Solving fractional Schrödinger-type spectral problems: Cauchy oscillator and Cauchy well
Żaba, Mariusz; Garbaczewski, Piotr
2014-09-01
This paper is a direct offspring of the work of Garbaczewski and Stephanovich ["Lévy flights and nonlocal quantum dynamics," J. Math. Phys. 54, 072103 (2013)] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions were made with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so-called fractional quantum mechanics spectral problems. Presently, we give a decisive computer-assisted proof, for an exemplary finite and ultimately infinite Cauchy well problem, that spectral solutions proposed so far were plainly wrong. As a constructive input, we provide an explicit spectral solution of the finite Cauchy well. The infinite well emerges as a limiting case in a sequence of deepening finite wells. The employed numerical methodology (algorithm based on the Strang splitting method) has been tested for an exemplary Cauchy oscillator problem, whose analytic solution is available. An impact of the inherent spatial nonlocality of motion generators upon computer-assisted outcomes (potentially defective, in view of various cutoffs), i.e., detailed eigenvalues and shapes of eigenfunctions, has been analyzed.
Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem
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Ya-Ning Li
2014-01-01
Full Text Available We firstly prove that β-times integrated α-resolvent operator function ((α,β-ROF satisfies a functional equation which extends that of β-times integrated semigroup and α-resolvent operator function. Secondly, for the inhomogeneous α-Cauchy problem cDtαu(t=Au(t+f(t, t∈(0,T, u(0=x0, u'(0=x1, if A is the generator of an (α,β-ROF, we give the relation between the function v(t=Sα,β(tx0+(g1*Sα,β(tx1+(gα-1*Sα,β*f(t and mild solution and classical solution of it. Finally, for the problem cDtαv(t=Av(t+gβ+1(tx, t>0, v(k(0=0, k=0,1,…,N-1, where A is a linear closed operator. We show that A generates an exponentially bounded (α,β-ROF on a Banach space X if and only if the problem has a unique exponentially bounded classical solution vx and Avx∈L loc 1(ℝ+,X. Our results extend and generalize some related results in the literature.
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Xiang-Chao Shi
2016-02-01
Full Text Available The fractional reaction diffusion equation is one of the popularly used fractional partial differential equations in recent years. The fast Adomian decomposition method is used to obtain the solution of the Cauchy problem. Also, the analytical scheme is extended to the fractional one where the Taylor series is employed. In comparison with the classical Adomian decomposition method, the ratio of the convergence is increased. The method is more reliable for the fractional partial differential equations.
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V. Vijayakumar
2014-09-01
Full Text Available In this article, we study the existence of mild solutions for nonlocal Cauchy problem for fractional neutral evolution equations with infinite delay. The results are obtained by using the Banach contraction principle. Finally, an application is given to illustrate the theory.
On the Cauchy problem for some fractional order partial differential equations
Moustafa, O L
2003-01-01
In the present paper, we study the Cauchy problem for a fractional order partial differential equation of the form D sub t supalpha((partial deriv u(x,t))/(partial deriv t))-SIGMA sub a sub b sub s sub ( sub q sub ) sub = sub 2 sub m a sub q (x,t)D sub x sup q u(x,t)=SIGMA sub a sub b sub s sub ( sub q sub ) subapprox sub 2 sub m b sub q (x,t)D sub x sup q u(x,t) where 0
On the Cauchy problem for some fractional order partial differential equations
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Moustafa, Osama L. E-mail: olmoustafa@yahoo.com
2003-09-01
In the present paper, we study the Cauchy problem for a fractional order partial differential equation of the form D{sub t}{sup {alpha}}(({partial_derivative}u(x,t))/({partial_derivative}t))-{sigma}{sub abs(q)=2m}a{sub q}(x,t)D{sub x}{sup q}u(x,t)={sigma}{sub abs(q){approx}}{sub 2m}b{sub q}(x,t)D{sub x}{sup q}u(x,t) where 0<{alpha}{<=}1 and {sigma}{sub abs(q)=2m}a{sub q}(x,t)D{sub x}{sup q} is a uniformly elliptic partial differential operator. The existence and uniqueness of the solution of the considered Cauchy problem is studied, also a continuity property is considered.
Abstract Cauchy problems three approaches
Melnikova, Irina V
2001-01-01
Although the theory of well-posed Cauchy problems is reasonably understood, ill-posed problems-involved in a numerous mathematical models in physics, engineering, and finance- can be approached in a variety of ways. Historically, there have been three major strategies for dealing with such problems: semigroup, abstract distribution, and regularization methods. Semigroup and distribution methods restore well-posedness, in a modern weak sense. Regularization methods provide approximate solutions to ill-posed problems. Although these approaches were extensively developed over the last decades by many researchers, nowhere could one find a comprehensive treatment of all three approaches.Abstract Cauchy Problems: Three Approaches provides an innovative, self-contained account of these methods and, furthermore, demonstrates and studies some of the profound connections between them. The authors discuss the application of different methods not only to the Cauchy problem that is not well-posed in the classical sense, b...
Beginnings of the Cauchy problem
Choquet-Bruhat, Yvonne
2014-01-01
Brief account of results on the Cauchy problem for the Einstein equations starting with early the works of Darmois and Lichnerowicz and going up to the proofs of the existence and uniqueness of solutions global in space and local in time, in Sobolev spaces, for the general equations either in vacuum or with classical sources.
Evolution semigroups for nonautonomous Cauchy problems
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Gregor Nickel
1997-01-01
Full Text Available In this paper, we characterize wellposedness of nonautonomous, linear Cauchy problems (NCP {u˙(t=A(tu(tu(s=x∈X on a Banach space X by the existence of certain evolution semigroups.
A Cauchy problem in nonlinear heat conduction
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De Lillo, S [Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Perugia (Italy); Lupo, G [Dipartimento di Matematica e Informatica, Universita degli Studi di Perugia, Via Vanvitelli, 1, 06123 Perugia (Italy); Sanchini, G [Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Perugia (Italy)
2006-06-09
A Cauchy problem on the semiline for a nonlinear diffusion equation is considered, with a boundary condition corresponding to a prescribed thermal conductivity at the origin. The problem is mapped into a moving boundary problem for the linear heat equation with a Robin-type boundary condition. Such a problem is then reduced to a linear integral Volterra equation of II type which admits a unique solution.
