or diminish the amount of radionuclides that are available for transport to dose receptors or otherwise can be shown to reduce exposure of receptors to radiological releases. The total release fraction includes a fraction of airborne material that is respirable and could result in inhalation doses; this subset of the total release fraction is referred to as the respirable release fraction. Accidents may involve waste forms characterized as: (1) bare unconfined intact fuel assemblies, (2) confined intact fuel assemblies, or (3) canistered failed commercial SNF. Confined intact commercial SNF assemblies at the repository are contained in shipping casks, canisters, or waste packages. Four categories of failed commercial SNF are identified: (1) mechanically and cladding-penetration damaged commercial SNF, (2) consolidated/reconstituted assemblies, (3) fuel rods, pieces, and debris, and (4) nonfuel components. It is assumed that failed commercial SNF is placed into waste packages with a mesh screen at each end (CRWMS M&O 1999). In contrast to bare unconfined fuel assemblies, the
Commercial SNF Accident Release Fractions
International Nuclear Information System (INIS)
Schulz, J.
2004-01-01
The purpose of this analysis is to specify and document the total and respirable fractions for radioactive materials that could be potentially released from an accident at the repository involving commercial spent nuclear fuel (SNF) in a dry environment. The total and respirable release fractions are used to support the preclosure licensing basis for the repository. The total release fraction is defined as the fraction of total commercial SNF assembly inventory, typically expressed as an activity inventory (e.g., curies), of a given radionuclide that is released to the environment from a waste form. Radionuclides are released from the inside of breached fuel rods (or pins) and from the detachment of radioactive material (crud) from the outside surfaces of fuel rods and other components of fuel assemblies. The total release fraction accounts for several mechanisms that tend to retain, retard, or diminish the amount of radionuclides that are available for transport to dose receptors or otherwise can be shown to reduce exposure of receptors to radiological releases. The total release fraction includes a fraction of airborne material that is respirable and could result in inhalation doses; this subset of the total release fraction is referred to as the respirable release fraction. Accidents may involve waste forms characterized as: (1) bare unconfined intact fuel assemblies, (2) confined intact fuel assemblies, or (3) canistered failed commercial SNF. Confined intact commercial SNF assemblies at the repository are contained in shipping casks, canisters, or waste packages. Four categories of failed commercial SNF are identified: (1) mechanically and cladding-penetration damaged commercial SNF, (2) consolidated/reconstituted assemblies, (3) fuel rods, pieces, and debris, and (4) nonfuel components. It is assumed that failed commercial SNF is placed into waste packages with a mesh screen at each end (CRWMS M andO 1999). In contrast to bare unconfined fuel assemblies, the
Jain, Sonal
2018-01-01
In this paper, we aim to use the alternative numerical scheme given by Gnitchogna and Atangana for solving partial differential equations with integer and non-integer differential operators. We applied this method to fractional diffusion model and fractional Buckmaster models with non-local fading memory. The method yields a powerful numerical algorithm for fractional order derivative to implement. Also we present in detail the stability analysis of the numerical method for solving the diffusion equation. This proof shows that this method is very stable and also converges very quickly to exact solution and finally some numerical simulation is presented.
Fractional Reserve Banking: Some Quibbles
Bagus, Philipp; Howden, David
2010-01-01
We explore several unaddressed issues in George Selgin’s (1988) claim that the best monetary system to maintain monetary equilibrium is a fractional reserve free banking one. The claim that adverse clearing balances would limit credit expansion in a fractional reserve free banking system is more troublesome than previously reckoned. Both lengthened clearing periods and interbank agreements render credit expansion unrestrained. “The theory of free banking” confuses increases in money held with...
Intelligent fractions learning system: implementation
CSIR Research Space (South Africa)
Smith, Andrew C
2011-05-01
Full Text Available Conference Proceedings Paul Cunningham and Miriam Cunningham (Eds) IIMC International Information Management Corporation, 2011 ISBN: 978-1-905824-24-3 An Intelligent Fractions Learning System: Implementation Andrew Cyrus SMITH1, Teemu H. LAINE2 1CSIR... to fractions. Our aim with the current research project is to extend the existing UFractions learning system to incorporate automatic data capturing. ?Intelligent UFractions? allows a teacher to remotely monitor the children?s progress during...
Xenon fractionation in porous planetesimals
Zahnle, Kevin; Pollack, James B.; Kasting, James F.
1990-01-01
The distinctively fractionated Xe on Mars and earth may have its root in a common source from which both planets accreted. Beginning with Ozima and Nakazawa's (1980) hypothesis that terrestrial Xe fractionation was caused by gravitational separation of adsorbed solar nebular gases inside large porous planetesimals, it is pointed out that Xe would have been trapped as the planetesimal grew and pores were squeezed shut by lithostatic pressure. It is shown that enough fractionated Xe to supply the earth could have been trapped this way. The degree of fractionation is controlled by the lithostatic pressure at the pore-closing front and so would have been roughly the same for all large planetesimals. The predicted degree of fractionation agrees well with that preserved in terrestrial and Martian Xe. Relative to Xe, this source is strongly depleted in other noble gases. In contrast to the original Ozima and Nakazawa hypothesis, the present hypothesis predicts the observed fractionation, and it allows planetary accretion to occur after the dissipation of the solar nebula.
Fractional Charge Definitions and Conditions
Energy Technology Data Exchange (ETDEWEB)
Goldhaber, A.S.
2004-06-04
Fractional charge is known through theoretical and experimental discoveries of isolable objects carrying fractions of familiar charge units--electric charge Q, spin S, and the difference of baryon and lepton numbers B-L. With a few simple assumptions all these effects may be described using a generalized version of charge renormalization for locally conserved charges, in which medium correlations yield familiar adiabatic, continuous renormalization, or sometimes nonadiabatic, discrete renormalization. Fractional charges may be carried by fundamental particles or fundamental solitons. Either picture works for the simplest fractional-quantum-Hall-effect quasiholes, though the particle description is far more general. The only known fundamental solitons in three or fewer space dimensions d are the kink (d = 1), the vortex (d = 2), and the magnetic monopole (d = 3). Further, for a charge not intrinsically coupled to the topological charge of a soliton, only the kink and the monopole may carry fractional values. The same reasoning enforces fractional values of B-L for electrically charged elementary particles.
Fractional Charge Definitions and Conditions
International Nuclear Information System (INIS)
Goldhaber, A.S.
2004-01-01
Fractional charge is known through theoretical and experimental discoveries of isolable objects carrying fractions of familiar charge units--electric charge Q, spin S, and the difference of baryon and lepton numbers B-L. With a few simple assumptions all these effects may be described using a generalized version of charge renormalization for locally conserved charges, in which medium correlations yield familiar adiabatic, continuous renormalization, or sometimes nonadiabatic, discrete renormalization. Fractional charges may be carried by fundamental particles or fundamental solitons. Either picture works for the simplest fractional-quantum-Hall-effect quasiholes, though the particle description is far more general. The only known fundamental solitons in three or fewer space dimensions d are the kink (d = 1), the vortex (d = 2), and the magnetic monopole (d = 3). Further, for a charge not intrinsically coupled to the topological charge of a soliton, only the kink and the monopole may carry fractional values. The same reasoning enforces fractional values of B-L for electrically charged elementary particles
Xenon fractionation in porous planetesimals
International Nuclear Information System (INIS)
Zahnle, K.; Pollack, J.B.; Kasting, J.F.
1990-01-01
The distinctively fractionated Xe on Mars and Earth may have its root in a common source from which both planets accreted. We begin with Ozima and Nakazawa's hypothesis that terrestrial Xe fractionation was caused by gravitational separation of adsorbed solar nebular gases inside large porous planetesimals. We point out that Xe would have been trapped as the planetesimal grew and pores were squeezed shut by lithostatic pressure. We show that enough fractionated Xe to supply the Earth could have been trapped this way. The degree of fractionation is controlled by the lithostatic pressure at the pore-closing front and so would have been roughly the same for all large planetesimals. The predicted degree of fractionation agrees well with that preserved in terrestrial and martian Xe. Relative to Xe, this source is strongly depleted in other noble gases. In contrast to the original Ozima and Nakazawa hypothesis, our hypothesis predicts the observed fractionation, and it allows planetary accretion to occur after the dissipation of the solar nebula. The required planetesimals are large, representing a class of object now extinct in the solar system
Implication of fractionated dose exposures in therapeutic gain
International Nuclear Information System (INIS)
Kim, Hye-Jin; Lee, Min-Ho; Kim, Eun-Hee
2016-01-01
Radiation therapy pursues killing tumor cells while sparing normal cells from the radiation exposure. Stereotactic radiosurgery (SRS) is a cancer treatment modality that delivers a high dose in a single operation. This high-dose single operation shortens the treatment course, but can increase the risk of normal cell damage. Normal cell damage can be reduced by employing multi-directional exposures for an increasing number of isocenters. In this study, we investigated whether therapeutic benefits would be expected by employing new dose fractionation patterns at a high-dose single operation. The conventional single-dose operation in brain tumor radiosurgery is performed by delivering fractionated uniform doses. According to Figs. 2 and 3, the conventional radiosurgery might have obtained some therapeutic benefit by employing the fractionated uniform-dose exposures instead of a single-dose exposure. We suggest that further therapeutic gain be expected by employing the fractionated radiation exposures in an increasing dose pattern. Until ensuring our suggestion, the significance in gain of cell surviving should be verified for all three dose patterns with both normal and tumor cells. The investigation whether normal and tumor cells show the same responses to the fractionated dose exposures at lower and higher than 15 Gy of total dose is also reserved for future work
Directory of Open Access Journals (Sweden)
Ariyadi Wijaya
2017-11-01
Full Text Available This paper reports an exploration into Indonesian fourth graders’ difficulties in fractions and their relation to the opportunity to learn fractions students got at schools. The concept of ‘opportunity to learn’ is often considered as a framework to investigate possible reasons for students’ difficulties. The data for this study was drawn from TIMSS 2015 that comprised test results and teachers’ responses to TIMSS Teacher Questionnaire. The test and questionnaire data were analysed by using descriptive statistics. In addition to test and questionnaire, this study also included an analysis of Indonesian textbooks in order to get a broader scope of the opportunity to learn. Qualitative approach was used to analyse the textbooks. The analysis of the TIMSS results shows Indonesian students’ low conceptual understanding of fractions. Three possible reasons for students’ low conceptual understanding were revealed. First, the content of Indonesian curriculum that gave low emphasis on basic concepts of fractions and introduced operations of fractions too early. Second, the Indonesian mathematics textbooks only presented one definition of fractions, i.e. fractions as parts of wholes. Third, there is a limited use of models or representations of fractions in the classroom practices.
Fractional statistics in 2+1 dimensions through the Gaussian model
International Nuclear Information System (INIS)
Murthy, G.
1986-01-01
The free massless field in 2+1 dimensions is written as an ''integral'' over free massless fields in 1+1 dimensions. Taking the operators with fractional dimension in the Gaussian model as a springboard we construct operators with fractional statistics in the former theory
Fractional Diffusion Equations and Anomalous Diffusion
Evangelista, Luiz Roberto; Kaminski Lenzi, Ervin
2018-01-01
Preface; 1. Mathematical preliminaries; 2. A survey of the fractional calculus; 3. From normal to anomalous diffusion; 4. Fractional diffusion equations: elementary applications; 5. Fractional diffusion equations: surface effects; 6. Fractional nonlinear diffusion equation; 7. Anomalous diffusion: anisotropic case; 8. Fractional Schrödinger equations; 9. Anomalous diffusion and impedance spectroscopy; 10. The Poisson–Nernst–Planck anomalous (PNPA) models; References; Index.
Conformable Fractional Bessel Equation and Bessel Functions
Gökdoğan, Ahmet; Ünal, Emrah; Çelik, Ercan
2015-01-01
In this work, we study the fractional power series solutions around regular singular point x=0 of conformable fractional Bessel differential equation and fractional Bessel functions. Then, we compare fractional solutions with ordinary solutions. In addition, we present certain property of fractional Bessel functions.
Can a sponge fractionate isotopes?
Patel, B; Patel, S; Balani, M C
1985-03-22
The study has unequivocally demonstrated that siliceous sponges Spirastrella cuspidifera and Prostylyssa foetida from the same microecological niche exhibit a high degree of species specificity, while accumulating a host of heavy metal ions (Ni, Cr, Cd, Sn, Ti, Mo, Zr). S. cuspidifera accumulated, in addition, 60Co and 63Ni, showing discrimination against other radionuclides, 137Cs and 131I, present in the ambient waters receiving controlled low level waste discharges from a B.W.R. nuclear power station. P. foetida, on the other hand, accumulated only 131I and showed discrimination against other radionuclides including 60Co, although the stable iodine concentrations in both the sponges were the same. The specific activity of 60Co (in becquerels per gram of 59Co) in S. cuspidifera and 131I (in becquerels per gram of 127I) in P. foetida were at least two orders of magnitude greater than in the ambient sea water. That of 63Ni (in becquerels per gram of 62Ni) in S. cuspidifera, on the other hand, was lower by two orders of magnitude than in either abiotic matrices from the same environment. Thus, not only did both the species show bioaccumulation of a specific element, but also preferential uptake of isotopes of the same element, though they were equally available for intake. Such differential uptake of isotopes can possibly be explained in terms of two quite different mechanisms operating, each applicable in a particular case. One is that the xenobiotic isotope enters the environment in a physicochemical form or as a complex different from that of its natural counterpart. If equilibration with the latter is slow, so that the organism acquires the xenobiotic in an unfamiliar chemical context, it may treat it as a chemically distinct entity so that its concentration factor differs from that of stable isotope, thus changing the specific activity. Alternatively, if the xenobiotic is present in the same chemical form as the stable isotope, the only way in which specific
M. L. Kavvas; T. Tu; A. Ercan; J. Polsinelli
2017-01-01
Using fractional calculus, a dimensionally consistent governing equation of transient, saturated groundwater flow in fractional time in a multi-fractional confined aquifer is developed. First, a dimensionally consistent continuity equation for transient saturated groundwater flow in fractional time and in a multi-fractional, multidimensional confined aquifer is developed. For the equation of water flux within a multi-fractional multidimensional confined aquifer, a dimensionally...
Void fraction instrument acceptance test procedure
International Nuclear Information System (INIS)
Stokes, T.I.; Pearce, K.L.
1994-01-01
This document presents the results of the acceptance test for the mechanical and electrical features (not specifically addressed by the software ATP) of the void fraction instrument (VFI). Acceptance testing of the VFI, control console, and decontamination spray assembly was conducted in the 306E building high bay and area adjacent to the facility. The VFI was tested in the horizontal position supported in multiple locations on rolling tables. The control console was located next to the VFI pneumatic control assembly. The VFI system was operated exactly as is expected in the tank farm, with the following exceptions: power was provided from a building outlet and the VFI was horizontal. The testing described in this document verifies that the mechanical and electrical features are operating as designed and that the unit is ready for field service
Fractional charge definitions and conditions
International Nuclear Information System (INIS)
Goldhaber, Alfred Scharff
2003-01-01
The phenomenon of fractional charge has come to prominence in recent decades through theoretical and experimental discoveries of isolable objects which carry fractions of familiar charge units--electric charge Q, spin S, baryon number B and lepton number L. It is shown here on the basis of a few simple assumptions that all these effects may be described using a generalized version of charge renormalization for locally conserved charges, in which many-body correlations can produce familiar adiabatic, continuous renormalization, and in some circumstances nonadiabatic, discrete renormalization. The fractional charges may be carried either by fundamental particles or by fundamental solitons. This excludes nontopological solitons and also skyrmions: The only known fundamental solitons in three or fewer space dimensions d are the kink (d=1), the vortex (d=2), and the magnetic monopole (d=3). Further, for a charge which is not intrinsically coupled to the topological charge of a soliton, only the kink and the monopole may carry fractional values. The same reasoning enforces fractional local values of B-L for electrically charged elementary particles
REFractions: The Representing Equivalent Fractions Game
Tucker, Stephen I.
2014-01-01
Stephen Tucker presents a fractions game that addresses a range of fraction concepts including equivalence and computation. The REFractions game also improves students' fluency with representing, comparing and adding fractions.
dimensional generalised time-fractional Hirota equation
Indian Academy of Sciences (India)
Youwei Zhang
2018-02-09
Feb 9, 2018 ... Fractional calculus has attracted much attention in ... cally proved that the fractional calculus theory is non- ... calculus and various definitions of fractional integration .... basic features of the tanh-expansion are outlined as.
Generalized Multiparameters Fractional Variational Calculus
Directory of Open Access Journals (Sweden)
Om Prakash Agrawal
2012-01-01
Full Text Available This paper builds upon our recent paper on generalized fractional variational calculus (FVC. Here, we briefly review some of the fractional derivatives (FDs that we considered in the past to develop FVC. We first introduce new one parameter generalized fractional derivatives (GFDs which depend on two functions, and show that many of the one-parameter FDs considered in the past are special cases of the proposed GFDs. We develop several parts of FVC in terms of one parameter GFDs. We point out how many other parts could be developed using the properties of the one-parameter GFDs. Subsequently, we introduce two new two- and three-parameter GFDs. We introduce some of their properties, and discuss how they can be used to develop FVC. In addition, we indicate how these formulations could be used in various fields, and how the generalizations presented here can be further extended.
Semi-infinite fractional programming
Verma, Ram U
2017-01-01
This book presents a smooth and unified transitional framework from generalised fractional programming, with a finite number of variables and a finite number of constraints, to semi-infinite fractional programming, where a number of variables are finite but with infinite constraints. It focuses on empowering graduate students, faculty and other research enthusiasts to pursue more accelerated research advances with significant interdisciplinary applications without borders. In terms of developing general frameworks for theoretical foundations and real-world applications, it discusses a number of new classes of generalised second-order invex functions and second-order univex functions, new sets of second-order necessary optimality conditions, second-order sufficient optimality conditions, and second-order duality models for establishing numerous duality theorems for discrete minmax (or maxmin) semi-infinite fractional programming problems. In the current interdisciplinary supercomputer-oriented research envi...
A componential view of children's difficulties in learning fractions
Gabriel, Florence; Coché, Frédéric; Szucs, Dénes; Carette, Vincent; Rey, Bernard; Content, Alain
2013-01-01
Fractions are well known to be difficult to learn. Various hypotheses have been proposed in order to explain those difficulties: fractions can denote different concepts; their understanding requires a conceptual reorganization with regard to natural numbers; and using fractions involves the articulation of conceptual knowledge with complex manipulation of procedures. In order to encompass the major aspects of knowledge about fractions, we propose to distinguish between conceptual and procedural knowledge. We designed a test aimed at assessing the main components of fraction knowledge. The test was carried out by fourth-, fifth- and sixth-graders from the French Community of Belgium. The results showed large differences between categories. Pupils seemed to master the part-whole concept, whereas numbers and operations posed problems. Moreover, pupils seemed to apply procedures they do not fully understand. Our results offer further directions to explain why fractions are amongst the most difficult mathematical topics in primary education. This study offers a number of recommendations on how to teach fractions. PMID:24133471
A componential view of children's difficulties in learning fractions.
Gabriel, Florence; Coché, Frédéric; Szucs, Dénes; Carette, Vincent; Rey, Bernard; Content, Alain
2013-01-01
Fractions are well known to be difficult to learn. Various hypotheses have been proposed in order to explain those difficulties: fractions can denote different concepts; their understanding requires a conceptual reorganization with regard to natural numbers; and using fractions involves the articulation of conceptual knowledge with complex manipulation of procedures. In order to encompass the major aspects of knowledge about fractions, we propose to distinguish between conceptual and procedural knowledge. We designed a test aimed at assessing the main components of fraction knowledge. The test was carried out by fourth-, fifth- and sixth-graders from the French Community of Belgium. The results showed large differences between categories. Pupils seemed to master the part-whole concept, whereas numbers and operations posed problems. Moreover, pupils seemed to apply procedures they do not fully understand. Our results offer further directions to explain why fractions are amongst the most difficult mathematical topics in primary education. This study offers a number of recommendations on how to teach fractions.
Some applications of the fractional Poisson probability distribution
International Nuclear Information System (INIS)
Laskin, Nick
2009-01-01
Physical and mathematical applications of the recently invented fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states has been introduced and studied. As mathematical applications, we have developed the fractional generalization of Bell polynomials, Bell numbers, and Stirling numbers of the second kind. The appearance of fractional Bell polynomials is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states. Fractional Stirling numbers of the second kind have been introduced and applied to evaluate the skewness and kurtosis of the fractional Poisson probability distribution function. A representation of the Bernoulli numbers in terms of fractional Stirling numbers of the second kind has been found. In the limit case when the fractional Poisson probability distribution becomes the Poisson probability distribution, all of the above listed developments and implementations turn into the well-known results of the quantum optics and the theory of combinatorial numbers.
On a Fractional Binomial Process
Cahoy, Dexter O.; Polito, Federico
2012-02-01
The classical binomial process has been studied by Jakeman (J. Phys. A 23:2815-2825, 1990) (and the references therein) and has been used to characterize a series of radiation states in quantum optics. In particular, he studied a classical birth-death process where the chance of birth is proportional to the difference between a larger fixed number and the number of individuals present. It is shown that at large times, an equilibrium is reached which follows a binomial process. In this paper, the classical binomial process is generalized using the techniques of fractional calculus and is called the fractional binomial process. The fractional binomial process is shown to preserve the binomial limit at large times while expanding the class of models that include non-binomial fluctuations (non-Markovian) at regular and small times. As a direct consequence, the generality of the fractional binomial model makes the proposed model more desirable than its classical counterpart in describing real physical processes. More statistical properties are also derived.
Riesz potential versus fractional Laplacian
Ortigueira, Manuel Duarte
2014-09-01
This paper starts by introducing the Grünwald-Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy-Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.
Geodesic continued fractions and LLL
Beukers, F
2014-01-01
We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to dd real numbers α1,…,αdα1,…,αd. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter tt as t↓0t↓0.
A graph with fractional revival
Bernard, Pierre-Antoine; Chan, Ada; Loranger, Érika; Tamon, Christino; Vinet, Luc
2018-02-01
An example of a graph that admits balanced fractional revival between antipodes is presented. It is obtained by establishing the correspondence between the quantum walk on a hypercube where the opposite vertices across the diagonals of each face are connected and, the coherent transport of single excitations in the extension of the Krawtchouk spin chain with next-to-nearest neighbour interactions.
Riesz potential versus fractional Laplacian
Ortigueira, Manuel Duarte; Laleg-Kirati, Taous-Meriem; Machado, José Antó nio Tenreiro
2014-01-01
This paper starts by introducing the Grünwald-Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy-Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.
What next in fractionated radiotherapy
International Nuclear Information System (INIS)
Fowler, J.F.
1984-01-01
Trends in models for predicting the total dose required to produce tolerable normal-tissue injury can be seen by the progression from the ''cube root law'', through Strandqvist's slope of 0.22, to NSD, TDF and CRE which have separate time and fraction number exponents, to even better approximations now available. The dose-response formulae that can be used to define the effect of fraction size (and number) include (1) the linear quadratic (LQ) model (2) the two-component (TC) multi-target model and (3) repair-misrepair models. The LQ model offers considerable convenience, requires only two parameters to be determined, and emphasizes the difference between late and early normal-tissue dependence on dose per fraction first shown by exponents greater than the NSD slope of 0.24. Exponents of overall time, e.g. Tsup(0.11), yield the wrong shape of time curve, suggesting that most proliferating occurs early, although it really occurs after a delay depending on the turnover time of the tissue. Improved clinical results are being sought by hyperfractionation, accelerated fractionation, or continuous low dose rate irradiation as in interstitial implants. (U.K.)
Fractional Laplace Transforms - A Perspective
Directory of Open Access Journals (Sweden)
Rudolf A. Treumann
2014-06-01
Full Text Available A new form of the Laplace transform is reviewed as a paradigm for an entire class of fractional functional transforms. Various of its properties are discussed. Such transformations should be useful in application to differential/integral equations or problems in non-extensive statistical mechanics.
Pythagorean Approximations and Continued Fractions
Peralta, Javier
2008-01-01
In this article, we will show that the Pythagorean approximations of [the square root of] 2 coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers…
Fractional Processes and Fractional-Order Signal Processing Techniques and Applications
Sheng, Hu; Qiu, TianShuang
2012-01-01
Fractional processes are widely found in science, technology and engineering systems. In Fractional Processes and Fractional-order Signal Processing, some complex random signals, characterized by the presence of a heavy-tailed distribution or non-negligible dependence between distant observations (local and long memory), are introduced and examined from the ‘fractional’ perspective using simulation, fractional-order modeling and filtering and realization of fractional-order systems. These fractional-order signal processing (FOSP) techniques are based on fractional calculus, the fractional Fourier transform and fractional lower-order moments. Fractional Processes and Fractional-order Signal Processing: • presents fractional processes of fixed, variable and distributed order studied as the output of fractional-order differential systems; • introduces FOSP techniques and the fractional signals and fractional systems point of view; • details real-world-application examples of FOSP techniques to demonstr...
Directory of Open Access Journals (Sweden)
M. L. Kavvas
2017-10-01
Full Text Available Using fractional calculus, a dimensionally consistent governing equation of transient, saturated groundwater flow in fractional time in a multi-fractional confined aquifer is developed. First, a dimensionally consistent continuity equation for transient saturated groundwater flow in fractional time and in a multi-fractional, multidimensional confined aquifer is developed. For the equation of water flux within a multi-fractional multidimensional confined aquifer, a dimensionally consistent equation is also developed. The governing equation of transient saturated groundwater flow in a multi-fractional, multidimensional confined aquifer in fractional time is then obtained by combining the fractional continuity and water flux equations. To illustrate the capability of the proposed governing equation of groundwater flow in a confined aquifer, a numerical application of the fractional governing equation to a confined aquifer groundwater flow problem was also performed.
Directory of Open Access Journals (Sweden)
Pooja Gupta Sidney
2017-07-01
Full Text Available When children learn about fractions, their prior knowledge of whole numbers often interferes, resulting in a whole number bias. However, many fraction concepts are generalizations of analogous whole number concepts; for example, fraction division and whole number division share a similar conceptual structure. Drawing on past studies of analogical transfer, we hypothesize that children’s whole number division knowledge will support their understanding of fraction division when their relevant prior knowledge is activated immediately before engaging with fraction division. Children in 5th and 6th grade modeled fraction division with physical objects after modeling a series of addition, subtraction, multiplication, and division problems with whole number operands and fraction operands. In one condition, problems were blocked by operation, such that children modeled fraction problems immediately after analogous whole number problems (e.g., fraction division problems followed whole number division problems. In another condition, problems were blocked by number type, such that children modeled all four arithmetic operations with whole numbers in the first block, and then operations with fractions in the second block. Children who solved whole number division problems immediately before fraction division problems were significantly better at modeling the conceptual structure of fraction division than those who solved all of the fraction problems together. Thus, implicit analogies across shared concepts can affect children’s mathematical thinking. Moreover, specific analogies between whole number and fraction concepts can yield a positive, rather than a negative, whole number bias.
Zeng, Shengda; Migórski, Stanisław
2018-03-01
In this paper a class of elliptic hemivariational inequalities involving the time-fractional order integral operator is investigated. Exploiting the Rothe method and using the surjectivity of multivalued pseudomonotone operators, a result on existence of solution to the problem is established. Then, this abstract result is applied to provide a theorem on the weak solvability of a fractional viscoelastic contact problem. The process is quasistatic and the constitutive relation is modeled with the fractional Kelvin-Voigt law. The friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals. The variational formulation of this problem leads to a fractional hemivariational inequality.
A conformal field theory description of fractional quantum Hall states
Ardonne, E.
2002-01-01
In this thesis, we give a description of fractional quantum Hall states in terms of conformal field theory (CFT). As was known for a long time, the Laughlin states could be written in terms of correlators of chiral vertex operators of a c=1 CFT. It was shown by G. Moore and N. Read that more general
A novel fractional sliding mode control configuration for ...
Indian Academy of Sciences (India)
Karima Rabah
2017-09-09
Sep 9, 2017 ... for synchronizing disturbed fractional-order chaotic systems. KARIMA RABAH1, SAMIR ... duced to realize chaos synchronization, such as PC control [19] .... where t0 is the initial time, α ∈ (0,1), λ > 0, b > 0, m(0) = 0, m(x) ≥ 0 .... where k = k2/k1 is a positive real number. Choosing .... operating conditions.
Regularity of spectral fractional Dirichlet and Neumann problems
DEFF Research Database (Denmark)
Grubb, Gerd
2016-01-01
Consider the fractional powers and of the Dirichlet and Neumann realizations of a second-order strongly elliptic differential operator A on a smooth bounded subset Ω of . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in ...
Generalized time fractional IHCP with Caputo fractional derivatives
International Nuclear Information System (INIS)
Murio, D A; MejIa, C E
2008-01-01
The numerical solution of the generalized time fractional inverse heat conduction problem (GTFIHCP) on a finite slab is investigated in the presence of measured (noisy) data when the time fractional derivative is interpreted in the sense of Caputo. The GTFIHCP involves the simultaneous identification of the heat flux and temperature transient functions at one of the boundaries of the finite slab together with the initial condition of the original direct problem from noisy Cauchy data at a discrete set of points on the opposite (active) boundary. A finite difference space marching scheme with adaptive regularization, using trigonometric mollification techniques and generalized cross validation is introduced. Error estimates for the numerical solution of the mollified problem and numerical examples are provided.
Use of Angle Model to Understand Addition and Subtraction of Fractions
Muzwangowenyu Mukwambo; Kenneth Ngcoza; Lineo Florence Ramasike
2018-01-01
Learners in lower primary and even some in upper primary grades grapple to perform mathematical operations which involve fractions. Failure to solve these mathematical operations creates a gap in the teaching and learning processes of mathematics. We opine that this is attributed to use of traditional mathematical approaches of teaching and learning (TMATL) of operations of fraction. With the hope of engaging the reformed mathematical approach of teaching and learning (RMATL) this study inves...
Some comparison of two fractional oscillators
International Nuclear Information System (INIS)
Kang Yonggang; Zhang Xiu'e
2010-01-01
The other form of fractional oscillator equation comparing to the widely discussed one is ushered in. The properties of vibration of two fractional oscillators are discussed under the influence of different initial conditions. The interpretation of the characteristics of the fractional oscillators using different method is illustrated. Based on two fractional oscillator equations, two linked bodies and the continuous system are studied.
9 CFR 113.7 - Multiple fractions.
2010-01-01
... 9 Animals and Animal Products 1 2010-01-01 2010-01-01 false Multiple fractions. 113.7 Section 113... § 113.7 Multiple fractions. (a) When a biological product contains more than one immunogenic fraction, the completed product shall be evaluated by tests applicable to each fraction. (b) When similar...
A fractional Dirac equation and its solution
International Nuclear Information System (INIS)
Muslih, Sami I; Agrawal, Om P; Baleanu, Dumitru
2010-01-01
This paper presents a fractional Dirac equation and its solution. The fractional Dirac equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange equations of motion. We present a Lagrangian and a Hamiltonian for the fractional Dirac equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Dirac equation which is the same as that obtained using the fractional variational principle. Eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard Dirac equation. We also provide expressions for the path integral quantization for the fractional Dirac field which, in the limit α → 1, approaches to the path integral for the regular Dirac field. It is hoped that the fractional Dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics.
On the fractional calculus of Besicovitch function
International Nuclear Information System (INIS)
Liang Yongshun
2009-01-01
Relationship between fractional calculus and fractal functions has been explored. Based on prior investigations dealing with certain fractal functions, fractal dimensions including Hausdorff dimension, Box dimension, K-dimension and Packing dimension is shown to be a linear function of order of fractional calculus. Both Riemann-Liouville fractional calculus and Weyl-Marchaud fractional derivative of Besicovitch function have been discussed.
12 CFR 5.67 - Fractional shares.
2010-01-01
... connection with fractional shares, a national bank issuing additional stock by stock dividend, upon... fair price upon the fraction not being issued through its sale, or the purchase of the additional... stock; (c) Remit the cash equivalent of the fraction not being issued to those to whom fractional shares...
Conceptual Knowledge of Fraction Arithmetic
Siegler, Robert S.; Lortie-Forgues, Hugues
2015-01-01
Understanding an arithmetic operation implies, at minimum, knowing the direction of effects that the operation produces. However, many children and adults, even those who execute arithmetic procedures correctly, may lack this knowledge on some operations and types of numbers. To test this hypothesis, we presented preservice teachers (Study 1),…
Fractional vector calculus and fluid mechanics
Lazopoulos, Konstantinos A.; Lazopoulos, Anastasios K.
2017-04-01
Basic fluid mechanics equations are studied and revised under the prism of fractional continuum mechanics (FCM), a very promising research field that satisfies both experimental and theoretical demands. The geometry of the fractional differential has been clarified corrected and the geometry of the fractional tangent spaces of a manifold has been studied in Lazopoulos and Lazopoulos (Lazopoulos KA, Lazopoulos AK. Progr. Fract. Differ. Appl. 2016, 2, 85-104), providing the bases of the missing fractional differential geometry. Therefore, a lot can be contributed to fractional hydrodynamics: the basic fractional fluid equations (Navier Stokes, Euler and Bernoulli) are derived and fractional Darcy's flow in porous media is studied.
FRACTIONATION AND CHARACTERISATION OF TECHNICAL AMMONIUM LIGNOSULPHONATE
Directory of Open Access Journals (Sweden)
Cheryl Ann Leger
2010-08-01
Full Text Available It is difficult to use lignin in any analytical methodology without reducing its considerable polydispersity by fractionation. An ammonium lignosulphonate sample was fractionated using a method of partial solubility in solutions of isopropanol increasingly diluted with distilled water, effectively fractionating by polarity. Selected fractions were characterised by gravimetric determination of the fractions, and determination of acid insoluble lignin, soluble lignin, and carbohydrate contents. Acid-insoluble lignin content was very low, and soluble lignin provided the majority of the lignin content, as should be expected from sulphonated lignin. Carbohydrate contents were also fairly low, the highest percentage at 14.5 being in Fraction 2, with the bulk lignin and Fraction 3 having 6.5% and 3.2%, respectively. Differences in the composition of each fraction support the efficacy of the fractionation process and permitted selection of fractions for use in subsequent studies.