The Cauchy problem for the Pavlov equation
Grinevich, P. G.; Santini, P. M.; Wu, D.
2015-10-01
Commutation of multidimensional vector fields leads to integrable nonlinear dispersionless PDEs that arise in various problems of mathematical physics and have been intensively studied in recent literature. This report aims to solve the scattering and inverse scattering problem for integrable dispersionless PDEs, recently introduced just at a formal level, concentrating on the prototypical example of the Pavlov equation, and to justify an existence theorem for global bounded solutions of the associated Cauchy problem with small data. An essential part of this work was made during the visit of the three authors to the Centro Internacional de Ciencias in Cuernavaca, Mexico in November-December 2012.
Meyer, J C; Needham, D J
2015-03-08
In this paper, we examine a semi-linear parabolic Cauchy problem with non-Lipschitz nonlinearity which arises as a generic form in a significant number of applications. Specifically, we obtain a well-posedness result and examine the qualitative structure of the solution in detail. The standard classical approach to establishing well-posedness is precluded owing to the lack of Lipschitz continuity for the nonlinearity. Here, existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem (Meyer & Needham 2015 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations. London Mathematical Society Lecture Note Series, vol. 419) which examines the difference of the maximal and minimal solutions to the problem. From this uniqueness result, the approach of Meyer & Needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset. The comparison and continuous dependence results obtained here are novel to this class of problem. This class of problem arises specifically in the study of a one-step autocatalytic reaction, which is schematically given by A→B at rate a(p)b(q) (where a and b are the concentrations of A and B, respectively, with 0
problem has been lacking up to the present.
A Parameter Property of Classical Solutions of Cauchy Problems
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Min He
2014-01-01
Full Text Available This work is concerned with the abstract Cauchy problems that depend on parameters. The goal is to study continuity in the parameters of the classical solutions of the Cauchy problems. The situation considered in this work is when the operator of the Cauchy problem is not densely defined. By applying integrated semigroup theory and the results on continuity in the parameters of C0-semigroup and integrated semigroup, we obtain the results on the existence and continuity in parameters of the classical solutions of the Cauchy problems. The application of the obtained abstract results in a parabolic partial differential equation is discussed in the last section of the paper.
The Cauchy problem for the Pavlov equation with large data
Wu, Derchyi
2017-08-01
We prove a local solvability of the Cauchy problem for the Pavlov equation with large initial data by the inverse scattering method. The Pavlov equation arises in studies Einstein-Weyl geometries and dispersionless integrable models. Our theory yields a local solvability of Cauchy problems for a quasi-linear wave equation with a characteristic initial hypersurface.
CAUCHY PROBLEM OF ONE TYPE OF ATMOSPHERE EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
HE Juan-xiong; HE You-hua
2006-01-01
One type of evolution atmosphere equations was discussed. It is found that according to the stratification theory, (i) the inertial force has no influence on the criterion of the well-posed Cauchy problem; (ii) the compressibility plays no role on the well-posed condition of the Cauchy problem of the viscid atmosphere equations, but changes the well-posed condition of the viscid atmosphere equations; (iii) this type of atmosphere evolution equations is ill-posed on the hyperplane t = 0 in spite of its compressibility and viscosity; (iv) the Cauchy problem of compressible viscosity atmosphere with still initial motion is ill-posed.
On the Cauchy Problem Describing an Electron-Phonon Interaction
Institute of Scientific and Technical Information of China (English)
Jo(a)o-Paulo DIAS; Mário FIGUEIRA; Filipe OLIVEIRA
2011-01-01
In this paper, a model is derived to describe a quartic anharmonic interatomic interaction with an external potential involving a pair electron-phonon. The authors study the corresponding Cauchy Problem in the semilinear and quasilinear cases.
Non-dense domain operator matrices and Cauchy problems
Lalaoui Rhali, S
2002-01-01
In this work, we study Cauchy problems with non-dense domain operator matrices. By assuming that the entries of an unbounded operator matrix are Hille-Yosida operators, we give a necessary and sufficient condition ensuring that the part of this operator matrix generates a semigroup in the closure of its domain. This allows us to prove the well-posedness of the corresponding Cauchy problem. Our results are applied to delay and neutral differential equations.
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Lucio R. Berrone
2005-01-01
Full Text Available The notion of invariance under transformations (changes of coordinates of the Cauchy mean-value expression is introduced and then used in furnishing a suitable two-variable version of a result by L. Losonczi on equality of many-variable Cauchy means. An assessment of the methods used by Losonczi and Matkowski is made and an alternative way is proposed to solve the problem of representation of two-variable Cauchy means.
Cauchy problem for Laplace equation: An observer based approach
Majeed, Muhammad Usman
2013-10-01
A method to solve Cauchy Problem for Laplace equation using state observers is proposed. It is known that this problem is ill-posed. The domain under consideration is simple lipschitz in 2 with a hole. The idea is to recover the solution over whole domain from the observations on outer boundary. Proposed approach adapts one of the space variables as a time variable. The observer developed to solve Cauchy problem for the Laplace\\'s equation is compuationally robust and accurate. © 2013 IEEE.
THE CAUCHY PROBLEM FOR SOME DISPERSIVE WAVE EQUATIONS
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Zhang Wenling
2004-01-01
In this paper, we consider the Cauchy problem for some dispersive equations. By means of nonlinear estimate in Besov spaces and fixed point theory, we prove the global well-posedness of the above problem. What's more, we improve the scattering result obtained in [1].
Laplace transform generation theorems and local Cauchy problems
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Claus Müller
2004-01-01
Full Text Available We give new criterions to decide if some vector-valued function is a local Laplace transform and apply this to the theory of local Cauchy problems. This leads to an improvement of known results and new Hille-Yosida-type theorems for local convoluted semigroups.