Fractional hydrodynamic equations for fractal media
International Nuclear Information System (INIS)
Tarasov, Vasily E.
2005-01-01
We use the fractional integrals in order to describe dynamical processes in the fractal medium. We consider the 'fractional' continuous medium model for the fractal media and derive the fractional generalization of the equations of balance of mass density, momentum density, and internal energy. The fractional generalization of Navier-Stokes and Euler equations are considered. We derive the equilibrium equation for fractal media. The sound waves in the continuous medium model for fractional media are considered
Some probabilistic properties of fractional point processes
Garra, Roberto
2017-05-16
In this article, the first hitting times of generalized Poisson processes N-f (t), related to Bernstein functions f are studied. For the spacefractional Poisson processes, N alpha (t), t > 0 ( corresponding to f = x alpha), the hitting probabilities P{T-k(alpha) < infinity} are explicitly obtained and analyzed. The processes N-f (t) are time-changed Poisson processes N( H-f (t)) with subordinators H-f (t) and here we study N(Sigma H-n(j= 1)f j (t)) and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form N( G(H,v) (t)) where G(H,v) (t) are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space-time Poisson process is no longer a renewal process.
International Nuclear Information System (INIS)
He, Ji-Huan; Elagan, S.K.; Li, Z.B.
2012-01-01
The fractional complex transform is suggested to convert a fractional differential equation with Jumarie's modification of Riemann–Liouville derivative into its classical differential partner. Understanding the fractional complex transform and the chain rule for fractional calculus are elucidated geometrically. -- Highlights: ► The chain rule for fractional calculus is invalid, a counter example is given. ► The fractional complex transform is explained geometrically. ► Fractional equations can be converted into differential equations.
Ultrasound speckle reduction based on fractional order differentiation.
Shao, Dangguo; Zhou, Ting; Liu, Fan; Yi, Sanli; Xiang, Yan; Ma, Lei; Xiong, Xin; He, Jianfeng
2017-07-01
Ultrasound images show a granular pattern of noise known as speckle that diminishes their quality and results in difficulties in diagnosis. To preserve edges and features, this paper proposes a fractional differentiation-based image operator to reduce speckle in ultrasound. An image de-noising model based on fractional partial differential equations with balance relation between k (gradient modulus threshold that controls the conduction) and v (the order of fractional differentiation) was constructed by the effective combination of fractional calculus theory and a partial differential equation, and the numerical algorithm of it was achieved using a fractional differential mask operator. The proposed algorithm has better speckle reduction and structure preservation than the three existing methods [P-M model, the speckle reducing anisotropic diffusion (SRAD) technique, and the detail preserving anisotropic diffusion (DPAD) technique]. And it is significantly faster than bilateral filtering (BF) in producing virtually the same experimental results. Ultrasound phantom testing and in vivo imaging show that the proposed method can improve the quality of an ultrasound image in terms of tissue SNR, CNR, and FOM values.
Continuous fractional distillation of petroleum
Energy Technology Data Exchange (ETDEWEB)
1921-11-05
This invention has for its object a process of distillation, fractional, and continuous, of shale oil, tar, etc., characterized by the vapors leaving the evaporation chamber being forced, before condensation, to go over a continuous circuit. The vapors traverse first a preheater then return to the vaporization chamber in which they are passed along large surfaces and by application of the counter-current principle in contact with the liquid to be distilled. They stream through the chamber in a continuous manner (the quantity of vapor emitted in the circuit being determined in a manner to advance the distillation just to completion); the excess of vapor formed being removed from the circuit and sent to a condensing apparatus for fractionation.
Search for free fractional charge
International Nuclear Information System (INIS)
Heilig, S.J.
1985-01-01
Recent results of searches for free fractional charge have been null with the exception of the experiment at Stanford under the leadership of W. Fairbank. His experiment, while claiming the observation of free fractional charge, has yet to show that this observation was not spurious. The need for a confirming experiment with a different physical system is the motivation for the current work. A torsional pendulum has been constructed of a fused silica fiber with an attached fused silica crossbar. A transverse electric field is applied to the end of the crossbar, and the resulting deflection of the crossbar is used to measure the torque applied by the field. To date the limit of measurement for the charge on the crossbar (without sample) is 0 +/- 24 electronic charges. The history of this experiment is discussed, along with plans for pushing the limits of measurement to below the single-charge level
Measuring condensate fraction in superconductors
International Nuclear Information System (INIS)
Chakravarty, Sudip; Kee, Hae-Young
2000-01-01
An analysis of off-diagonal long-range order in superconductors shows that the spin-spin correlation function is significantly influenced by the order if the order parameter is anisotropic on a microscopic scale. Thus, magnetic neutron scattering can provide a direct measurement of the condensate fraction of a superconductor. It is also argued that recent measurements in high-temperature superconductors come very close to achieving this goal. (c) 2000 The American Physical Society
Microfluidic Devices for Blood Fractionation
Hou, Han Wei; Bhagat, Ali Asgar S.; Lee, Wong Cheng J.; Huang, Sha; Han, Jongyoon; Lim, Chwee Teck
2011-01-01
Blood, a complex biological fluid, comprises 45% cellular components suspended in protein rich plasma. These different hematologic components perform distinct functions in vivo and thus the ability to efficiently fractionate blood into its individual components has innumerable applications in both clinical diagnosis and biological research. Yet, processing blood is not trivial. In the past decade, a flurry of new microfluidic based technologies has emerged to address this compelling problem. ...
Surfaces allowing for fractional statistics
International Nuclear Information System (INIS)
Aneziris, Charilaos.
1992-07-01
In this paper we give a necessary condition in order for a geometrical surface to allow for Abelian fractional statistics. In particular, we show that such statistics is possible only for two-dimentional oriented surfaces of genus zero, namely the sphere S 2 , the plane R 2 and the cylindrical surface R 1 *S 1 , and in general the connected sum of n planes R 2 -R 2 -R 2 -...-R 2 . (Author)
TECHNOLOGICAL FEATURES OF MILLING AND FRACTIONATION OF FLAXSEEDS
Directory of Open Access Journals (Sweden)
A. Feskova
2015-01-01
Full Text Available Summary. The optimal parameters of milling and fractionation of flaxseeds were substantiated. It was found that the hull fraction with the highest content of lignan secoisolariciresinol diglucoside SDG was obtained when flaxseeds were grinded using a rotatory impact continuous operation mill at the rotation 1380-1640 rpm. Studies have shown that with the increasing of the rotor speed the number of unbriken seeds decreased. However, due to the fact that the shells are crushed more, they become more difficult to separate from the cotyledons. For identification and quantification of SDG the HPLC-MS method was used. It is found that the optimum separation membranes and cotyledon fraction occurs at sifting milled seeds sequentially through the sieves having meshes of 1 and 0.5 mm. The technology of industrial production of lignans-containing fraction and flour on the basis of flaxseeds processing were proposed. This technology includes milling flaxseeds at the rotation 1380-1640 rpm, with subsequent 2% silicon dioxide addition and stepwise sieving using sieves with the mesh size 2 mm. To use a fraction membranes high in lignans as raw material for biologically active additives to food it needed additional enforcement-ground to a size not more than 0.4 mm (technological features of capsulation. The developed technology allowed getting with maximum yields of lignans-containing fraction (10% yield and flaxseed flour (80% yield.
Directory of Open Access Journals (Sweden)
Yanning Wang
2016-01-01
Full Text Available Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for a p-Laplacian conformable fractional differential equation boundary value problem on time scale T: Tα(Tαup-2Tα(u(t=∇F(σ(t,u(σ(t, Δ-a.e. t∈a,bTκ2, u(a-u(b=0, Tα(u(a-Tα(u(b=0, where Tα(u(t denotes the conformable fractional derivative of u of order α at t, σ is the forward jump operator, a,b∈T, 01, and F:[0,T]T×RN→R. By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.
Isotopic fractionation of soil water during evaporation
Energy Technology Data Exchange (ETDEWEB)
Leopoldo, P R [Faculdade de Ciencias Medicas e Biologicas de Botucatu (Brazil); Salati, E; Matsui, E [Centro de Energia Nuclear na Agricultura, Piracicaba (Brazil)
1974-07-01
The study of the variation of D/H relation in soil water during evaporation is studied. The isotopic fractionation of soil water has been observed in two soils of light and heavy texture. Soil columns were utilized. Soil water was extracted in a system operated under low pressure and the gaseous hydrogen was obtained by decomposition of the water and was analyzed in a GD-150 mass spectrometer for deuterium content. The variation of the delta sub(eta) /sup 0///sub 00/ value during evaporation showed that for water held at potentials below 15 atm, the deuterium content of soil water stays practically constant. For water held at potentials higher than 15 atm, corresponding to the third stage of evaporation, there is a strong tendency of a constant increase of delta sub(eta) /sup 0///sub 00/ of the remaining water.
Electrochemically controlled iron isotope fractionation
Black, Jay R.; Young, Edward D.; Kavner, Abby
2010-02-01
Variations in the stable isotope abundances of transition metals have been observed in the geologic record and trying to understand and reconstruct the physical/environmental conditions that produced these signatures is an area of active research. It is clear that changes in oxidation state lead to large fractionations of the stable isotopes of many transition metals such as iron, suggesting that transition metal stable isotope signatures could be used as a paleo-redox proxy. However, the factors contributing to these observed stable isotope variations are poorly understood. Here we investigate how the kinetics of iron redox electrochemistry generates isotope fractionation. Through a combination of electrodeposition experiments and modeling of electrochemical processes including mass-transport, we show that electron transfer reactions are the cause of a large isotope separation, while mass transport-limited supply of reactant to the electrode attenuates the observed isotopic fractionation. Furthermore, the stable isotope composition of electroplated transition metals can be tuned in the laboratory by controlling parameters such as solution chemistry, reaction overpotential, and solution convection. These methods are potentially useful for generating isotopically-marked metal surfaces for tracking and forensic purposes. In addition, our studies will help interpret stable isotope data in terms of identifying underlying electron transfer processes in laboratory and natural samples.
Eden, W. T.; Alighiri, D.; Cahyono, E.; Supardi, K. I.; Wijayati, N.
2018-04-01
The aim of this work was to assess the performance of a vacuum fractionating column for the fractionation of Java Citronella Oil (Cymbopogon winterianus) and citronellal purification during batch mode operation at vacuum -76 cmHg and reflux ratios 5:1. Based on GC-MS analysis of Java Citronella Oil is known that citronellal, citronellol, and geraniol has yielded 21,59%; 7,43%; and 34,27%, respectively. Fractional distillation under reduced pressure and continued redistilled are needed to isolate the component of Java Citronella Oil. Redistilled can improve the purity, then distillate collected while the temperature changed. In the first distillate yielded citronellal with a purity of 75.67%. The first distillate obtained residue rhodinol product will then be carried back to separation into citronellol and geraniol. The purity of citronellol reached 80,65% purity, whereas geraniol reached 76.63% purity. Citronellal Purification resulting citronellal to 95.10% purity and p-menthane-3,8-diol reached 75.95% purity.
ANALISYS OF FRACTIONAL-N FREQUENCY SYNTHESIZERS
Directory of Open Access Journals (Sweden)
Boris I. Shakhtarin
2018-01-01
Full Text Available Modern information and control systems cannot be imagined without synchronization subsystems. These are the basic elements that provide tracking of the frequency and phase of reference and information signals, the evaluation of information parameters, and the synthesis of reference and clock signals. Frequency synthesizers (FS are widely used due to the high speed of frequency setting, a wide range of frequency grids and minimal phase noise in the operating frequency range. Since with the mass appearance of specialized microprocessors and with the improvement of automatic design systems, the feasibility and repeatability of products has become simpler, digital FS are increasingly being used. The most widely used are FS with a frequency divider on digital elements, which serves to convert the signal of a reference oscillator and a controlled generator. For FS using a divisor with an integer division factor in the feedback loop, there are a number of limitations, such as the lower frequency of the FS and the frequency step of the FS. To solve this problem, divisors with fractional-variable division factors in the feedback loop are used, which allow to obtain the required range and the grid frequency step of the FS. The methods of improving the quality of spectral and dynamic characteristics of digital synthesizers in a given band of frequency detuning are analyzed. The principles of the FS operation with a divisor with a fractionalvariable fission coefficient are described, and structural schemes are given. The results of imitation simulation in the Simulink system of the software package MATLAB of frequency synthesizers with a divisor with a fractional-variable fission factor implemented in various ways are presented, and a comparative analysis of the spectral characteristics of the obtained models is carried out.
Fractal physiology and the fractional calculus: a perspective
Directory of Open Access Journals (Sweden)
Bruce J West
2010-10-01
Full Text Available This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. We review the allometric aggregation approach to the processing of physiologic time series as a way of determining the fractal character of the underlying phenomena. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. Fractional operators acting on fractal functions yield fractal functions, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process. Control of physiologic complexity is one of the goals of medicine. Allometric control incorporates long-time memory, inverse power-law (IPL correlations, and long-range interactions in complex phenomena as manifest by IPL distributions. We hypothesize that allometric control, rather than homeostatic control, maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can be described using the fractional calculus to capture the dynamics of complex physiologic networks. This hypothesis is supported by a number of physiologic time series data.
Transient void fraction measurements in rod bundle geometries
International Nuclear Information System (INIS)
Chan, A.M.C.
1998-01-01
A new gamma densitometer with a Ba-133 source and a Nal(TI) scintillator operated in the count mode has been designed for transient void fraction measurements in the RD-14M heated channels containing a seven-element heater bundle. The device was calibrated dynamically in the laboratory using an air-water flow loop. The void fraction measured was found to compare well with values obtained using the trapped-water method. The device was also found to follow very well the passage of air slugs in pulsating flow with slug passing frequencies of up to about 1.5 hz. (author)
Implementation of fractional order integrator/differentiator on field programmable gate array
K.P.S. Rana; V. Kumar; N. Mittra; N. Pramanik
2016-01-01
Concept of fractional order calculus is as old as the regular calculus. With the advent of high speed and cost effective computing power, now it is possible to model the real world control and signal processing problems using fractional order calculus. For the past two decades, applications of fractional order calculus, in system modeling, control and signal processing, have grown rapidly. This paper presents a systematic procedure for hardware implementation of the basic operators of fractio...
Directory of Open Access Journals (Sweden)
Marco A. P. Orrico Júnior
2009-01-01
Full Text Available O objetivo foi avaliar o potencial poluidor remanescente dos efluentes de biodigestores abastecidos com dejetos de suínos com separação da fração sólida (CSFS e sem separação da fração sólida (SSFS, e conduzidos sob diferentes tempos de retenção hidráulica (TRH. Os efluentes utilizados eram de biodigestores semicontínuos manejados com TRH de 15; 22; 29 e 36 dias, com e sem separação da fração sólida. Foram utilizados biodigestores batelada, que permaneceram em operação por todo o tempo em que houve produção de biogás (60 dias. Foram avaliadas a produção e a qualidade do biogás, bem como os potenciais de produção por kg de sólidos totais e sólidos voláteis, e as demandas química e bioquímica de oxigênio. Utilizou-se do delineamento inteiramente casualisado, em esquema fatorial 2x4, com três repetições por tratamento. Foram encontrados potenciais de produção de 385 e 117 litros de CH4kg-1 de SV adicionados no material SSFS e CSFS, respectivamente, no menor TRH (15 dias, e potenciais de produção de 74 e 18 litros de CH4kg-1 de SV adicionados no material SSFS e CSFS, respectivamente, no maior TRH (36 dias.The objective of this work was to evaluate the polluting potential from the remainings of effluents from biodigesters that operate with swine manure with the separation of the solid fraction and without the separation of the solid fraction, both under different hydraulic retention times (HRT. For the biodigestion trial, the effluents from semi-continuous biodigesters were processed with 15; 22; 29 and 36 days of hydraulic retention, with and without the separation of the solid fraction. In this part of the work batch biodigesters were used, which were kept in the operation as long as biogas was produced (60 days. It was evaluated: biogas production and quality and yield potential, the potential production per kg of total solids and volatile solids and chemical and biochemical demands for oxygen. Production
LITERATURE SURVEY FOR FRACTIONAL CRYSTALLIZATION STUDY
International Nuclear Information System (INIS)
PERSON, J.C.
2004-01-01
The literature survey for the fractional crystallization study of material from tank 241-S-112 is completed, fulfilling the requirements of the Test Plan for Tank 241-S-112 Fractional Crystallization Study (Herting 2003). Crystallization involves the formation of one or more solid phases from a fluid phase or an amorphous solid phase. It is applied extensively in the chemical industry, both as a purification process and a separation process. The main advantage of crystallization over distillation is the production of substances with a very high purity, at a low level of energy consumption, and at relatively mild process conditions. Crystallization is one of the older operations in the chemical industry; therefore, practical experience can usually be used for the design and operation of industrial crystallizers. In addition, advances in the understanding of crystallization kinetics can be useful in the control, design, and scale-up of industrial crystallizers. Research work is currently underway; e.g., the CrysCODE (Crystallizer Control and Design) project, littu://www.aui.tudelft.nl/uroiect/Cn/scode/crvscode.htm, at the Delft University of Technology, with the goal of improving the performance and controllability of industrial crystallizers by means of better control and improved design methodologies. Recent developments in fluid dynamics and reactor technology (e.g., compartment approaches) have led to a better understanding of processes and scale-up phenomena. The ultimate aim of such research is to develop a knowledge-based design frame for optimization of industrial crystallization units. Development work is in progress on a rigorous design analysis model for the description of the crystallization process as a function of the reactor geometry, crystallization kinetics, and operating conditions. One modeling effort is aimed at improving the predictive crystallizer model by implementing a population balance equation that depends on two variables: the size and
Directory of Open Access Journals (Sweden)
Fukang Yin
2013-01-01
Full Text Available A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs. The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs. The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.