Uniqueness Result in the Cauchy Dirichlet Problem via Mehler Kernel
Dhungana, Bishnu P.
2014-09-01
Using the Mehler kernel, a uniqueness theorem in the Cauchy Dirichlet problem for the Hermite heat equation with homogeneous Dirichlet boundary conditions on a class P of bounded functions U( x, t) with certain growth on U x ( x, t) is established.
Applications of elliptic Carleman inequalities to Cauchy and inverse problems
Choulli, Mourad
2016-01-01
This book presents a unified approach to studying the stability of both elliptic Cauchy problems and selected inverse problems. Based on elementary Carleman inequalities, it establishes three-ball inequalities, which are the key to deriving logarithmic stability estimates for elliptic Cauchy problems and are also useful in proving stability estimates for certain elliptic inverse problems. The book presents three inverse problems, the first of which consists in determining the surface impedance of an obstacle from the far field pattern. The second problem investigates the detection of corrosion by electric measurement, while the third concerns the determination of an attenuation coefficient from internal data, which is motivated by a problem encountered in biomedical imaging.
The Cauchy problem for the Pavlov equation with large data
Wu, Derchyi
2015-01-01
The Pavlov equation is one of the simplest integrable systems of vector fields arising from various problems of mathematical physics and differential geometry which are intensively studied in recent literature. In this report, solving a nonlinear Riemann-Hilbert problem via a Newtonian iteration scheme, we complete the inverse scattering theory and prove a short time unique solvability of the Cauchy problem of the Pavlov equation with large initial data.
The stability for the Cauchy problem for elliptic equations
Alessandrini, Giovanni; Rosset, Edi; Vessella, Sergio
2009-01-01
We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.
Laneev, E. B.; Mouratov, M. N.; Zhidkov, E. P.
2008-05-01
Cauchy problem for the Laplace equation with inaccurately given Cauchy conditions on an inaccurately defined arbitrary surface is considered. Discretization was performed and proved to obtain a numerical solution. An economic algorithm is proposed.
Application of homotopy analysis method for solving nonlinear Cauchy problem
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V.G. Gupta
2012-11-01
Full Text Available In this paper, by means of the homotopy analysis method (HAM, the solutions of some nonlinear Cauchy problem of parabolic-hyperbolic type are exactly obtained in the form of convergent Taylor series. The HAM contains the auxiliary parameter \\hbar that provides a convenient way of controlling the convergent region of series solutions. This analytical method is employed to solve linear examples to obtain the exact solutions. The results reveal that the proposed method is very effective and simple.
The Cauchy problem for higher order abstract differential equations
Xiao, Ti-Jun
1998-01-01
This monograph is the first systematic exposition of the theory of the Cauchy problem for higher order abstract linear differential equations, which covers all the main aspects of the developed theory. The main results are complete with detailed proofs and established recently, containing the corresponding theorems for first and incomplete second order cases and therefore for operator semigroups and cosine functions. They will find applications in many fields. The special power of treating the higher order problems directly is demonstrated, as well as that of the vector-valued Laplace transforms in dealing with operator differential equations and operator families. The reader is expected to have a knowledge of complex and functional analysis.
Institute of Scientific and Technical Information of China (English)
Chen Guowang; Xue Hongxia
2008-01-01
In this article, the existence, uniqueness and regularities of the global gener-alized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equation utt -uxx-auxxtt+bux4 - duxxt= f(u)xx are proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.
Oscillatory solutions of the Cauchy problem for linear differential equations
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Gro Hovhannisyan
2015-06-01
Full Text Available We consider the Cauchy problem for second and third order linear differential equations with constant complex coefficients. We describe necessary and sufficient conditions on the data for the existence of oscillatory solutions. It is known that in the case of real coefficients the oscillatory behavior of solutions does not depend on initial values, but we show that this is no longer true in the complex case: hence in practice it is possible to control oscillatory behavior by varying the initial conditions. Our Proofs are based on asymptotic analysis of the zeros of solutions, represented as linear combinations of exponential functions.
A formal infinite dimensional Cauchy problem and its relation to integrable hierarchies
Helminck, G.F.; Panasenko, E.A.; Sergeeva, A.O.; Ge, M.L.; Bai, C.; Jing, N.
2012-01-01
In this paper it is shown under mild assumptions that the local solvability of an infinite dimensional formal Cauchy problem is equivalent to a set of zero curvature relations. The role of this type of Cauchy problems plays in integrable systems is illustrated at the hand of lower triangular Toda hi
Continuous Dependence on Modeling in the Cauchy Problem for Nonlinear Elliptic Equations.
1987-04-01
problema di Cauchy per le equazione di tipo ellitico, Ann. Mat. Pura Appl., 46 (1958), pp. 131-153 [18] P. W. Schaefer, On the Cauchy problem for an...Continued) PP 438 PP 448 Fletcher, Jean W. Supply Problems in the Naval Reserve, Cymrot, Donald J., Military Retiremnt and Social Security: A 14 pp
The Cauchy problem for non-autonomous nonlinear Schr(o)dinger equations
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Peter Y. H. Pang; TANG Hongyan; WANG Youde
2005-01-01
In this paper we study the Cauchy problem for cubic nonlinear Schr(o)dinger equation with space-and time-dependent coefficients on Rm and Tm. By an approximation argument we prove that for suitable initial values, the Cauchy problem admits unique local solutions. Global existence is discussed in the cases of m=1,2.
Estimates for mild solutions to semilinear Cauchy problems
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Kresimir Burazin
2014-09-01
Full Text Available The existence (and uniqueness results on mild solutions of the abstract semilinear Cauchy problems in Banach spaces are well known. Following the results of Tartar (2008 and Burazin (2008 in the case of decoupled hyperbolic systems, we give an alternative proof, which enables us to derive an estimate on the mild solution and its time of existence. The nonlinear term in the equation is allowed to be time-dependent. We discuss the optimality of the derived estimate by testing it on three examples: the linear heat equation, the semilinear heat equation that models dynamic deflection of an elastic membrane, and the semilinear Schrodinger equation with time-dependent nonlinearity, that appear in the modelling of numerous physical phenomena.