The fractional quantum Hall effect
International Nuclear Information System (INIS)
Stormer, H.L.
1988-01-01
The fractional quantum Hall effect (FQHE), is the manifestation of a new, highly correlated, many-particle ground state that forms in a two-dimensional electron system at low temperatures and in high magnetic fields. It is an example of the new physics that has grown out of the tremendous recent advances in semiconductor material science, which has provided us with high-quality, lower-dimensional carrier systems. The novel electronic state exposes itself in transport experiments through quantization of the Hall resistance to an exact rational fraction of h/e, and concomitantly vanishing longitudinal resistivity. Its relevant energy scale is only a few degrees kelvin. The quantization is a consequence of the spontaneous formation of an energy gap separating the condensed ground state from its rather elusive quasiparticle excitations. The theoretical understanding of the novel quantum liquids which underlie the FQHE has predominantly emerged from an ingenious many-particle wave function strongly supported by numerous few-particle simulations. Theory has now constructed a complex model for ideal two-dimensional electron systems in the presence of high magnetic fields and makes definitive, often fascinating predictions. Experiments have successively uncovered odd-denominator fractional states reaching presently to 7/13. The application of new experimental tools to the FQHE, such as optics, microwaves, and phonon techniques promises the direct observation of such parameters as the gap energy and possibly even some of the more elusive quantities in the future. While theory and experiment in the FQHE appear to be converging, there remains considerable room for challenging surprises. This paper provides a concise overview of the FQHE. It focuses on the experimental aspects and states, but does not expand on the theoretical advances. 70 refs., 11 figs
Low power constant fraction discriminator
International Nuclear Information System (INIS)
Krishnan, Shanti; Raut, S.M.; Mukhopadhyay, P.K.
2001-01-01
This paper describes the design of a low power ultrafast constant fraction discriminator, which significantly reduces the power consumption. A conventional fast discriminator consumes about 1250 MW of power whereas this low power version consumes about 440 MW. In a multi detector system, where the number of discriminators is very large, reduction of power is of utmost importance. This low power discriminator is being designed for GRACE (Gamma Ray Atmospheric Cerenkov Experiments) telescope where 1000 channels of discriminators are required. A novel method of decreasing power consumption has been described. (author)
Natural fractionation of uranium isotopes
International Nuclear Information System (INIS)
Noordmann, Janine
2015-01-01
The topic of this thesis was the investigation of U (n( 238 U) / n( 235 U)) isotope variations in nature with a focus on samples (1) that represent the continental crust and its weathering products (i.e. granites, shales and river water) (2) that represent products of hydrothermal alteration on mid-ocean ridges (i.e. altered basalts, carbonate veins and hydrothermal water) and (3) from restricted euxinic basins (i.e. from the water column and respective sediments). The overall goal was to explore the environmental conditions and unravel the mechanisms that fractionate the two most abundant U isotopes, n( 238 U) and n( 235 U), on Earth.
Boundedness of the Segal-Bargmann Transform on Fractional Hermite-Sobolev Spaces
Directory of Open Access Journals (Sweden)
Hong Rae Cho
2017-01-01
Full Text Available Let s∈R and 2≤p≤∞. We prove that the Segal-Bargmann transform B is a bounded operator from fractional Hermite-Sobolev spaces WHs,pRn to fractional Fock-Sobolev spaces FRs,p.
Competence with Fractions in Fifth or Sixth Grade as a Unique Predictor of Algebraic Thinking?
Pearn, Catherine; Stephens, Max
2016-01-01
Researchers have argued that there are strong links between primary school students' competence with fraction concepts and operations and their algebraic readiness. This study involving 162 Years 5/6 students in three primary schools examined the strength of that relationship using a test based on familiar fraction tasks and a test of algebraic…
Strategies for Solving Fraction Tasks and Their Link to Algebraic Thinking
Pearn, Catherine; Stephens, Max
2015-01-01
Many researchers argue that a deep understanding of fractions is important for a successful transition to algebra. Teaching, especially in the middle years, needs to focus specifically on those areas of fraction knowledge and operations that support subsequent solution processes for algebraic equations. This paper focuses on the results of Year 6…
A generalized Gronwall inequality and its application to fractional neutral evolution inclusions
Directory of Open Access Journals (Sweden)
Zufeng Zhang
2016-02-01
Full Text Available Abstract This paper deals with the fractional neutral evolution differential inclusions. The existence results are established by using the fractional power of operators and a fixed point theorem for multivalued map. Moreover, we present a new generalized Gronwall inequality with singularity, which is an important tool in the proof of solvability.
Fractional Wigner Crystal in the Helical Luttinger Liquid.
Traverso Ziani, N; Crépin, F; Trauzettel, B
2015-11-13
The properties of the strongly interacting edge states of two dimensional topological insulators in the presence of two-particle backscattering are investigated. We find an anomalous behavior of the density-density correlation functions, which show oscillations that are neither of Friedel nor of Wigner type: they, instead, represent a Wigner crystal of fermions of fractional charge e/2, with e the electron charge. By studying the Fermi operator, we demonstrate that the state characterized by such fractional oscillations still bears the signatures of spin-momentum locking. Finally, we compare the spin-spin correlation functions and the density-density correlation functions to argue that the fractional Wigner crystal is characterized by a nontrivial spin texture.
International Nuclear Information System (INIS)
Huang, Chih-Hsien; Hsieh, Wen-Feng; Wu, Jing-Nuo; Cheng, Szu-Cheng; Li, Yen-Yin
2011-01-01
Fractional time derivative, an abstract mathematical operator of fractional calculus, is used to describe the real optical system of a V-type three-level atom embedded in a photonic crystal. A fractional kinetic equation governing the dynamics of the spontaneous emission from this optical system is obtained as a fractional Langevin equation. Solving this fractional kinetic equation by fractional calculus leads to the analytical solutions expressed in terms of fractional exponential functions. The accuracy of the obtained solutions is verified through reducing the system into the special cases whose results are consistent with the experimental observation. With accurate physical results and avoiding the complex integration for solving this optical system, we propose fractional calculus with fractional time derivative as a better mathematical method to study spontaneous emission dynamics from the optical system with non-Markovian dynamics.
International Nuclear Information System (INIS)
Aoun, M.; Aribi, A.; Najar, S.; Abdelkrim, M.N.
2011-01-01
This paper shows the interest of extending the dynamic parity space fault detection method for fractional systems. Accordingly, a comparison between fractional and rational residual generators using the later method is presented. An analysis of fractional and rational residuals sensitivity shows the merits of the fractional residual generators. A numerical example illustrating the advantage of using fractional residual generators for fractional systems diagnosis is given.
THE NEW SOLUTION OF TIME FRACTIONAL WAVE EQUATION WITH CONFORMABLE FRACTIONAL DERIVATIVE DEFINITION
Çenesiz, Yücel; Kurt, Ali
2015-01-01
– In this paper, we used new fractional derivative definition, the conformable fractional derivative, for solving two and three dimensional time fractional wave equation. This definition is simple and very effective in the solution procedures of the fractional differential equations that have complicated solutions with classical fractional derivative definitions like Caputo, Riemann-Liouville and etc. The results show that conformable fractional derivative definition is usable and convenient ...
Nonlinear dynamics of fractional order Duffing system
International Nuclear Information System (INIS)
Li, Zengshan; Chen, Diyi; Zhu, Jianwei; Liu, Yongjian
2015-01-01
In this paper, we analyze the nonlinear dynamics of fractional order Duffing system. First, we present the fractional order Duffing system and the numerical algorithm. Second, nonlinear dynamic behaviors of Duffing system with a fixed fractional order is studied by using bifurcation diagrams, phase portraits, Poincare maps and time domain waveforms. The fractional order Duffing system shows some interesting dynamical behaviors. Third, a series of Duffing systems with different fractional orders are analyzed by using bifurcation diagrams. The impacts of fractional orders on the tendency of dynamical motion, the periodic windows in chaos, the bifurcation points and the distance between the first and the last bifurcation points are respectively studied, in which some basic laws are discovered and summarized. This paper reflects that the integer order system and the fractional order one have close relationship and an integer order system is a special case of fractional order ones.
A fractional model for dye removal
Directory of Open Access Journals (Sweden)
Ji-Huan He
2016-01-01
Full Text Available The adsorption process has a fractional property, and a fractional model is suggested to study a transport model of direct textile industry wastewater. An approximate solution of the concentration is obtained by the variational iteration method.
Boundary Controllability of Nonlinear Fractional Integrodifferential Systems
Directory of Open Access Journals (Sweden)
Ahmed HamdyM
2010-01-01
Full Text Available Sufficient conditions for boundary controllability of nonlinear fractional integrodifferential systems in Banach space are established. The results are obtained by using fixed point theorems. We also give an application for integropartial differential equations of fractional order.
Fractional Order Element Based Impedance Matching
Radwan, Ahmed Gomaa
2014-06-24
Disclosed are various embodiments of methods and systems related to fractional order element based impedance matching. In one embodiment, a method includes aligning a traditional Smith chart (|.alpha.|=1) with a fractional order Smith chart (|.alpha.|.noteq.1). A load impedance is located on the traditional Smith chart and projected onto the fractional order Smith chart. A fractional order matching element is determined by transitioning along a matching circle of the fractional order Smith chart based at least in part upon characteristic line impedance. In another embodiment, a system includes a fractional order impedance matching application executed in a computing device. The fractional order impedance matching application includes logic that obtains a first set of Smith chart coordinates at a first order, determines a second set of Smith chart coordinates at a second order, and determines a fractional order matching element from the second set of Smith chart coordinates.
Chaos in discrete fractional difference equations
Indian Academy of Sciences (India)
2016-09-07
Sep 7, 2016 ... chaotic behaviour of fractional difference equations for the tent map, Gauss map and 2x(mod 1) map are studied ..... (4) No significant change is observed by changing .... (3) In fractional case, the rational initial condition.
Fractional Order Element Based Impedance Matching
Radwan, Ahmed Gomaa; Salama, Khaled N.; Shamim, Atif
2014-01-01
Disclosed are various embodiments of methods and systems related to fractional order element based impedance matching. In one embodiment, a method includes aligning a traditional Smith chart (|.alpha.|=1) with a fractional order Smith chart (|.alpha
Fractionated stereotactic radiotherapy for craniopharyngiomas
International Nuclear Information System (INIS)
Schulz-Ertner, Daniela; Frank, Claudia; Herfarth, Klaus K.; Rhein, Bernhard; Wannenmacher, Michael; Debus, Juergen
2002-01-01
Purpose: To investigate outcome and toxicity after fractionated stereotactic radiation therapy (FSRT) in patients with craniopharyngiomas. Methods and Materials: Twenty-six patients with craniopharyngiomas were treated with FSRT between May 1989 and February 2001. Median age was 33.5 years (range: 5-57 years). Nine patients received FSRT after surgery as primary treatment, and 17 patients were irradiated for recurrent tumor or progressive growth after initial surgery. Median target dose was 52.2 Gy (range: 50.0-57.6 Gy) with conventional fractionation. Follow-up included MRI and neurologic, ophthalmologic, and endocrinologic examinations. Results: The median follow-up was 43 months (range: 7-143 months). The actuarial local control rate and actuarial overall survival rates were 100% and 100%, respectively, at 5 years and 100% and 83%, respectively, at 10 years. Four patients showed complete response, 14 patients showed partial response, and 8 patients remained stable. In 5 patients, vision improved after radiation therapy. Acute toxicity was mild. One patient required cyst drainage 3 months after radiotherapy. Late toxicity after radiotherapy included impairment of hormone function in 3 out of 18 patients at risk. We did not observe any vision impairment, radionecrosis, or secondary malignancies. Conclusions: FSRT is effective and safe in the treatment of cystic craniopharyngiomas. Toxicity is extremely low using this conformal technique
Microfluidic Devices for Blood Fractionation
Directory of Open Access Journals (Sweden)
Chwee Teck Lim
2011-07-01
Full Text Available Blood, a complex biological fluid, comprises 45% cellular components suspended in protein rich plasma. These different hematologic components perform distinct functions in vivo and thus the ability to efficiently fractionate blood into its individual components has innumerable applications in both clinical diagnosis and biological research. Yet, processing blood is not trivial. In the past decade, a flurry of new microfluidic based technologies has emerged to address this compelling problem. Microfluidics is an attractive solution for this application leveraging its numerous advantages to process clinical blood samples. This paper reviews the various microfluidic approaches realized to successfully fractionate one or more blood components. Techniques to separate plasma from hematologic cellular components as well as isolating blood cells of interest including certain rare cells are discussed. Comparisons based on common separation metrics including efficiency (sensitivity, purity (selectivity, and throughput will be presented. Finally, we will provide insights into the challenges associated with blood-based separation systems towards realizing true point-of-care (POC devices and provide future perspectives.
The synchronization of three fractional differential systems
International Nuclear Information System (INIS)
Li Changpin; Yan Jianping
2007-01-01
In this paper, a new method is proposed and applied to the synchronization of fractional differential systems (or 'differential systems with fractional orders'), where both drive and response systems have the same dimensionality and are coupled by the driving signal. The present technique is based on the stability criterion of linear fractional systems. This method is implemented in (chaos) synchronization of the fractional Lorenz system, Chen system and Chua circuit. Numerical simulations show the present synchronization method works well
Caratelli, Diego; Mescia, Luciano; Bia, Pietro; Stukach, Oleg V.
2016-01-01
A novel finite-difference time-domain algorithm for modeling ultrawideband electromagnetic pulse propagation in arbitrary multirelaxed dispersive media is presented. The proposed scheme is based on a general, yet computationally efficient, series representation of the fractional derivative operators
A recursion relation for coefficients of fractional parentage in the seniority scheme
International Nuclear Information System (INIS)
Evans, T.
1985-01-01
A recursion relations for coefficients as fractional parentage in the seniority scheme are discussed. Determinated dependence of recursion relations from the particle number permit to evaluate matrix elements of creation and annihilation operators for fermions or bosons. 10 refs. (author)
Early Predictors of Middle School Fraction Knowledge
Bailey, Drew H.; Siegler, Robert S.; Geary, David C.
2014-01-01
Recent findings that earlier fraction knowledge predicts later mathematics achievement raise the question of what predicts later fraction knowledge. Analyses of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic…
2010-01-01
... 16 Commercial Practices 1 2010-01-01 2010-01-01 false Fractions. 500.17 Section 500.17 Commercial... LABELING ACT § 500.17 Fractions. (a) SI metric declarations of net quantity of contents of any consumer commodity may contain only decimal fractions. Other declarations of net quantity of contents may contain...
Teaching Fractions. Educational Practices Series-22
Fazio, Lisa; Siegler, Robert
2011-01-01
Students around the world have difficulties in learning about fractions. In many countries, the average student never gains a conceptual knowledge of fractions. This research guide provides suggestions for teachers and administrators looking to improve fraction instruction in their classrooms or schools. The recommendations are based on a…
On varitional iteration method for fractional calculus
Directory of Open Access Journals (Sweden)
Wu Hai-Gen
2017-01-01
Full Text Available Modification of the Das’ variational iteration method for fractional differential equations is discussed, and its main shortcoming involved in the solution process is pointed out and overcome by using fractional power series. The suggested computational procedure is simple and reliable for fractional calculus.
An Alternative Starting Point for Fraction Instruction
Cortina, José Luis; Višnovská, Jana; Zúñiga, Claudia
2015-01-01
We analyze the results of a study conducted for the purpose of assessing the viability of an alternative starting point for teaching fractions. The alternative is based on Freudenthal's insights about fraction as comparison. It involves portraying the entities that unit fractions quantify as always being apart from the reference unit, instead of…
Fractional Euler Limits and Their Applications
MacNamara, Shev; Henry, Bruce I; McLean, William
2016-01-01
Generalisations of the classical Euler formula to the setting of fractional calculus are discussed. Compound interest and fractional compound interest serve as motivation. Connections to fractional master equations are highlighted. An application to the Schlogl reactions with Mittag-Leffler waiting times is described.
Time-fractional particle deposition in porous media
Xu, Jianping
2017-05-01
In the percolation process where fluids carry small solid particles, particle deposition causes a real-time permeability change of the medium as the swarm of particles propagates along the medium. Then the permeability change influences percolation and deposition behaviors as a feedback. This fact triggers memory effect in the deposition dynamics, which means the particulate transport and deposition behaviors become history-dependent. In this paper, we conduct the time-fractional generalization of the classical phenomenological model of particle deposition in porous media to incorporate the memory effect. We tested and compared the effects of employing different types of fractional operators, i.e. the Riemann-Liouville type, the Hadamard type and the Prabhakar type. Numerical simulation results show that the system behaviors vary according to the change of distinct memory kernels in an expected way. We then discuss the physical meaning of the time-fractional generalization. It is shown that different types of fractional operators unanimously ground themselves on the local-Newtonian time transformation in a complex system, which is equivalent to a class of history integrals. By the introduction of various memory kernels, it enables the model to more powerfully fit and approximate observed data. Further, the fundamental meaning of this work is not to show which fractional operator is ‘better’, but to argue collectively the legitimacy and practicality of a non-Markovian particle deposition dynamics in porous media, and in fact it is admissible to a bunch of memory kernels which differ greatly from each other in functional forms. Hopefully the presented generalized mass conservation formalism offers a broader framework to investigate transport problems in porous media.
Time-fractional particle deposition in porous media
International Nuclear Information System (INIS)
Xu, Jianping
2017-01-01
In the percolation process where fluids carry small solid particles, particle deposition causes a real-time permeability change of the medium as the swarm of particles propagates along the medium. Then the permeability change influences percolation and deposition behaviors as a feedback. This fact triggers memory effect in the deposition dynamics, which means the particulate transport and deposition behaviors become history-dependent. In this paper, we conduct the time-fractional generalization of the classical phenomenological model of particle deposition in porous media to incorporate the memory effect. We tested and compared the effects of employing different types of fractional operators, i.e. the Riemann–Liouville type, the Hadamard type and the Prabhakar type. Numerical simulation results show that the system behaviors vary according to the change of distinct memory kernels in an expected way. We then discuss the physical meaning of the time-fractional generalization. It is shown that different types of fractional operators unanimously ground themselves on the local-Newtonian time transformation in a complex system, which is equivalent to a class of history integrals. By the introduction of various memory kernels, it enables the model to more powerfully fit and approximate observed data. Further, the fundamental meaning of this work is not to show which fractional operator is ‘better’, but to argue collectively the legitimacy and practicality of a non-Markovian particle deposition dynamics in porous media, and in fact it is admissible to a bunch of memory kernels which differ greatly from each other in functional forms. Hopefully the presented generalized mass conservation formalism offers a broader framework to investigate transport problems in porous media. (paper)
Analysis of Student Errors on Division of Fractions
Maelasari, E.; Jupri, A.
2017-02-01
This study aims to describe the type of student errors that typically occurs at the completion of the division arithmetic operations on fractions, and to describe the causes of students’ mistakes. This research used a descriptive qualitative method, and involved 22 fifth grade students at one particular elementary school in Kuningan, Indonesia. The results of this study showed that students’ error answers caused by students changing their way of thinking to solve multiplication and division operations on the same procedures, the changing of mix fractions to common fraction have made students confused, and students are careless in doing calculation. From student written work, in solving the fraction problems, we found that there is influence between the uses of learning methods and student response, and some of student responses beyond researchers’ prediction. We conclude that the teaching method is not only the important thing that must be prepared, but the teacher should also prepare about predictions of students’ answers to the problems that will be given in the learning process. This could be a reflection for teachers to be better and to achieve the expected learning goals.
Numerical solutions of multi-order fractional differential equations by Boubaker polynomials
Directory of Open Access Journals (Sweden)
Bolandtalat A.
2016-01-01
Full Text Available In this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.
An experimental and theoretical analysis of void fraction dynamics in a boiling channel
International Nuclear Information System (INIS)
Romberg, T.M.
1977-01-01
This paper describes an experimental and theoretical investigation of the void fraction dynamics at the exit of a test boiling channel which is operated near the 'instability threshold power' (the power level at which coolant flow instabilities occur). Dynamic measurements of the perturbations in channel inlet flow-rate, power input and exit void fraction are analysed using multivariate spectral analysis. The resulting experimental cross-spectral density functions between flow-rate/exit void fraction and power input/exit void fraction agree favourably with those calculated by a linearised hydrodynamic model in the frequency domain. (Author)
A representation theory for a class of vector autoregressive models for fractional processes
DEFF Research Database (Denmark)
Johansen, Søren
2008-01-01
Based on an idea of Granger (1986), we analyze a new vector autoregressive model defined from the fractional lag operator 1-(1-L)^{d}. We first derive conditions in terms of the coefficients for the model to generate processes which are fractional of order zero. We then show that if there is a un...... root, the model generates a fractional process X(t) of order d, d>0, for which there are vectors ß so that ß'X(t) is fractional of order d-b, 0...
Natural fractionation of uranium isotopes
Energy Technology Data Exchange (ETDEWEB)
Noordmann, Janine
2015-01-24
The topic of this thesis was the investigation of U (n({sup 238}U) / n({sup 235}U)) isotope variations in nature with a focus on samples (1) that represent the continental crust and its weathering products (i.e. granites, shales and river water) (2) that represent products of hydrothermal alteration on mid-ocean ridges (i.e. altered basalts, carbonate veins and hydrothermal water) and (3) from restricted euxinic basins (i.e. from the water column and respective sediments). The overall goal was to explore the environmental conditions and unravel the mechanisms that fractionate the two most abundant U isotopes, n({sup 238}U) and n({sup 235}U), on Earth.
Second Study of Hyper-Fractionated Radiotherapy
Directory of Open Access Journals (Sweden)
R. Jacob
1999-01-01
Full Text Available Purpose and Method. Hyper-fractionated radiotherapy for treatment of soft tissue sarcomas is designed to deliver a higher total dose of radiation without an increase in late normal tissue damage. In a previous study at the Royal Marsden Hospital, a total dose of 75 Gy using twice daily 1.25 Gy fractions resulted in a higher incidence of late damage than conventional radiotherapy using 2 Gy daily fractions treating to a total of 60 Gy. The current trial therefore used a lower dose per fraction of 1.2 Gy and lower total dose of 72 Gy, with 60 fractions given over a period of 6 weeks.
Hamiltonian Chaos and Fractional Dynamics
International Nuclear Information System (INIS)
Combescure, M
2005-01-01
This book provides an introduction and discussion of the main issues in the current understanding of classical Hamiltonian chaos, and of its fractional space-time structure. It also develops the most complex and open problems in this context, and provides a set of possible applications of these notions to some fundamental questions of dynamics: complexity and entropy of systems, foundation of classical statistical physics on the basis of chaos theory, and so on. Starting with an introduction of the basic principles of the Hamiltonian theory of chaos, the book covers many topics that can be found elsewhere in the literature, but which are collected here for the readers' convenience. In the last three parts, the author develops topics which are not typically included in the standard textbooks; among them are: - the failure of the traditional description of chaotic dynamics in terms of diffusion equations; - he fractional kinematics, its foundation and renormalization group analysis; - 'pseudo-chaos', i.e. kinetics of systems with weak mixing and zero Lyapunov exponents; - directional complexity and entropy. The purpose of this book is to provide researchers and students in physics, mathematics and engineering with an overview of many aspects of chaos and fractality in Hamiltonian dynamical systems. In my opinion it achieves this aim, at least provided researchers and students (mainly those involved in mathematical physics) can complement this reading with comprehensive material from more specialized sources which are provided as references and 'further reading'. Each section contains introductory pedagogical material, often illustrated by figures coming from several numerical simulations which give the feeling of what's going on, and thus is very useful to the reader who is not very familiar with the topics presented. Some problems are included at the end of most sections to help the reader to go deeper into the subject. My one regret is that the book does not
Transformation of fractions into simple fractions in divisive meadows
Bergstra, J.A.; Middelburg, C.A.
Meadows are alternatives for fields with a purely equational axiomatization. At the basis of meadows lies the decision to make the multiplicative inverse operation total by imposing that the multiplicative inverse of zero is zero. Divisive meadows are meadows with the multiplicative inverse
Utilization of Different Corn Fractions by Broilers
Directory of Open Access Journals (Sweden)
SIFR Costa
2015-09-01
Full Text Available ABSTRACTThis study was conducted to evaluate the nutritional values of fractions of damaged corn. One hundred and eighty 22-d-old Cobb 500 male broilers were distributed in batteries according to a completely randomized design with six treatments of six replicates each. The treatments consisted of diets containing five corn fractions, classified as sound, fermented, insect-damaged, mold-damaged, or reference corn. The test diets consisted of 60% of reference diet + 40% of each corn fraction. Only the reference corn fraction included all the fractions at different proportions (0.8% fermented, 0.05% insect-damaged, 3.3% mold-damaged, and 95.85% sound grains. The method of total excreta collection was used to determine AMEn values and metabolizability coefficients of dry matter (MDM, crude protein (MCP, ether extract (MEE, and gross energy (MGE of the reference corn and its fractions. The density values of the corn fractions were used to calculate the correlations among the evaluated parameters. The evaluated corn fractions presented different compositions values. The insect-damaged and mold-damaged grains presented higher CP level, lower density, and MDM and MCP coefficients compared with the other fractions. However, calculated AMEn values were not significantly different (p>0.05 among corn fractions. A low correlation between density and AMEn content (r0.8 were calculated. Although the evaluated corn fractions presented different nutritional values, there were no marked differences in their utilization by broilers.
The fractional dynamics of quantum systems
Lu, Longzhao; Yu, Xiangyang
2018-05-01
The fractional dynamic process of a quantum system is a novel and complicated problem. The establishment of a fractional dynamic model is a significant attempt that is expected to reveal the mechanism of fractional quantum system. In this paper, a generalized time fractional Schrödinger equation is proposed. To study the fractional dynamics of quantum systems, we take the two-level system as an example and derive the time fractional equations of motion. The basic properties of the system are investigated by solving this set of equations in the absence of light field analytically. Then, when the system is subject to the light field, the equations are solved numerically. It shows that the two-level system described by the time fractional Schrödinger equation we proposed is a confirmable system.
The Uneasy Case for Fractional-Reserve Free Banking
van den Hauwe, Ludwig
2006-01-01
Since a few decades several sub-disciplines within economics have witnessed a reorientation towards institutional analysis. This development has in particular also affected the fields of macroeconomics and monetary theory where it has led to several proposals for far-reaching financial and monetary reform. One of the more successful of these proposals advocates a fractional-reserve free banking system, that is, a system with no central bank, but with permission for the banks to operate with a...
A fractional calculus perspective of distributed propeller design
Tenreiro Machado, J.; Galhano, Alexandra M.
2018-02-01
A new generation of aircraft with distributed propellers leads to operational performances superior to those exhibited by standard designs. Computational simulations and experimental tests show a reduction of fuel consumption and noise. This paper proposes an analogy between aerodynamics and electrical circuits. The model reveals properties similar to those of fractional-order systems and gives a deeper insight into the dynamics of multi-propeller coupling.
Fractal physiology and the fractional calculus: a perspective.
West, Bruce J
2010-01-01
This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a
Directory of Open Access Journals (Sweden)
Ai-Min Yang
2014-01-01
Full Text Available The local fractional Laplace variational iteration method was applied to solve the linear local fractional partial differential equations. The local fractional Laplace variational iteration method is coupled by the local fractional variational iteration method and Laplace transform. The nondifferentiable approximate solutions are obtained and their graphs are also shown.
Introduction to fractional and pseudo-differential equations with singular symbols
Umarov, Sabir
2015-01-01
The book systematically presents the theories of pseudo-differential operators with symbols singular in dual variables, fractional order derivatives, distributed and variable order fractional derivatives, random walk approximants, and applications of these theories to various initial and multi-point boundary value problems for pseudo-differential equations. Fractional Fokker-Planck-Kolmogorov equations associated with a large class of stochastic processes are presented. A complex version of the theory of pseudo-differential operators with meromorphic symbols based on the recently introduced complex Fourier transform is developed and applied for initial and boundary value problems for systems of complex differential and pseudo-differential equations.
Lie symmetry analysis and conservation laws for the time fractional fourth-order evolution equation
Directory of Open Access Journals (Sweden)
Wang Li
2017-06-01
Full Text Available In this paper, we study Lie symmetry analysis and conservation laws for the time fractional nonlinear fourth-order evolution equation. Using the method of Lie point symmetry, we provide the associated vector fields, and derive the similarity reductions of the equation, respectively. The method can be applied wisely and efficiently to get the reduced fractional ordinary differential equations based on the similarity reductions. Finally, by using the nonlinear self-adjointness method and Riemann-Liouville time-fractional derivative operator as well as Euler-Lagrange operator, the conservation laws of the equation are obtained.
DEFF Research Database (Denmark)
Farvin Habebullah, Sabeena; Andersen, Lisa Lystbæk; Otte, Jeanette
2016-01-01
This study aimed to characterise peptide fractions (>5 kDa, 3–5 kDa and fractions were dominated by Ala, Gly, Glu and Ser. The total amino acid composition had high proportions of Lys, Ala...... and Glu. The 3–5 kDa and fractions were further fractionated by size exclusion chromatography. All sub-fractions showed high Fe2+ chelating activity. The DPPH radical-scavenging activity of the 3–5 kDa fraction was exerted mainly by one sub-fraction dominated by peptides with masses below 600 Da....... The DPPH radical-scavenging activity of the fraction was exerted by sub-fractions with low molecular weight. The highest reducing power was found in a sub-fraction containing peptides rich in Arg, Tyr and Phe. Both free amino acids and low molecular weight peptides thus seemed to contribute...
Directory of Open Access Journals (Sweden)
Qiang Yu
Full Text Available Texture enhancement is one of the most important techniques in digital image processing and plays an essential role in medical imaging since textures discriminate information. Most image texture enhancement techniques use classical integral order differential mask operators or fractional differential mask operators using fixed fractional order. These masks can produce excessive enhancement of low spatial frequency content, insufficient enhancement of large spatial frequency content, and retention of high spatial frequency noise. To improve upon existing approaches of texture enhancement, we derive an improved Variable Order Fractional Centered Difference (VOFCD scheme which dynamically adjusts the fractional differential order instead of fixing it. The new VOFCD technique is based on the second order Riesz fractional differential operator using a Lagrange 3-point interpolation formula, for both grey scale and colour image enhancement. We then use this method to enhance photographs and a set of medical images related to patients with stroke and Parkinson's disease. The experiments show that our improved fractional differential mask has a higher signal to noise ratio value than the other fractional differential mask operators. Based on the corresponding quantitative analysis we conclude that the new method offers a superior texture enhancement over existing methods.
Boundedness of positive operators on weighted amalgams
Directory of Open Access Journals (Sweden)
Aguilar Cañestro María Isabel
2011-01-01
Full Text Available Abstract In this article, we characterize the pairs (u, v of positive measurable functions such that T maps the weighted amalgam in (Lp (u, ℓ q for all , where T belongs to a class of positive operators which includes Hardy operators, maximal operators, and fractional integrals. 2000 Mathematics Subject Classification 26D10, 26D15 (42B35
Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Ahmet Bekir
2013-01-01
Full Text Available The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.
Directory of Open Access Journals (Sweden)
Wei Li
2014-01-01
Full Text Available Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.
Directory of Open Access Journals (Sweden)
Ping Zhou
2012-01-01
Full Text Available The unstable equilibrium points of the fractional-order Lorenz chaotic system can be controlled via fractional-order derivative, and chaos synchronization for the fractional-order Lorenz chaotic system can be achieved via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.
Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey
Directory of Open Access Journals (Sweden)
Chunye Gong
2015-01-01
Full Text Available We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science. The computational complexities of time fractional, space fractional, and space-time fractional equations are O(N2M, O(NM2, and O(NM(M + N compared with O(MN for the classical partial differential equations with finite difference methods, where M, N are the number of space grid points and time steps. The potential solutions for this challenge include, but are not limited to, parallel computing, memory access optimization (fractional precomputing operator, short memory principle, fast Fourier transform (FFT based solutions, alternating direction implicit method, multigrid method, and preconditioner technology. The relationships of these solutions for both space fractional derivative and time fractional derivative are discussed. The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcome this challenge, and some attention should be paid to the fractional killer applications, high performance iteration methods, high order schemes, and Monte Carlo methods. Since the computation of fractional equations with high dimension and variable order is even heavier, the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in the area of fractional calculus.
Fractions Learning in Children With Mathematics Difficulties.
Tian, Jing; Siegler, Robert S
Learning fractions is difficult for children in general and especially difficult for children with mathematics difficulties (MD). Recent research on developmental and individual differences in fraction knowledge of children with MD and typically achieving (TA) children has demonstrated that U.S. children with MD start middle school behind their TA peers in fraction understanding and fall further behind during middle school. In contrast, Chinese children, who like the MD children in the United States score in the bottom one third of the distribution in their country, possess reasonably good fraction understanding. We interpret these findings within the framework of the integrated theory of numerical development. By emphasizing the importance of fraction magnitude knowledge for numerical understanding in general, the theory proved useful for understanding differences in fraction knowledge between MD and TA children and for understanding how knowledge can be improved. Several interventions demonstrated the possibility of improving fraction magnitude knowledge and producing benefits that generalize to fraction arithmetic learning among children with MD. The reasonably good fraction understanding of Chinese children with MD and several successful interventions with U.S. students provide hope for the improvement of fraction knowledge among American children with MD.
The Initial Conditions of Fractional Calculus
International Nuclear Information System (INIS)
Trigeassou, J. C.; Maamri, N.