A NOTE ON "THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS"
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Hongxia GUO; Guowang CHEN
2013-01-01
In this article,we prove that the Cauchy problem for a N-dimensional system of nonlinear wave equations utt-a△utt =△f(u,v),x ∈ RN,t ＞ 0,vtt-a△vtt =△g(u,v),x ∈ RN,t ＞ 0 admits a unique global generalized solution in C3([0,∞); Wm,p(RN) ∩ L∞(RN) f∩ L2(RN))(m ≥ 0 is an integer,1 ≤ p ≤ ∞) and a unique global classical solution in C3([0,∞); Wm,p ∩ L∞ ∩ L2) (m ＞ 2 + N/p),the sufficient conditions of the blow up of the solution in finite time are given,and also two examples are given.
The Cauchy problem of scalar-tensor theories of gravity
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Salgado, Marcelo [Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Apdo. Postal 70-543 Mexico 04510 DF (Mexico)
2006-07-21
The 3 + 1 formulation of scalar-tensor theories of gravity (STT) is obtained in the physical (Jordan) frame departing from the 4 + 0 covariant field equations. Contrary to common belief (folklore), the new system of ADM-like equations shows that the Cauchy problem of STT is well formulated (in the sense that the whole system of evolution equations is of first order in the time derivative). This is the first step towards a full first-order (in time and space) formulation from which a subsequent hyperbolicity analysis (a well-posedness determination) can be performed. Several gauge (lapse and shift) conditions are considered and implemented for STT. In particular, a generalization of the harmonic gauge for STT allows us to prove the well posedness of the STT using a second-order analysis which is very similar to the one employed in general relativity. Several appendices complement the ideas of the main part of the paper.
GLOBALLY SMOOTH SOLUTIONS TO CAUCHY PROBLEM OF A QUASILINEAR HYPERBOLIC SYSTEM ARISING IN BIOLOGY
Institute of Scientific and Technical Information of China (English)
郑永树
2001-01-01
This article considers Cauchy problemut- (uv)x = 0,vt- ux =0, (E)v(x,0) = u0(x) ＞ 0, v(x,0) = v0(x). (Ⅰ)A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global G1-solution on t ＞ 0 is obtained.
A Generalized Cauchy Distribution Framework for Problems Requiring Robust Behavior
Carrillo, Rafael E.; Aysal, Tuncer C.; Barner, Kenneth E.
2010-12-01
Statistical modeling is at the heart of many engineering problems. The importance of statistical modeling emanates not only from the desire to accurately characterize stochastic events, but also from the fact that distributions are the central models utilized to derive sample processing theories and methods. The generalized Cauchy distribution (GCD) family has a closed-form pdf expression across the whole family as well as algebraic tails, which makes it suitable for modeling many real-life impulsive processes. This paper develops a GCD theory-based approach that allows challenging problems to be formulated in a robust fashion. Notably, the proposed framework subsumes generalized Gaussian distribution (GGD) family-based developments, thereby guaranteeing performance improvements over traditional GCD-based problem formulation techniques. This robust framework can be adapted to a variety of applications in signal processing. As examples, we formulate four practical applications under this framework: (1) filtering for power line communications, (2) estimation in sensor networks with noisy channels, (3) reconstruction methods for compressed sensing, and (4) fuzzy clustering.
A Generalized Cauchy Distribution Framework for Problems Requiring Robust Behavior
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Carrillo RafaelE
2010-01-01
Full Text Available Statistical modeling is at the heart of many engineering problems. The importance of statistical modeling emanates not only from the desire to accurately characterize stochastic events, but also from the fact that distributions are the central models utilized to derive sample processing theories and methods. The generalized Cauchy distribution (GCD family has a closed-form pdf expression across the whole family as well as algebraic tails, which makes it suitable for modeling many real-life impulsive processes. This paper develops a GCD theory-based approach that allows challenging problems to be formulated in a robust fashion. Notably, the proposed framework subsumes generalized Gaussian distribution (GGD family-based developments, thereby guaranteeing performance improvements over traditional GCD-based problem formulation techniques. This robust framework can be adapted to a variety of applications in signal processing. As examples, we formulate four practical applications under this framework: (1 filtering for power line communications, (2 estimation in sensor networks with noisy channels, (3 reconstruction methods for compressed sensing, and (4 fuzzy clustering.
THE CAUCHY PROBLEM FOR A CLASS OF DOUBLY DEGENERATE NONLINEAR PARABOLIC EQUATION
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
This article studies the Cauchy problem for a class of doubly nonlinear deauthor considers its regularized problem and establishes some estimates. On the basis of the estimates, the existence and uniqueness of the generalized solutions in BV space are proved.
INITIAL VALUE PROBLEM FOR FRACTIONAL ORDER EQUATION WITH CONSTANT COEFFICIENTS
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Bogatyreva F. T.
2016-12-01
Full Text Available In this paper we construct an explicit representation of the solution of the Cauchy problem for ordinary differential equation of fractional order with Dzhrbashyan-Nersesyan operators.
The Cauchy Problem of Scalar Tensor Theories of Gravity
Salgado, M
2006-01-01
The 3+1 formulation of scalar-tensor theories of gravity (STT) is obtained in the physical (Jordan) frame departing from the 4+0 covariant field equations. Contrary to the common believe (folklore), the new system of ADM-like equations shows that the Cauchy problem of STT is well formulated (in the sense that the whole system is of first order in the time-derivative). This is the first step towards a full first order (in time and space) formulation from which a subsequent hyperbolicity analysis (a well-posedness determination) can be performed. Several gauge (lapse and shift) conditions are considered and implemented for STT. In particular, a generalization of the harmonic gauge for STT allows to prove the well posedness of the STT using a second order analysis which is very similar to the one used in general relativity. Some spacetimes of astrophysical and cosmological interest are considered as specific applications. Several appendices complement the ideas of the main part of the paper.
Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow
Zhijian, Yang
2006-01-01
The paper studies the existence, both locally and globally in time, stability, decay estimates and blowup of solutions to the Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow. Under the assumption that the nonlinear term of the equations is of polynomial growth order, say [alpha], it proves that when [alpha]>1, the Cauchy problem admits a unique local solution, which is stable and can be continued to a global solution under rather mild conditions; when [alpha][greater-or-equal, slanted]5 and the initial data is small enough, the Cauchy problem admits a unique global solution and its norm in L1,p(R) decays at the rate for 2
Cauchy problem blow up in finite time.
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Hongwu Zhang
2011-08-01
Full Text Available In this article, we study a Cauchy problem for an elliptic equation with variable coefficients. It is well-known that such a problem is severely ill-posed; i.e., the solution does not depend continuously on the Cauchy data. We propose a modified quasi-boundary value regularization method to solve it. Convergence estimates are established under two a priori assumptions on the exact solution. A numerical example is given to illustrate our proposed method.
WEAK REGULARIZATION FOR A CLASS OF ILL-POSED CAUCHY PROBLEMS
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contained in a sector of right-half complex plane and its resolvent is polynomially bounded, the weak regularization for such ill-posed Cauchy problem can be shown by using the quasi-reversibility method and regularized semigroups. Finally, an example is given.
The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions
Choquet-Bruhat, Yvonne; Martín-García, José M
2010-01-01
We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary dimensions. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.
L~p-L~q decay estimates of solutions to Cauchy problems of thermoviscoelastic systems
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YANG Lin; HUANG Li-hong; KUANG Feng-lian
2009-01-01
Lp-Lq decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. Then with the help of the information of characteristic roots for the coefficient matrix of the system, L~p-L~q decay estimate of parabolic type of solution to the Cauchy problem is obtained.
Cauchy-Riemann方程解的延拓问题%The Extended Problems of Solution for Cauchy-Riemann Equation
Institute of Scientific and Technical Information of China (English)
马忠泰; 赵红玲
2005-01-01
This paper, we discuss the solutions' characterize of Cauchy-Riemann equation and the extension phenomenon of Hartogs in Cn and, a series of new extended results of the solutions for Cauchy-Riemann equations is obtained by using the latest developments of the solutions' extension. Furthermore, the case of the extension's limitation for the solutions is also given.
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Eva Lowen-Colebunders
1982-01-01
Full Text Available A family C of filters on a set X is uniformizable if there is a uniformity on X such that C is its collection of Cauchy filters. Using the theory of completions and Cauchy continuous maps for Cauchy spaces, we obtain characterizations of uniformizable Cauchy spaces. In particular, given a Cauchy structure C on X we investigate under what conditions the filter u(C=⋂F∈CF×F is a uniformity and C is its collection of Cauchy filters. This problem is treated using Cauchy covering systems.
A Fourth-Order Modified Method for the Cauchy Problem of the Modified Helmholtz Equation
Institute of Scientific and Technical Information of China (English)
R. Shi; H. H. Qin
2009-01-01
This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0 < x≤ 1, y ∈ R. The Cauchy data at x = 0 is given and the solution is then sought for the interval 0 < x ≤1. This problem is highly ill-posed and the solution (if it exists) does not depend continuously on the given data. In this paper, we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution. Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.
Integral Formula for the Characteristic Cauchy Problem on a curved Background
Joudioux, Jérémie
2009-01-01
We give a local integral formula, valid on general curved space-times, for the characteristic Cauchy problem for the Dirac equation with arbitrary spin using the method developed by Friedlander in his book "the wave equation on a curved spacetime" (1975). The results obtained by Penrose in the flat case in "Null hypersurface initial data for classical fields of arbitrary spin for general relativity" (Gen. Rel. Grav 1980) are recovered directly. It is expected that this method can be used to obtain sharp estimates for the characteristic Cauchy problem for the Dirac equation.
THE CAUCHY PROBLEM FOR THE SYSTEM OF EQUATIONS OF THERMOELASTICITY IN E^n
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Ikbol E. Niyozov
2014-05-01
Full Text Available ABSTRACT: In this paper we consider the problem of analytical continuation of solutions to the system of equations of thermoelasticity in a bounded domain from their values and values of their strains on a part of the boundary of this domain, i.e., we study the Cauchy problem. ABSTRAK: Di dalam kajian ini, kami menyelidiki masalah keselanjaran analitik bagi penyelesaian-penyelesaian terhadap sistem persamaan-persamaan termoelastik di dalam domain bersempadan berdasarkan nilai-nilainya dan nilai tegasannya bagi sebahagian daripada sempadan domain tersebut, iaitu kami mengkaji masalah Cauchy.
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Danuta Jaruszewska-Walczak
1994-05-01
Full Text Available We formulate a criterion of uniqueness of solutions of a Cauchy problem using the comparison function of the Kamke type. This will be a generalization of classical results concerning first order equations with partial derivatives. We prove that the uniqueness criteria of Perron and Kamke type for differential-function problems are equivalent if given functions are continuous.
Danuta Jaruszewska-Walczak
1994-01-01
We formulate a criterion of uniqueness of solutions of a Cauchy problem using the comparison function of the Kamke type. This will be a generalization of classical results concerning first order equations with partial derivatives. We prove that the uniqueness criteria of Perron and Kamke type for differential-function problems are equivalent if given functions are continuous.
An optimal iterative algorithm to solve Cauchy problem for Laplace equation
Majeed, Muhammad Usman
2015-05-25
An optimal mean square error minimizer algorithm is developed to solve severely ill-posed Cauchy problem for Laplace equation on an annulus domain. The mathematical problem is presented as a first order state space-like system and an optimal iterative algorithm is developed that minimizes the mean square error in states. Finite difference discretization schemes are used to discretize first order system. After numerical discretization algorithm equations are derived taking inspiration from Kalman filter however using one of the space variables as a time-like variable. Given Dirichlet and Neumann boundary conditions are used on the Cauchy data boundary and fictitious points are introduced on the unknown solution boundary. The algorithm is run for a number of iterations using the solution of previous iteration as a guess for the next one. The method developed happens to be highly robust to noise in Cauchy data and numerically efficient results are illustrated.
The Cauchy Problem for Schrödinger Equations with Time-Dependent Hamiltonian
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Massimo Cicognani
2014-12-01
Full Text Available We consider the Cauchy problem for a Schrödinger equation with an Hamiltonian depending also on the time variable and that may vanish at t = 0. We find optimal Levi conditions for well-posedness in Sobolev and Gevrey spaces.
Cauchy Problem of Some Doubly Degenerate Parabolic Equations with Initial Datum a Measure
Institute of Scientific and Technical Information of China (English)
Hui Jun FAN
2004-01-01
This paper discusses the Cauchy problem of the equation ut=△·（|△um|p-2△um）-uq with initial datum a measure. Under the assumption of the parameters, one proves the existence and non-existence of the non-negative generalized solution.
CAUCHY PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS WITH CHARACTERISTICS WITH CONSTANT MULTIPLICITY
Institute of Scientific and Technical Information of China (English)
Wang Libin
2004-01-01
For quasilinear hyperbolic systems with characteristics of constant multiplicity, suppose that characteristics of constant multiplicity(＞1) are linearly degenerate, by means of generalized normalized coordinates we get the global existence and the blow-up phenomenon of the C1 solution to the Cauchy problem under an additional hypothesis.
POINTWISE ESTIMATES OF SOLUTIONS TO CAUCHY PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS
Institute of Scientific and Technical Information of China (English)
WANG WEIKE; YANG XIONGFENG
2003-01-01
This paper considers the pointwise estimate of the solutions to Cauchy problem for quasilin-ear hyperbolic systems, which bases on the existence of the solutions by using the fundamentalsolutions. It gives a sharp pointwise estimates of the solutions on domain under consideration.Specially, the estimate is precise near each characteristic direction.
EXISTENCE AND UNIQUENESS OF THE CAUCHY PROBLEM FOR A GENERALIZED NAVIER-STOKES EQUATIONS
Institute of Scientific and Technical Information of China (English)
刘晓风
2003-01-01
We consider the Cauchy problem for a generalized Navier-Stokes equations with hyperdissipation, with the initial data in Lpσ. We follow the theme of [1] but with more complicated analysis on the symbol and obtain the existence and uniqueness results.
The Cauchy problem for a model of immiscible gas flow with large data
Energy Technology Data Exchange (ETDEWEB)
Sande, Hilde
2008-12-15
The thesis consists of an introduction and two papers; 1. The solution of the Cauchy problem with large data for a model of a mixture of gases. 2. Front tracking for a model of immiscible gas flow with large data. (AG) refs, figs
Cauchy Problem for Quasilinear Hyperbolic Systems with Higher Order Dissipative Terms
Institute of Scientific and Technical Information of China (English)
Wei-guo Zhang
2003-01-01
In this paper, the author studies the global existence, singularities and life span of smooth solutions of the Cauchy problem for a class of quasilinear hyperbolic systems with higher order dissipative terms and gives their applications to nonlinear wave equations with higher order dissipative terms.
Cauchy problem and initial traces for a doubly nonlinear degenerate parabolic equation
Institute of Scientific and Technical Information of China (English)
赵俊宁; 徐中海
1996-01-01
The Cauchy problem and initial traces for the doubly degenerate parabolic equationsare studied. Under certain growth condition on the initial datum u0(x) as the existence of solution is proved. The results obtained are optimal in the dass of nonnegative locally bounded solution, for which a Harnack-type inequality holds.
Global existence and decay of solutions of the Cauchy problem in thermoelasticity with second sound
Kasimov, Aslan R.
2013-06-04
We consider the one-dimensional Cauchy problem in non-linear thermoelasticity with second sound, where the heat conduction is modelled by Cattaneo\\'s law. After presenting decay estimates for solutions to the linearized problem, including refined estimates for data in weighted Lebesgue-spaces, we prove a global existence theorem for small data together with improved decay estimates, in particular for derivatives of the solutions. © 2013 Taylor & Francis.
An inverse source problem of the Poisson equation with Cauchy data
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Ji-Chuan Liu
2017-05-01
Full Text Available In this article, we study an inverse source problem of the Poisson equation with Cauchy data. We want to find iterative algorithms to detect the hidden source within a body from measurements on the boundary. Our goal is to reconstruct the location, the size and the shape of the hidden source. This problem is ill-posed, regularization techniques should be employed to obtain the regularized solution. Numerical examples show that our proposed algorithms are valid and effective.
A robust data completion method for 2D Laplacian Cauchy problems
Energy Technology Data Exchange (ETDEWEB)
Delvare, F [Laboratoire Energetique Explosions Structures and Institut PRISME, Universite d' Orleans and ENSI de Bourges, 88 Boulevard Lahitolle, 18020 Bourges Cedex (France); Cimetiere, A [Laboratoire de Metallurgie Physique, Universite de Poitiers and ENSMA Poitiers, Boulevard Marie et Pierre CURIE, Teleport 2, BP 30179, 86962 Futuroscope Chasseneuil Cedex (France)], E-mail: franck.delvare@ensi-bourges.fr, E-mail: alain.cimetiere@univ-poitiers.fr
2008-11-01
The purpose is to propose an improved regularization method for data completion problems. This method is presented on the Cauchy problem for the Laplace equation in 2D situations. Many numerical simulations using finite element method highlight the efficiency of this new approach. In particular, it gives reconstructions with an increased accuracy, it is stable with respect to strong perturbations on the data and is able to deblur noisy data.
Global solution to the Cauchy problem on a universe fireworks model
Zhenglu Jiang
2008-01-01
We prove existence and uniqueness of the global solution to the Cauchy problem on a universe fireworks model with finite total mass at the initial state when the ratio of the mass surviving the explosion, the probability of the explosion of fragments and the probability function of the velocity change of a surviving particle satisfy the corresponding physical conditions. Although the nonrelativistic Boltzmann-like equation modeling the universe fireworks is mathematically easy, this paper leads rather theoretically to an understanding of how to construct contractive mappings in a Banach space for the proof of the existence and uniqueness by means of methods taken from the famous work by DiPerna & Lions about the Boltzmann equation. We also show both the regularity and the time-asymptotic behavior of solution to the Cauchy problem.