2011-01-01
During the past fifty years , Fractional Calculus has become an original and renowned mathematical tool for the modelling of diffusion Partial Differential Equations and the design of robust control algorithms. However, in spite of these celebrated results, some theoretical problems have not yet received a satisfying solution. The mastery of initial conditions, either for Fractional Differential Equations (FDEs) or for the Caputo and Riemann-Liouville fractional derivatives, remains an open research domain. The solution of this fundamental problem, also related to the long range memory property, is certainly the necessary prerequisite for a satisfying approach to modelling and control applications. The fractional integrator and its continuously frequency distributed differential model is a valuable tool for the simulation of fractional systems and the solution of initial condition problems. Indeed, the infinite dimensional state vector of fractional integrators allows the direct generalization to fractional calculus of the theoretical results of integer order systems. After a reminder of definitions and properties related to fractional derivatives and systems, this presentation is intended to show, based on the results of two recent publications [1,2], how the fractional integrator provides the solution of the initial condition problem of FDEs and of Caputo and Riemann-Liouville fractional derivatives. Numerical simulation examples illustrate and validate these new theoretical concepts.
Estimation of the Void Fraction in the moderator cell of the Cold Neutron Source
International Nuclear Information System (INIS)
Choi, Jungwoon; Kim, Young-ki
2015-01-01
To estimate the average void fraction in the liquid hydrogen, the Kazimi and Chen correlation is used with its modified method suggested by R.E. Williams in NBSR. Since the multiplying number can be changed along the operation condition and working fluid, the different figure is applied to estimate the average void fraction in the different moderator cell shape. This approach is checked with the void fraction measurement results from the HANARO-CNS mock-up test. Owing to national research demands on cold neutron beam utilization, the Cold Neutron Research Facility had been and operated for neuron scientists all over the world. In HANARO, the CNS facility has been operated since 2009. The actual void fraction, which is the one of dominant factors affecting the cold neutron flux, is difficult to know without the real measurement performed at the cryogenic temperature using the same moderator medium. Accordingly, the two-phase mock-up test in the CNS-IPA (In-pool assembly) had been performed using the liquid hydrogen in terms of the fluidity check, void fraction measurement, operation procedure set-up, and so on for the development of the HANARO-CNS. This paper presents the estimated void fraction in the different operating conditions and geometrical shape in the comparison with the measurement data of the void fraction in the full-scale mockup test based on the Kazimi and Chen correlation. This approach is applied to estimate the average void fraction in the newly designed moderator cell using the liquid hydrogen as a working fluid in the two-phase thermosiphon. From this calculation result, the estimated average void fraction will be used to design the optimized cold neutron source to produce the maximum cold neutron flux within the desired wavelength
Estimation of the Void Fraction in the moderator cell of the Cold Neutron Source
Energy Technology Data Exchange (ETDEWEB)
Choi, Jungwoon; Kim, Young-ki [Korea Atomic Energy Research Institute, Daejeon (Korea, Republic of)
2015-10-15
To estimate the average void fraction in the liquid hydrogen, the Kazimi and Chen correlation is used with its modified method suggested by R.E. Williams in NBSR. Since the multiplying number can be changed along the operation condition and working fluid, the different figure is applied to estimate the average void fraction in the different moderator cell shape. This approach is checked with the void fraction measurement results from the HANARO-CNS mock-up test. Owing to national research demands on cold neutron beam utilization, the Cold Neutron Research Facility had been and operated for neuron scientists all over the world. In HANARO, the CNS facility has been operated since 2009. The actual void fraction, which is the one of dominant factors affecting the cold neutron flux, is difficult to know without the real measurement performed at the cryogenic temperature using the same moderator medium. Accordingly, the two-phase mock-up test in the CNS-IPA (In-pool assembly) had been performed using the liquid hydrogen in terms of the fluidity check, void fraction measurement, operation procedure set-up, and so on for the development of the HANARO-CNS. This paper presents the estimated void fraction in the different operating conditions and geometrical shape in the comparison with the measurement data of the void fraction in the full-scale mockup test based on the Kazimi and Chen correlation. This approach is applied to estimate the average void fraction in the newly designed moderator cell using the liquid hydrogen as a working fluid in the two-phase thermosiphon. From this calculation result, the estimated average void fraction will be used to design the optimized cold neutron source to produce the maximum cold neutron flux within the desired wavelength.
Fractional ablative erbium YAG laser
DEFF Research Database (Denmark)
Taudorf, Elisabeth H; Haak, Christina S; Erlendsson, Andrés M
2014-01-01
laser parameters with tissue effects. MATERIALS AND METHODS: Ex vivo pig skin was exposed to a miniaturized 2,940 nm AFXL, spot size 225 µm, density 5%, power levels 1.15-2.22 W, pulse durations 50-225 microseconds, pulse repetition rates 100-500 Hz, and 2, 20, or 50 stacked pulses, resulting in pulse......BACKGROUND AND OBJECTIVES: Treatment of a variety of skin disorders with ablative fractional lasers (AFXL) is driving the development of portable AFXLs. This study measures micropore dimensions produced by a small 2,940 nm AFXL using a variety of stacked pulses, and determines a model correlating...... 190 to 347 µm. CONCLUSIONS: Pulse stacking with a small, low power 2,940 nm AFXL created reproducible shallow to deep micropores, and influenced micropore configuration. Mathematical modeling established relations between laser settings and micropore dimensions, which assists in choosing laser...
Parareal algorithms with local time-integrators for time fractional differential equations
Wu, Shu-Lin; Zhou, Tao
2018-04-01
It is challenge work to design parareal algorithms for time-fractional differential equations due to the historical effect of the fractional operator. A direct extension of the classical parareal method to such equations will lead to unbalance computational time in each process. In this work, we present an efficient parareal iteration scheme to overcome this issue, by adopting two recently developed local time-integrators for time fractional operators. In both approaches, one introduces auxiliary variables to localized the fractional operator. To this end, we propose a new strategy to perform the coarse grid correction so that the auxiliary variables and the solution variable are corrected separately in a mixed pattern. It is shown that the proposed parareal algorithm admits robust rate of convergence. Numerical examples are presented to support our conclusions.
New method dynamically models hydrocarbon fractionation
Energy Technology Data Exchange (ETDEWEB)
Kesler, M.G.; Weissbrod, J.M.; Sheth, B.V. [Kesler Engineering, East Brunswick, NJ (United States)
1995-10-01
A new method for calculating distillation column dynamics can be used to model time-dependent effects of independent disturbances for a range of hydrocarbon fractionation. It can model crude atmospheric and vacuum columns, with relatively few equilibrium stages and a large number of components, to C{sub 3} splitters, with few components and up to 300 equilibrium stages. Simulation results are useful for operations analysis, process-control applications and closed-loop control in petroleum, petrochemical and gas processing plants. The method is based on an implicit approach, where the time-dependent variations of inventory, temperatures, liquid and vapor flows and compositions are superimposed at each time step on the steady-state solution. Newton-Raphson (N-R) techniques are then used to simultaneously solve the resulting finite-difference equations of material, equilibrium and enthalpy balances that characterize distillation dynamics. The important innovation is component-aggregation and tray-aggregation to contract the equations without compromising accuracy. This contraction increases the N-R calculations` stability. It also significantly increases calculational speed, which is particularly important in dynamic simulations. This method provides a sound basis for closed-loop, supervisory control of distillation--directly or via multivariable controllers--based on a rigorous, phenomenological column model.
Generalized hydrodynamic correlations and fractional memory functions
Rodríguez, Rosalio F.; Fujioka, Jorge
2015-12-01
A fractional generalized hydrodynamic (GH) model of the longitudinal velocity fluctuations correlation, and its associated memory function, for a complex fluid is analyzed. The adiabatic elimination of fast variables introduces memory effects in the transport equations, and the dynamic of the fluctuations is described by a generalized Langevin equation with long-range noise correlations. These features motivate the introduction of Caputo time fractional derivatives and allows us to calculate analytic expressions for the fractional longitudinal velocity correlation function and its associated memory function. Our analysis eliminates a spurious constant term in the non-fractional memory function found in the non-fractional description. It also produces a significantly slower power-law decay of the memory function in the GH regime that reduces to the well-known exponential decay in the non-fractional Navier-Stokes limit.
The fractional oscillator process with two indices
International Nuclear Information System (INIS)
Lim, S C; Teo, L P
2009-01-01
We introduce a new fractional oscillator process which can be obtained as a solution of a stochastic differential equation with two fractional orders. Basic properties such as fractal dimension and short-range dependence of the process are studied by considering the asymptotic properties of its covariance function. By considering the fractional oscillator process as the velocity of a diffusion process, we derive the corresponding diffusion constant, fluctuation-dissipation relation and mean-square displacement. The fractional oscillator process can also be regarded as a one-dimensional fractional Euclidean Klein-Gordon field, which can be obtained by applying the Parisi-Wu stochastic quantization method to a nonlocal Euclidean action. The Casimir energy associated with the fractional field at positive temperature is calculated by using the zeta function regularization technique
Fractional RC and LC Electrical Circuits
Directory of Open Access Journals (Sweden)
Gómez-Aguilar José Francisco
2014-04-01
Full Text Available In this paper we propose a fractional differential equation for the electrical RC and LC circuit in terms of the fractional time derivatives of the Caputo type. The order of the derivative being considered is 0 < ɣ ≤1. To keep the dimensionality of the physical parameters R, L, C the new parameter σ is introduced. This parameter characterizes the existence of fractional structures in the system. A relation between the fractional order time derivative ɣ and the new parameter σ is found. The numeric Laplace transform method was used for the simulation of the equations results. The results show that the fractional differential equations generalize the behavior of the charge, voltage and current depending of the values of ɣ. The classical cases are recovered by taking the limit when ɣ = 1. An analysis in the frequency domain of an RC circuit shows the application and use of fractional order differential equations.
Directory of Open Access Journals (Sweden)
Anda VELICANU
2010-09-01
Full Text Available This paper contains a brief description of the most important operations that can be performed on spatial data such as spatial queries, create, update, insert, delete operations, conversions, operations on the map or analysis on grid cells. Each operation has a graphical example and some of them have code examples in Oracle and PostgreSQL.
Dostal, Jiri
1993-01-01
This book provides the reader with the practical knowledge necessary to select and use operational amplifier devices. It presents an extensive treatment of applications and a practically oriented, unified theory of operational circuits.Provides the reader with practical knowledge necessary to select and use operational amplifier devices. Presents an extensive treatment of applications and a practically oriented, unified theory of operational circuits
Fractional Bateman—Feshbach Tikochinsky Oscillator
Dumitru, Baleanu; Jihad, H. Asad; Ivo, Petras
2014-02-01
In the last few years the numerical methods for solving the fractional differential equations started to be applied intensively to real world phenomena. Having these things in mind in this manuscript we focus on the fractional Lagrangian and Hamiltonian of the complex Bateman—Feshbach Tikochinsky oscillator. The numerical analysis of the corresponding fractional Euler-Lagrange equations is given within the Grünwald—Letnikov approach, which is power series expansion of the generating function.
Early Predictors of Middle School Fraction Knowledge
Bailey, Drew H.; Siegler, Robert S.; Geary, David C.
2014-01-01
Recent findings that earlier fraction knowledge predicts later mathematics achievement raise the question of what predicts later fraction knowledge. Analyses of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender. Similarly, knowledge of whole number arithmetic in first...
Utilization of Different Corn Fractions by Broilers
Costa, SIFR; Stringhini, JH; Ribeiro, AML; Pontalti, G; MacManus, C
2015-01-01
ABSTRACTThis study was conducted to evaluate the nutritional values of fractions of damaged corn. One hundred and eighty 22-d-old Cobb 500 male broilers were distributed in batteries according to a completely randomized design with six treatments of six replicates each. The treatments consisted of diets containing five corn fractions, classified as sound, fermented, insect-damaged, mold-damaged, or reference corn. The test diets consisted of 60% of reference diet + 40% of each corn fraction. ...
On the Conformable Fractional Quantum Mechanics
Mozaffari, F. S.; Hassanabadi, H.; Sobhani, H.; Chung, W. S.
2018-05-01
In this paper, a conformable fractional quantum mechanic has been introduced using three postulates. Then in such a formalism, Schr¨odinger equation, probability density, probability flux and continuity equation have been derived. As an application of considered formalism, a fractional-radial harmonic oscillator has been considered. After obtaining its wave function and energy spectrum, effects of the conformable fractional parameter on some quantities have been investigated and plotted for different excited states.
Fractional Order Models of Industrial Pneumatic Controllers
Directory of Open Access Journals (Sweden)
Abolhassan Razminia
2014-01-01
Full Text Available This paper addresses a new approach for modeling of versatile controllers in industrial automation and process control systems such as pneumatic controllers. Some fractional order dynamical models are developed for pressure and pneumatic systems with bellows-nozzle-flapper configuration. In the light of fractional calculus, a fractional order derivative-derivative (FrDD controller and integral-derivative (FrID are remodeled. Numerical simulations illustrate the application of the obtained theoretical results in simple examples.
Fractional Bateman—Feshbach Tikochinsky Oscillator
International Nuclear Information System (INIS)
Baleanu, Dumitru; Asad, Jihad H.; Petras Ivo
2014-01-01
In the last few years the numerical methods for solving the fractional differential equations started to be applied intensively to real world phenomena. Having these things in mind in this manuscript we focus on the fractional Lagrangian and Hamiltonian of the complex Bateman—Feshbach Tikochinsky oscillator. The numerical analysis of the corresponding fractional Euler-Lagrange equations is given within the Grünwald—Letnikov approach, which is power series expansion of the generating function. (physics of elementary particles and fields)
On some fractional order hardy inequalities
Directory of Open Access Journals (Sweden)
Kufner Alois
1997-01-01
Full Text Available Weighted inequalities for fractional derivatives ( fractional order Hardy-type inequalities have recently been proved in [4] and [1]. In this paper, new inequalities of this type are proved and applied. In particular, the general mixed norm case and a general twodimensional weight are considered. Moreover, an Orlicz norm version and a multidimensional fractional order Hardy inequality are proved. The connections to related results are pointed out.
A Fractional Micro-Macro Model for Crowds of Pedestrians Based on Fractional Mean Field Games
Institute of Scientific and Technical Information of China (English)
Kecai Cao; Yang Quan Chen; Daniel Stuart
2016-01-01
Modeling a crowd of pedestrians has been considered in this paper from different aspects. Based on fractional microscopic model that may be much more close to reality, a fractional macroscopic model has been proposed using conservation law of mass. Then in order to characterize the competitive and cooperative interactions among pedestrians, fractional mean field games are utilized in the modeling problem when the number of pedestrians goes to infinity and fractional dynamic model composed of fractional backward and fractional forward equations are constructed in macro scale. Fractional micromacro model for crowds of pedestrians are obtained in the end.Simulation results are also included to illustrate the proposed fractional microscopic model and fractional macroscopic model,respectively.
The fractional scaling methodology (FSM) Part 1. methodology development
International Nuclear Information System (INIS)
Novak Zuber; Ivan Catton; Upendra S Rohatgi; Wolfgang Wulff
2005-01-01
Full text of publication follows: a quantitative methodology is developed, based on the concepts of hierarchy and synthesis, to integrate and organize information and data. The methodology uses scaling to synthesize experimental data and analytical results, and to provide quantitative criteria for evaluating the effects of various design and operating parameters that influence processes in a complex system such as a nuclear power plant or a related test facility. Synthesis and scaling are performed on three hierarchical levels: the process, component and system levels. Scaling on the process level determines the effect of a selected process on a particular state variable during a selected scenario. At the component level this scaling determines the effects various processes have on a state variable, and it ranks the processes according to their importance by the magnitude of the fractional change they cause on that state variable. At the system level the scaling determines the governing processes and corresponding components, ranking these in the order of importance according to their effect on the fractional change of system-wide state variables. The scaling methodology reveals on all levels the fractional change of state variables and is called therefore the Fractional Scaling Methodology (FSM). FSM synthesizes process parameters and assigns to each thermohydraulic process a dimensionless effect metric Ω = ωt, that is the product of the specific rate of fractional change ω and the characteristic time t. The rate of fractional change ω is the ratio of process transport rate over content of a preserved quantity in a component. The effect metric Ω quantifies the contribution of the process to the fractional change of a state variable in a given component. Ordering of a component effect metrics provides the hierarchy of processes in a component, then in all components and the system. FSM separates quantitatively dominant from minor processes and components and
Vector continued fractions using a generalized inverse
International Nuclear Information System (INIS)
Haydock, Roger; Nex, C M M; Wexler, Geoffrey
2004-01-01
A real vector space combined with an inverse (involution) for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse permits construction of vector analogues of the Jacobi continued fraction. These vector Jacobi fractions are related to vector and scalar-valued polynomial functions of the vectors, which satisfy recurrence relations similar to those of orthogonal polynomials. The vector Jacobi fraction has strong convergence properties which are demonstrated analytically, and illustrated numerically
Improving Children's Knowledge of Fraction Magnitudes.
Directory of Open Access Journals (Sweden)
Lisa K Fazio
Full Text Available We examined whether playing a computerized fraction game, based on the integrated theory of numerical development and on the Common Core State Standards' suggestions for teaching fractions, would improve children's fraction magnitude understanding. Fourth and fifth-graders were given brief instruction about unit fractions and played Catch the Monster with Fractions, a game in which they estimated fraction locations on a number line and received feedback on the accuracy of their estimates. The intervention lasted less than 15 minutes. In our initial study, children showed large gains from pretest to posttest in their fraction number line estimates, magnitude comparisons, and recall accuracy. In a more rigorous second study, the experimental group showed similarly large improvements, whereas a control group showed no improvement from practicing fraction number line estimates without feedback. The results provide evidence for the effectiveness of interventions emphasizing fraction magnitudes and indicate how psychological theories and research can be used to evaluate specific recommendations of the Common Core State Standards.
Fractional calculus phenomenology in two-dimensional plasma models
Gustafson, Kyle; Del Castillo Negrete, Diego; Dorland, Bill
2006-10-01
Transport processes in confined plasmas for fusion experiments, such as ITER, are not well-understood at the basic level of fully nonlinear, three-dimensional kinetic physics. Turbulent transport is invoked to describe the observed levels in tokamaks, which are orders of magnitude greater than the theoretical predictions. Recent results show the ability of a non-diffusive transport model to describe numerical observations of turbulent transport. For example, resistive MHD modeling of tracer particle transport in pressure-gradient driven turbulence for a three-dimensional plasma reveals that the superdiffusive (2̂˜t^α where α> 1) radial transport in this system is described quantitatively by a fractional diffusion equation Fractional calculus is a generalization involving integro-differential operators, which naturally describe non-local behaviors. Our previous work showed the quantitative agreement of special fractional diffusion equation solutions with numerical tracer particle flows in time-dependent linearized dynamics of the Hasegawa-Mima equation (for poloidal transport in a two-dimensional cold-ion plasma). In pursuit of a fractional diffusion model for transport in a gyrokinetic plasma, we now present numerical results from tracer particle transport in the nonlinear Hasegawa-Mima equation and a planar gyrokinetic model. Finite Larmor radius effects will be discussed. D. del Castillo Negrete, et al, Phys. Rev. Lett. 94, 065003 (2005).
Rowing Sport in Learning Fractions of the Fourth Grade Students
Directory of Open Access Journals (Sweden)
Marhamah Fajriyah Nasution
2017-06-01
Full Text Available This study aimed to produce learning trajectory with rowing context that can help students understand addition and subtraction of fractions. Subject of the research were students IV MIN 2 Palembang. The method used was research design with three stages, those are preparing for the experiment, the design experiments, and the retrospective analysis. Learning trajectory was designed from in-formal stage to the formal stage. At the informal stage, Rowing was used as a starting point to explore the students’ knowledge of fractions. Data collection conducted through video recordings and photos to see the learning process in the classroom, written tests, observation and interviews during the learning process with the students which is the subject of research. Research produced learning trajectory consisting of a series of learning addition and subtraction of fractions dealing with the rowing. The results showed that the use of the rowing can be a bridge of students' thinking and help students in understanding the operation of addition and subtraction of fractions.
Modeling and analysis of fractional order DC-DC converter.
Radwan, Ahmed G; Emira, Ahmed A; AbdelAty, Amr M; Azar, Ahmad Taher
2017-07-11
Due to the non-idealities of commercial inductors, the demand for a better model that accurately describe their dynamic response is elevated. So, the fractional order models of Buck, Boost and Buck-Boost DC-DC converters are presented in this paper. The detailed analysis is made for the two most common modes of converter operation: Continuous Conduction Mode (CCM) and Discontinuous Conduction Mode (DCM). Closed form time domain expressions are derived for inductor currents, voltage gain, average current, conduction time and power efficiency where the effect of the fractional order inductor is found to be strongly present. For example, the peak inductor current at steady state increases with decreasing the inductor order. Advanced Design Systems (ADS) circuit simulations are used to verify the derived formulas, where the fractional order inductor is simulated using Valsa Constant Phase Element (CPE) approximation and Generalized Impedance Converter (GIC). Different simulation results are introduced with good matching to the theoretical formulas for the three DC-DC converter topologies under different fractional orders. A comprehensive comparison with the recently published literature is presented to show the advantages and disadvantages of each approach. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Using LEGO for learning fractions, supporting or distracting?
Rejeki, Sri; Setyaningsih, Nining; Toyib, Muhamad
2017-05-01
The role of games used for learning mathematics is still in debate. However, many research revealed that it gave positive effects on both students' motivation and performance in mathematics. Therefore, this study aims at investigating the effects of using LEGO-as one of games which students are familiar with, for learning mathematics, on both students' conceptual knowledge of fractions and students' attitude in learning mathematics. A set of learning activities consisting three meetings of fractions learning was designed for this study. The activities were mainly about solving word-context problems using LEGO as the model. Thirty students of seven grade with high-ability in mathematics and thirty two students with low-ability in mathematics were involved in this study. The data were collected through students' written works, video registration and field notes during the teaching and learning activities. The results indicate that in general the use of LEGO in learning activities support the conceptual understanding on fractions for both students with high-ability and low-ability in mathematics. Moreover, for students with low-ability in mathematics, it promotes the computational skill of fractions operation. The evidences also suggest that bringing LEGO into classroom activities improve students' motivation and engagement. However, in some cases, students were more focus on playing than learning. Therefore, teachers play important roles on providing clear pedagogical instructions about the way to use LEGO properly.
Fractional-order adaptive fault estimation for a class of nonlinear fractional-order systems
N'Doye, Ibrahima; Laleg-Kirati, Taous-Meriem
2015-01-01
This paper studies the problem of fractional-order adaptive fault estimation for a class of fractional-order Lipschitz nonlinear systems using fractional-order adaptive fault observer. Sufficient conditions for the asymptotical convergence of the fractional-order state estimation error, the conventional integer-order and the fractional-order faults estimation error are derived in terms of linear matrix inequalities (LMIs) formulation by introducing a continuous frequency distributed equivalent model and using an indirect Lyapunov approach where the fractional-order α belongs to 0 < α < 1. A numerical example is given to demonstrate the validity of the proposed approach.
Fractional-order adaptive fault estimation for a class of nonlinear fractional-order systems
N'Doye, Ibrahima
2015-07-01
This paper studies the problem of fractional-order adaptive fault estimation for a class of fractional-order Lipschitz nonlinear systems using fractional-order adaptive fault observer. Sufficient conditions for the asymptotical convergence of the fractional-order state estimation error, the conventional integer-order and the fractional-order faults estimation error are derived in terms of linear matrix inequalities (LMIs) formulation by introducing a continuous frequency distributed equivalent model and using an indirect Lyapunov approach where the fractional-order α belongs to 0 < α < 1. A numerical example is given to demonstrate the validity of the proposed approach.
Chromium Stable Isotope Fractionation - An Indicator of Hexavalent Chromium Reduction.
Ellis, A.; Johnson, T. M.; Bullen, T. D.
2001-12-01
fractionation during plating operations during up to 5 years of use. These results demonstrate that Cr stable isotope analyses should be a highly practical indicator of the critical chromate reduction reaction, and an otherwise useful geologic and oceanographic tool.
The dynamics of contract plasma fractionation.
Farrugia, Albert; Scaramuccia, Daniela
2017-03-01
Plasma Derived Medicinal Products (PMDPs) are an essential component of the modern therapeutic armamentarium. They are differentiated from most other medicines in several ways, particularly the unique nature of the raw material used for their manufacture. Human plasma has been fractionated to PDMPs for the past 75 years, and the economics of manufacturing requires currently that as many products are harvested from each litre as is feasible and reflective of clinical needs. PDMPs may be purchased on the open market from the various commercial and not-for-profit (NFP) manufacturers. They may also be manufactured under contract (CM) from plasma supplied by government and similar agencies as a product of blood transfusion services. Clients for CM aspire to make full use of donated plasma, hence maximizing the donors' gift after the standard components of transfusion have been harvested. Many such countries also aspire to making their national clinical needs self-sufficient in PDMPs, attempting to acquire strategic independence from the vagaries of the commercial open market. The increasing commercial imperatives operating in the PMDP sector generate a tension with such ethical aspirations which are not easily resolved. In particular, the need to harvest as many proteins as possible may generate products which are surplus to national needs, necessitating an ethical paradigm for the optimal provision of such products. In addition, traditional relationships between blood services and domestic fractionation agencies may come under stress as a result of the competitive processes underpinning such transactions, which are now subject to international norms of free trade. Blood services engaged in the supply of hospital transfusion components are detached from the pharmaceutical Good Manufacturing Practices (GMP) culture needed for the production of plasma for CM, while the generation of such plasma through extraction from whole blood donations deflects the focus from that of
14 CFR 91.503 - Flying equipment and operating information.
2010-01-01
... Turbine-Powered Multiengine Airplanes and Fractional Ownership Program Aircraft § 91.503 Flying equipment...) Emergency operation of fuel, hydraulic, electrical, and mechanical systems. (2) Emergency operation of...
The come-back of hypo fractionation?
International Nuclear Information System (INIS)
Cosset, J.M.
2005-01-01
Hypo-fractionation (i.e. the use of fewer higher fractional doses than usual) is not a new concept. It had actually been proposed in the early year of Radiotherapy by the German and Austrian specialists. In the seventy's, supported by the - wrong - hypotheses which gave birth to the NSD (Nominal Standard Dose), hypo-fractionation reappears. The consequential increase of late complications which was observed led the radiation oncologists to give up again using large doses per fraction, except for a few specific situations, such as palliative treatments. We are recently facing a new 'come-back' of hypo-fractionation, in particular for breast and prostate cancers. In the case of breast cancer, the aim is clearly to look for more 'convenience' for both the patients and the physicians, proposing shorter irradiation schedules including a lesser number of fractions. Some 'modestly' hypo-fractionated schemes have been proposed and used, without apparently altering the efficacy/toxicity ratio, but these results have been seriously questioned. As for prostate cancer, the situation is different, since in that case new radiobiological data are at the origin of the newly proposed hypo-fractionation schedules. A number of papers actually strongly suggested that the fractionation sensitivity of prostate cancer could be higher than the one of the tissues responsible for late toxicity (i.e the exact opposite of the classical dogma). Based on those data, several hypo-fractionated schemes have been proposed, with a few preliminary results looking similar to the ones obtained by the classical schedules. However, no randomized study is available so far, and a few recent radiobiological data are now questioning the new dogma of the high fractionation sensitivity of prostate cancer. For those two - frequent - cancers, it seems therefore that prudence should prevail before altering classical irradiation schedules which have proven their efficacy, while staying open to new concepts and
Influence of the void fraction in the linear reactivity model
International Nuclear Information System (INIS)
Castillo, J.A.; Ramirez, J.R.; Alonso, G.
2003-01-01
The linear reactivity model allows the multicycle analysis in pressurized water reactors in a simple and quick way. In the case of the Boiling water reactors the void fraction it varies axially from 0% of voids in the inferior part of the fuel assemblies until approximately 70% of voids to the exit of the same ones. Due to this it is very important the determination of the average void fraction during different stages of the reactor operation to predict the burnt one appropriately of the same ones to inclination of the pattern of linear reactivity. In this work a pursuit is made of the profile of power for different steps of burnt of a typical operation cycle of a Boiling water reactor. Starting from these profiles it builds an algorithm that allows to determine the voids profile and this way to obtain the average value of the same one. The results are compared against those reported by the CM-PRESTO code that uses another method to carry out this calculation. Finally, the range in which is the average value of the void fraction during a typical cycle is determined and an estimate of the impact that it would have the use of this value in the prediction of the reactivity produced by the fuel assemblies is made. (Author)
Matrix formulation of fractional supersymmetry and q-deformation
Energy Technology Data Exchange (ETDEWEB)
Benkaddour, I.
2006-02-24
Supersymmetry, which is the only non-trivial Z{sub 2} extension of the Poincare algebra, can be generalized to fractional supersymmetry, when the space time is smaller than 3. Since symmetries play an important role in physics; the principal task of quantum groups consist in extanding these standard symmetries to the deformed ones, which might be used in physics as well. This two aspects will be the main focus of this thesis. In this work, we discuss the matrix formulation of fractional supersymmetry, the q-deformation of KdV hierarchy systems and noncommutative geometry. In the first part fractional supersymmetry generated by more than one charge operator and those which can be described as a matrix model are studied. Using parafermionic field-theoretical methods, the fundamentals of two-dimensional fractional supersymmetry Q{sup k}=P are set up. Known difficulties induced by methods based on the U{sub q}(sl(2)) quantum group representations and noncommutative geometry are avoided in the parafermionic approach. Moreover, we find that fractional supersymmetric algebras are naturally realized as matrix models. The k=3 case is studied in detail. In the second part we will study the q-deformed algebra and the q-analogues of the generalised KdV hierarchy. We construct in this part the algebra of q-deformed pseudo-differential operators, shown to be an essential step toward setting up a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the q-analogues of the generalised KdV hierarchy. We focus in particular on the first leading orders of this q-deformed hierarchy, namely the q-KdV and q-Boussinesq integrable systems. We also present the q-generalisation of the conformal transformations of the currents u{sub n}, n{>=}2, and discuss the primary condition of the fields w{sub n}, n{>=}2, by using the Volterra gauge group transformations for the q-covariant Lax operators. In the last part we will discuss quantum groups and
Power filtering of n-th order in the fractional Fourier domain
Alieva, T.; Calvo, M.L.; Bastiaans, M.J.
2002-01-01
The main properties of the power filtering operation in the fractional Fourier domain and its relationship to the differentiation operation are considered. The application of linear power filtering for solving the phase retrieval problem from only intensity distributions is proposed. The optical
Use of Angle Model to Understand Addition and Subtraction of Fractions
Mukwambo, Muzwangowenyu; Ngcoza, Kenneth; Ramasike, Lineo Florence
2018-01-01
Learners in lower primary and even some in upper primary grades grapple to perform mathematical operations which involve fractions. Failure to solve these mathematical operations creates a gap in the teaching and learning processes of mathematics. We opine that this is attributed to use of traditional mathematical approaches of teaching and…
Representations of the Magnitudes of Fractions
Schneider, Michael; Siegler, Robert S.
2010-01-01
We tested whether adults can use integrated, analog, magnitude representations to compare the values of fractions. The only previous study on this question concluded that even college students cannot form such representations and instead compare fraction magnitudes by representing numerators and denominators as separate whole numbers. However,…
A Computational Model of Fraction Arithmetic
Braithwaite, David W.; Pyke, Aryn A.; Siegler, Robert S.
2017-01-01
Many children fail to master fraction arithmetic even after years of instruction, a failure that hinders their learning of more advanced mathematics as well as their occupational success. To test hypotheses about why children have so many difficulties in this area, we created a computational model of fraction arithmetic learning and presented it…
Safety of protein hydrolysates, fractions thereof and
Gertjan Schaafsma
2009-01-01
This paper evaluates the safety for humans with regard to consumption of protein hydrolysates and fractions thereof, including bioactive peptides. The available literature on the safety of protein, protein hydrolysates, fractions thereof and free amino acids on relevant food legislation is reviewed
Fractional ablative laser skin resurfacing: a review.
Tajirian, Ani L; Tarijian, Ani L; Goldberg, David J
2011-12-01
Ablative laser technology has been in use for many years now. The large side effect profile however has limited its use. Fractional ablative technology is a newer development which combines a lesser side effect profile along with similar efficacy. In this paper we review fractional ablative laser skin resurfacing.
Forced splitting of fractions in CE
Zalewski, D.R.; Schlautmann, Stefan; Gardeniers, Johannes G.E.
2008-01-01
In order to increase the electrophoretic separation between fractions of analytes on a microfluidic chip, without the need for a longer separation channel, we propose and demonstrate a preparative electrokinetic procedure by which overlapping or closely spaced fractions are automatically split. The
Fractions Learning in Children with Mathematics Difficulties
Tian, Jing; Siegler, Robert S.
2017-01-01
Learning fractions is difficult for children in general and especially difficult for children with mathematics difficulties (MD). Recent research on developmental and individual differences in fraction knowledge of children with MD and typically achieving (TA) children has demonstrated that U.S. children with MD start middle school behind their TA…
Mass fractionation processes of transition metal isotopes
Zhu, X. K.; Guo, Y.; Williams, R. J. P.; O'Nions, R. K.; Matthews, A.; Belshaw, N. S.; Canters, G. W.; de Waal, E. C.; Weser, U.; Burgess, B. K.; Salvato, B.
2002-06-01
Recent advances in mass spectrometry make it possible to utilise isotope variations of transition metals to address some important issues in solar system and biological sciences. Realisation of the potential offered by these new isotope systems however requires an adequate understanding of the factors controlling their isotope fractionation. Here we show the results of a broadly based study on copper and iron isotope fractionation during various inorganic and biological processes. These results demonstrate that: (1) naturally occurring inorganic processes can fractionate Fe isotope to a detectable level even at temperature ˜1000°C, which challenges the previous view that Fe isotope variations in natural system are unique biosignatures; (2) multiple-step equilibrium processes at low temperatures may cause large mass fractionation of transition metal isotopes even when the fractionation per single step is small; (3) oxidation-reduction is an importation controlling factor of isotope fractionation of transition metal elements with multiple valences, which opens a wide range of applications of these new isotope systems, ranging from metal-silicate fractionation in the solar system to uptake pathways of these elements in biological systems; (4) organisms incorporate lighter isotopes of transition metals preferentially, and transition metal isotope fractionation occurs stepwise along their pathways within biological systems during their uptake.
Gauge invariance and fractional quantized Hall effect
International Nuclear Information System (INIS)
Tao, R.; Wu, Y.S.