Institute of Scientific and Technical Information of China (English)
2008-01-01
In this article, we consider the existence of local and global solution to the Cauchy problem of a doubly nonlinear equation. By introducing the norms |||f|||h and
On the Blow-up Phenomena of Cauchy Problem for the Camassa- Holm Equation
Institute of Scientific and Technical Information of China (English)
LIU Yongqin; WANG Weike
2006-01-01
We focus on the blow-up phenomena of Cauchy problem for the Camassa-Holm equation. Blow-up can occur only in the form of wave-breaking, i.e. the solution is bounded but its slope becomes unbounded in finite time. We proved that there is such a point that its slope becomes infinite exactly at breaking time. We also gave the precise blow-up rate and the blow-up set.
GENERAL CAUCHY PROBLEM FOR THE LINEAR SHALLOW -WATER EQUATIONS ON AN EQUATORIAL BETA-PLANE
Institute of Scientific and Technical Information of China (English)
SHEN Chun; SHI Wei-hui
2006-01-01
Based on the theory of stratification, the well-posedness of the initial value problem for the linear shallow-water equations on an equatorial beta-plane was discussed. The sufficient and necessary conditions of the existence and uniqueness for the local solution of the equations were presented and the existence conditions for formal solutions of the equations were also given. For the Cauchy problem on the hyper-plane, the local analytic solution were worked out and a special case was discussed. Finally, an example was used to explain the variety of formal solutions for the ill-posed problem.
Fourier Moment Method with Regularization for the Cauchy Problem of Helmholtz Equation
Institute of Scientific and Technical Information of China (English)
MA YUN-YUN; MA FU-MING
2012-01-01
In this paper,we consider the reconstruction of the wave field in a bounded domain.By choosing a special family of functions,the Cauchy problem can be transformed into a Fourier moment problem.This problem is ill-posed.We propose a regularization method for obtaining an approximate solution to the wave field on the unspecified boundary.We also give the convergence analysis and error estimate of the numerical algorithm.Finally,we present some numerical examples to show the effectiveness of this method.
Waves attractors in rotating fluids a paradigm for ill-posed Cauchy problems
Rieutord, M; Valdettaro, L
2000-01-01
In the limit of low viscosity, we show that the amplitude of the modes of oscillation of a rotating fluid, namely inertial modes, concentrate along an attractor formed by a periodic orbit of characteristics of the underlying hyperbolic Poincar\\'e equation. The dynamics of characteristics is used to elaborate a scenario for the asymptotic behaviour of the eigenmodes and eigenspectrum in the physically relevant r\\'egime of very low viscosities which are out of reach numerically. This problem offers a canonical ill-posed Cauchy problem which has applications in other fields.
The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps
DEFF Research Database (Denmark)
Brander, David; Svensson, Martin
2013-01-01
The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces...... representation for surfaces associated with Lorentzian harmonic maps (1-1 wave maps) into symmetric spaces, developed since the 1990's. Explicit formulae for the potentials in terms of the prescribed data are given, and some applications are considered....
Global well-posedness for nonlinear nonlocal Cauchy problems arising in elasticity
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Hantaek Bae
2017-02-01
Full Text Available In this article, we prove global well-posedness for a family of one dimensional nonlinear nonlocal Cauchy problems arising in elasticity. We consider the equation $$ u_{tt}-\\delta Lu_{xx}=\\big(\\beta \\ast [(1-\\deltau+u^{2n+1}]\\big_{xx}\\,, $$ where $L$ is a differential operator, $\\beta$ is an integral operator, and $\\delta =0$ or 1. (Here, the case $\\delta=1$ represents the additional doubly dispersive effect. We prove the global well-posedness of the equation in energy spaces.
On existence of weak solutions to a Cauchy problem for one class of conservation laws
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P. I. Kogut
2015-02-01
Full Text Available We discuss the existence of weak solutions to the Cauchy problem for one classof hyperbolic conservation laws that models a highly re-entrant production system.The output of the factory is described as a function of the work in progress and theposition of the so-called push-pull point (PPP where we separate the beginning ofthe factory employing a push policy from the end of the factory, which uses a pullpolicy. The main question we discuss in this paper is about the optimal choice ofthe input in-ux, push and pull constituents, and the position of PPP.
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Rubio Gerardo
2011-03-01
Full Text Available We consider the Cauchy problem in ℝd for a class of semilinear parabolic partial differential equations that arises in some stochastic control problems. We assume that the coefficients are unbounded and locally Lipschitz, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution by approximation with linear parabolic equations. The linear equations involved can not be solved with the traditional results. Therefore, we construct a classical solution to the linear Cauchy problem under the same hypotheses on the coefficients for the semilinear equation. Our approach is using stochastic differential equations and parabolic differential equations in bounded domains. Finally, we apply the results to a stochastic optimal consumption problem. Nous considérons le problème de Cauchy dans ℝd pour une classe d’équations aux dérivées partielles paraboliques semi linéaires qui se pose dans certains problèmes de contrôle stochastique. Nous supposons que les coefficients ne sont pas bornés et sont localement Lipschitziennes, pas nécessairement différentiables, avec des données continues et ellipticité local uniforme. Nous construisons une solution classique par approximation avec les équations paraboliques linéaires. Les équations linéaires impliquées ne peuvent être résolues avec les résultats traditionnels. Par conséquent, nous construisons une solution classique au problème de Cauchy linéaire sous les mêmes hypothèses sur les coefficients pour l’équation semi-linéaire. Notre approche utilise les équations différentielles stochastiques et les équations différentielles paraboliques dans les domaines bornés. Enfin, nous appliquons les résultats à un problème stochastique de consommation optimale.