1984-01-01
It is shown that gauge invariance arguments imply the possibility of fractional quantized Hall effect; the Hall conductance is accurately quantized to a rational value. The ground state of a system showing the fractional quantized Hall effect must be degenerate; the non-degenerate ground state can only produce the integral quantized Hall effect. 12 references
Fractional Josephson vortices: oscillating macroscopic spins
Energy Technology Data Exchange (ETDEWEB)
Gaber, T.; Buckenmaier, K.; Koelle, D.; Kleiner, R.; Goldobin, E. [Universitaet Tuebingen, Physikalisches Institut - Experimentalphysik II, Tuebingen (Germany)
2007-11-15
Fractional Josephson vortices carry a magnetic flux {phi}, which is a fraction of the magnetic flux quantum {phi}{sub 0}{approx}2.07 x 10{sup -15} Wb. We consider a fractional vortex which spontaneously appears at a phase discontinuity. Its properties are very different from the properties of the usual integer fluxon. In particular, a fractional vortex is pinned and may have one of two possible polarities - just like a usual spin 1/2 particle. The fractional vortex may also oscillate around its equilibrium position with an eigenfrequency which is expected to be within the Josephson plasma gap. Using microwave spectroscopy, we investigate the dependence of the eigenfrequency of a fractional Josephson vortex on its magnetic flux {phi} and on the bias current. The experimental results are in good agreement with theoretical predictions. Positive result of this experiment is a cornerstone for further investigation of more complex fractional vortex systems such as fractional vortex molecules and tunable bandgap materials. (orig.)
Making Sense of Fractions and Percentages
Whitin, David J.; Whitin, Phyllis
2012-01-01
Because fractions and percentages can be difficult for children to grasp, connecting them whenever possible is beneficial. Linking them can foster representational fluency as children simultaneously see the part-whole relationship expressed numerically (as a fraction and as a percentage) and visually (as a pie chart). NCTM advocates these…
Stieltjes' continued fraction for the gamma function
International Nuclear Information System (INIS)
Cha, B.W.
1980-01-01
The first forty-one coefficients of a continued fraction for 1n GAMMA(z)+z-(z-1/2) 1n z-1n√2π, are given. The computation, based on Wall's algorithm for converting a function's power series representation to a continued fraction representation, was run on the algebraic manipulation system MACSYMA
On Fractional Order Hybrid Differential Equations
Directory of Open Access Journals (Sweden)
Mohamed A. E. Herzallah
2014-01-01
Full Text Available We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order 0<α<1. Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.
Cell fractionation of parasitic protozoa: a review
Directory of Open Access Journals (Sweden)
Souza Wanderley de
2003-01-01
Full Text Available Cell fractionation, a methodological strategy for obtaining purified organelle preparations, has been applied successfully to parasitic protozoa by a number of investigators. Here we present and discuss the work of several groups that have obtained highly purified subcellular fractions from trypanosomatids, Apicomplexa and trichomonads, and whose work have added substantially to our knowledge of the cell biology of these parasites.
Wigner distribution and fractional Fourier transform
Alieva, T.; Bastiaans, M.J.; Boashash, B.
2003-01-01
We have described the relationship between the fractional Fourier transform and the Wigner distribution by using the Radon-Wigner transform, which is a set of projections of the Wigner distribution as well as a set of squared moduli of the fractional Fourier transform. We have introduced the concept
Fractional-Fourier-domain weighted Wigner distribution
Stankovic, L.; Alieva, T.; Bastiaans, M.J.
2001-01-01
A fractional-Fourier-domain realization of the weighted Wigner distribution (or S-method), producing auto-terms close to the ones in the Wigner distribution itself, but with reduced cross-terms, is presented. The computational cost of this fractional-domain realization is the same as the
International Nuclear Information System (INIS)
1994-12-01
This document contains compiled data from the DOE Handbook on Airborne Release Fractions/Rates and Respirable Fractions for Nonreactor Nuclear facilities. Source data and example facilities utilized, such as the Plutonium Recovery Facility, are included
Reply to "Comment on 'Fractional quantum mechanics' and 'Fractional Schrödinger equation' ".
Laskin, Nick
2016-06-01
The fractional uncertainty relation is a mathematical formulation of Heisenberg's uncertainty principle in the framework of fractional quantum mechanics. Two mistaken statements presented in the Comment have been revealed. The origin of each mistaken statement has been clarified and corrected statements have been made. A map between standard quantum mechanics and fractional quantum mechanics has been presented to emphasize the features of fractional quantum mechanics and to avoid misinterpretations of the fractional uncertainty relation. It has been shown that the fractional probability current equation is correct in the area of its applicability. Further studies have to be done to find meaningful quantum physics problems with involvement of the fractional probability current density vector and the extra term emerging in the framework of fractional quantum mechanics.
Exact solutions of time-fractional heat conduction equation by the fractional complex transform
Directory of Open Access Journals (Sweden)
Li Zheng-Biao
2012-01-01
Full Text Available The Fractional Complex Transform is extended to solve exactly time-fractional differential equations with the modified Riemann-Liouville derivative. How to incorporate suitable boundary/initial conditions is also discussed.
Comparison of some peptidic and proteic ovine pineal fractions with a bovine pineal E5 fraction
Energy Technology Data Exchange (ETDEWEB)
Noteborn, H P; Ebels, I; Salemink, C A [State Univ. of Utrecht, Utrecht (Netherlands). Department of Organic Chemistry; Pevet, P [The Netherlands Institute for Brain Research, Amsterdam (Netherlands).; Reinharz, A C [Hopital Cantonal, Geneva (Switzerland). Department of Medicine, Division of Endocrinology; Neacsu, C [Institute of Cellular Biology and Pathology, Bucharest (Romania).
1982-01-01
Using rather simple and mild extraction and separation methods, three ovine pineal fractions (XM 300R - PP 7.2, PP 7.2' and PP 7.2S) were obtained, which contain peptidic/proteic substances and which show fluorescence characteristics of indoles. The ovine fractions were compared with the bovine pineal E5-fraction. The ovine fractions are chemically sensitive to normal laboratory light and stable in red light (..lambda.. > 600 nm). Immunologically, these fractions and the bovine E5 fraction are stable. From the results of radioimmunological experiments it was concluded that the bovine pineal E5 fraction as well as the ovine pineal fraction XM 300R - PP 7.2 and PP 7.2S may contain (a) peptide(s) ending by the same carboxy terminal tripeptide Pro-Arg-Gly(NH/sub 2/).
Identification of fractional-order systems with time delays using block pulse functions
Tang, Yinggan; Li, Ning; Liu, Minmin; Lu, Yao; Wang, Weiwei
2017-07-01
In this paper, a novel method based on block pulse functions is proposed to identify continuous-time fractional-order systems with time delays. First, the operational matrices of block pulse functions for fractional integral operator and time delay operator are derived. Then, these operational matrices are applied to convert the continuous-time fractional-order systems with time delays to an algebraic equation. Finally, the system's parameters along with the differentiation orders and the time delays are all simultaneously estimated through minimizing a quadric error function. The proposed method reduces the computation complexity of the identification process, and also it does not require the system's differentiation orders to be commensurate. The effectiveness of the proposed method are demonstrated by several numerical examples.
Spectroscopy of fractional Josephson vortex molecules
Energy Technology Data Exchange (ETDEWEB)
Goldobin, Edward; Gaber, Tobias; Buckenmaier, Kai; Kienzle, Uta; Sickinger, Hanna; Koelle, Dieter; Kleiner, Reinhold [Physikalisches Institut - Experimentalphysik II, Center for Collective Quantum Phenomena, Universitaet Tuebingen, Auf der Morgenstelle 14, D-72076 Tuebingen (Germany)
2010-07-01
Using tiny current injectors we create {kappa} discontinuities of the Josephson phase in a long Josephson junction. The junction reacts at the discontinuities by creating fractional Josephson vortices of size {lambda}{sub J} pinned at them. Such vortices carry the flux {phi}, which is a fraction of the magnetic flux quantum {phi}{sub 0}{approx}2.07 x 10{sup -15} Wb. Being pinned, a fractional vortex has an eigenfrequency (localized mode), which depends on {kappa} and applied bias current, and which lays within the plasma gap. If one considers a molecule consisting of several coupled fractional vortices, the eigenfrequency will split into several modes. We report on spectroscopy of a fractional vortex molecule performed in the thermal regime.
Generalized Fractional Derivative Anisotropic Viscoelastic Characterization
Directory of Open Access Journals (Sweden)
Harry H. Hilton
2012-01-01
Full Text Available Isotropic linear and nonlinear fractional derivative constitutive relations are formulated and examined in terms of many parameter generalized Kelvin models and are analytically extended to cover general anisotropic homogeneous or non-homogeneous as well as functionally graded viscoelastic material behavior. Equivalent integral constitutive relations, which are computationally more powerful, are derived from fractional differential ones and the associated anisotropic temperature-moisture-degree-of-cure shift functions and reduced times are established. Approximate Fourier transform inversions for fractional derivative relations are formulated and their accuracy is evaluated. The efficacy of integer and fractional derivative constitutive relations is compared and the preferential use of either characterization in analyzing isotropic and anisotropic real materials must be examined on a case-by-case basis. Approximate protocols for curve fitting analytical fractional derivative results to experimental data are formulated and evaluated.
Kumar, Sanjay
2018-01-01
In this paper, a new variant to fractional signal processing is proposed known as the Reduced Order Fractional Fourier Transform. Various properties satisfied by its transformation kernel is derived. The properties associated with the proposed Reduced Order Fractional Fourier Transform like shift, modulation, time-frequency shift property are also derived and it is shown mathematically that when the rotation angle of Reduced Order Fractional Fourier Transform approaches 90 degrees, the propos...
Integral transform method for solving time fractional systems and fractional heat equation
Directory of Open Access Journals (Sweden)
Arman Aghili
2014-01-01
Full Text Available In the present paper, time fractional partial differential equation is considered, where the fractional derivative is defined in the Caputo sense. Laplace transform method has been applied to obtain an exact solution. The authors solved certain homogeneous and nonhomogeneous time fractional heat equations using integral transform. Transform method is a powerful tool for solving fractional singular Integro - differential equations and PDEs. The result reveals that the transform method is very convenient and effective.
The realization problem for positive and fractional systems
Kaczorek, Tadeusz
2014-01-01
This book addresses the realization problem of positive and fractional continuous-time and discrete-time linear systems. Roughly speaking the essence of the realization problem can be stated as follows: Find the matrices of the state space equations of linear systems for given their transfer matrices. This first book on this topic shows how many well-known classical approaches have been extended to the new classes of positive and fractional linear systems. The modified Gilbert method for multi-input multi-output linear systems, the method for determination of realizations in the controller canonical forms and in observer canonical forms are presented. The realization problem for linear systems described by differential operators, the realization problem in the Weierstrass canonical forms and of the descriptor linear systems for given Markov parameters are addressed. The book also presents a method for the determination of minimal realizations of descriptor linear systems and an extension for cone linear syste...
Microwave Determination of Water Mole Fraction in Humid Gas Mixtures
Cuccaro, R.; Gavioso, R. M.; Benedetto, G.; Madonna Ripa, D.; Fernicola, V.; Guianvarc'h, C.
2012-09-01
A small volume (65 cm3) gold-plated quasi-spherical microwave resonator has been used to measure the water vapor mole fraction x w of H2O/N2 and H2O/air mixtures. This experimental technique exploits the high precision achievable in the determination of the cavity microwave resonance frequencies and is particularly sensitive to the presence of small concentrations of water vapor as a result of the high polarizability of this substance. The mixtures were prepared using the INRIM standard humidity generator for frost-point temperatures T fp in the range between 241 K and 270 K and a commercial two-pressure humidity generator operated at a dew-point temperature between 272 K and 291 K. The experimental measurements compare favorably with the calculated molar fractions of the mixture supplied by the humidity generators, showing a normalized error lower than 0.8.
Abro, Kashif Ali; Memon, Anwar Ahmed; Uqaili, Muhammad Aslam
2018-03-01
This research article is analyzed for the comparative study of RL and RC electrical circuits by employing newly presented Atangana-Baleanu and Caputo-Fabrizio fractional derivatives. The governing ordinary differential equations of RL and RC electrical circuits have been fractionalized in terms of fractional operators in the range of 0 ≤ ξ ≤ 1 and 0 ≤ η ≤ 1. The analytic solutions of fractional differential equations for RL and RC electrical circuits have been solved by using the Laplace transform with its inversions. General solutions have been investigated for periodic and exponential sources by implementing the Atangana-Baleanu and Caputo-Fabrizio fractional operators separately. The investigated solutions have been expressed in terms of simple elementary functions with convolution product. On the basis of newly fractional derivatives with and without singular kernel, the voltage and current have interesting behavior with several similarities and differences for the periodic and exponential sources.
Directory of Open Access Journals (Sweden)
Dipanjan Majumder
2012-01-01
Conclusions: Different fractionation of radiation has same response and toxicity in treatment of vertebral bone metastasis. Single fraction RT may be safely used to treat these cases as this is more cost effective and less time consuming. Studies may be conducted to find out particular subgroup of patients to be benefitted more by either fractionation schedule; however, our study cannot comment on that issue.
van Dongen, Joris A.; Stevens, Hieronymus P.; Parvizi, Mojtaba; van der Lei, Berend; Harmsen, Martin C.
2016-01-01
Autologous adipose tissue transplantation is clinically used to reduce dermal scarring and to restore volume loss. The therapeutic benefit on tissue damage more likely depends on the stromal vascular fraction of adipose tissue than on the adipocyte fraction. This stromal vascular fraction can be
Universal block diagram based modeling and simulation schemes for fractional-order control systems.
Bai, Lu; Xue, Dingyü
2017-05-08
Universal block diagram based schemes are proposed for modeling and simulating the fractional-order control systems in this paper. A fractional operator block in Simulink is designed to evaluate the fractional-order derivative and integral. Based on the block, the fractional-order control systems with zero initial conditions can be modeled conveniently. For modeling the system with nonzero initial conditions, the auxiliary signal is constructed in the compensation scheme. Since the compensation scheme is very complicated, therefore the integrator chain scheme is further proposed to simplify the modeling procedures. The accuracy and effectiveness of the schemes are assessed in the examples, the computation results testify the block diagram scheme is efficient for all Caputo fractional-order ordinary differential equations (FODEs) of any complexity, including the implicit Caputo FODEs. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Using wavelet multi-resolution nature to accelerate the identification of fractional order system
International Nuclear Information System (INIS)
Li Yuan-Lu; Meng Xiao; Ding Ya-Qing
2017-01-01
Because of the fractional order derivatives, the identification of the fractional order system (FOS) is more complex than that of an integral order system (IOS). In order to avoid high time consumption in the system identification, the least-squares method is used to find other parameters by fixing the fractional derivative order. Hereafter, the optimal parameters of a system will be found by varying the derivative order in an interval. In addition, the operational matrix of the fractional order integration combined with the multi-resolution nature of a wavelet is used to accelerate the FOS identification, which is achieved by discarding wavelet coefficients of high-frequency components of input and output signals. In the end, the identifications of some known fractional order systems and an elastic torsion system are used to verify the proposed method. (paper)
Ding, Xiao-Li; Nieto, Juan J.
2017-11-01
In this paper, we consider the analytical solutions of coupling fractional partial differential equations (FPDEs) with Dirichlet boundary conditions on a finite domain. Firstly, the method of successive approximations is used to obtain the analytical solutions of coupling multi-term time fractional ordinary differential equations. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the coupling FPDEs to the coupling multi-term time fractional ordinary differential equations. By applying the obtained analytical solutions to the resulting multi-term time fractional ordinary differential equations, the desired analytical solutions of the coupling FPDEs are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.
International Nuclear Information System (INIS)
Pierantozzi, T.; Vazquez, L.
2005-01-01
Through fractional calculus and following the method used by Dirac to obtain his well-known equation from the Klein-Gordon equation, we analyze a possible interpolation between the Dirac and the diffusion equations in one space dimension. We study the transition between the hyperbolic and parabolic behaviors by means of the generalization of the D'Alembert formula for the classical wave equation and the invariance under space and time inversions of the interpolating fractional evolution equations Dirac like. Such invariance depends on the values of the fractional index and is related to the nonlocal property of the time fractional differential operator. For this system of fractional evolution equations, we also find an associated conserved quantity analogous to the Hamiltonian for the classical Dirac case
Tsichritzis, Dionysios C; Rheinboldt, Werner
1974-01-01
Operating Systems deals with the fundamental concepts and principles that govern the behavior of operating systems. Many issues regarding the structure of operating systems, including the problems of managing processes, processors, and memory, are examined. Various aspects of operating systems are also discussed, from input-output and files to security, protection, reliability, design methods, performance evaluation, and implementation methods.Comprised of 10 chapters, this volume begins with an overview of what constitutes an operating system, followed by a discussion on the definition and pr
Boehme, Thomas K
1987-01-01
Operational Calculus, Volume II is a methodical presentation of operational calculus. An outline of the general theory of linear differential equations with constant coefficients is presented. Integral operational calculus and advanced topics in operational calculus, including locally integrable functions and convergence in the space of operators, are also discussed. Formulas and tables are included.Comprised of four sections, this volume begins with a discussion on the general theory of linear differential equations with constant coefficients, focusing on such topics as homogeneous and non-ho