CAUCHY PROBLEM FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS
Institute of Scientific and Technical Information of China (English)
Lian Ruxu; Liu Jian; Li Hailiang; Xiao Ling
2012-01-01
We consider the Cauchy problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient.For regular initial data,we show that the unique strong solution exits globally in time and converges to the equilibrium state time asymptotically.When initial density is piecewise regular with jump discontinuity,we show that there exists a unique global piecewise regular solution. In particular,the jump discontinuity of the density decays exponentially and the piecewise regular solution tends to the equilibrium state as t→ +∞.
On the Cauchy problem for nonlinear Schrödinger equations with rotation
Antonelli, Paolo
2011-10-01
We consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superuid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in [11, 12]. Moreover, we find that the rotation term has a considerable in fiuence in proving finite time blow-up in the focusing case.
The Cauchy Boundary Value Problems on Closed Piecewise Smooth Manifolds in Cn
Institute of Scientific and Technical Information of China (English)
Liang Yu LIN; Chun Hui QIU
2004-01-01
Suppose that D is a bounded domain with a piecewise C1 smooth boundary in Cn. Let ψ∈ C1+α((б)D). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Bochner-Martinelli kernel, which has integral density ψ. Moreover,by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauchy boundary value problem with the Bochner-Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.
A dimension decomposition approach based on iterative observer design for an elliptic Cauchy problem
Majeed, Muhammad Usman
2015-07-13
A state observer inspired iterative algorithm is presented to solve boundary estimation problem for Laplace equation using one of the space variables as a time-like variable. Three dimensional domain with two congruent parallel surfaces is considered. Problem is set up in cartesian co-ordinates and Laplace equation is re-written as a first order state equation with state operator matrix A and measurements are provided on the Cauchy data surface with measurement operator C. Conditions for the existence of strongly continuous semigroup generated by A are studied. Observability conditions for pair (C, A) are provided in infinite dimensional setting. In this given setting, special observability result obtained allows to decompose three dimensional problem into a set of independent two dimensional sub-problems over rectangular cross-sections. Numerical simulation results are provided.
A Series Solution of the Cauchy Problem for Turing Reaction-diffusion Model
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L. Päivärinta
2011-12-01
Full Text Available In this paper, the series pattern solution of the Cauchy problem for Turing reaction-diffusion model is obtained by using the homotopy analysis method (HAM. Turing reaction-diffusion model is nonlinear reaction-diffusion system which usually has power-law nonlinearities or may be rewritten in the form of power-law nonlinearities. Using the HAM, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a series of functions which converges rapidly to the exact solution of the problem. The efficiency of the approach will be shown by applying the procedure on two problems. Furthermore, the so-called homotopy-Pade technique (HPT is applied to enlarge the convergence region and rate of solution series given by the HAM.
The Cauchy problem for a class of pseudodifferential equations over p-adic field
Chuong, Nguyen Minh; Co, Nguyen Van
2008-04-01
In this paper, some classes much more general than the one in [N.M. Chuong, Yu.V. Egorov, A. Khrennikov, Y. Meyer, D. Mumford (Eds.), Harmonic, Wavelet and p-Adic Analysis, World Scientific, Singapore, 2007] of Cauchy problems for an interesting class of pseudodifferential equations over p-adic fields are studied. The used functions belong to mixed classes of real and p-adic functions. Even for p-adic partial differential equations such problems in such function spaces have not been discussed yet. The established mathematical foundation requires very complicated and very difficult proofs. Days after days, these equations occur increasingly in mathematical physics, quantum mechanics. Explicit solutions of such problems are very needed for specialists on applied mathematics, physics, and engineering.
The Cauchy problem in hybrid metric-Palatini f(X)-gravity
Capozziello, Salvatore; Lobo, Francisco S N; Olmo, Gonzalo J; Vignolo, Stefano
2013-01-01
The well-formulation and the well-posedness of the Cauchy problem is discussed for {\\it hybrid metric-Palatini gravity}, a recently proposed modified gravitational theory consisting of adding to the Einstein-Hilbert Lagrangian an $f(R)$ term constructed {\\it \\`{a} la} Palatini. The theory can be recast as a scalar-tensor one predicting the existence of a light long-range scalar field that evades the local Solar System tests and is able to modify galactic and cosmological dynamics, leading to the late-time cosmic acceleration. In this work, adopting generalized harmonic coordinates, we show that the initial value problem can always be {\\it well-formulated} and, furthermore, can be {\\it well-posed} depending on the adopted matter sources.
Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems
Ducrot, A.; Liu, Z.; Magal, P.
2008-05-01
This paper is devoted to the study of the essential growth rate of some class of semigroup generated by bounded perturbation of some non-densely defined problem. We extend some previous results due to Thieme [H.R. Thieme, Quasi-compact semigroups via bounded perturbation, in: Advances in Mathematical Population Dynamics--Molecules, Cells and Man, Houston, TX, 1995, in: Ser. Math. Biol. Med., vol. 6, World Sci. Publishing, River Edge, NJ, 1997, pp. 691-711] to a class of non-densely defined Cauchy problems in Lp. In particular in the context the integrated semigroup is not operator norm locally Lipschitz continuous. We overcome the lack of Lipschitz continuity of the integrated semigroup by deriving some weaker properties that are sufficient to give information on the essential growth rate.
Institute of Scientific and Technical Information of China (English)
2000-01-01
This article considers Cauchy problem for quasilinear hyperbolic systems in diagonal form.A necessary and sufficient condition in guaranteeing that Cauchy problem admits a unique global classical solution on t 0 is obtained,and a sharp estimate of the life span for the classical solution is given.
Directory of Open Access Journals (Sweden)
Dhaou Lassoued
2013-04-01
Full Text Available We prove that a family of $q$-periodic continuous matrix valued function ${A(t}_{tin mathbb{R}}$ has an exponential dichotomy with a projector $P$ if and only if $int_0^t e^{imu s}U(t,sPds$ is bounded uniformly with respect to the parameter $mu$ and the solution of the Cauchy operator Problem $$displaylines{ dot{Y}(t=-Y(tA(t+ e^{i mu t}(I-P ,quad tgeq s cr Y(s=0, }$$ has a limit in $mathcal{L}(mathbb{C}^n$ as s tends to $-infty$ which is bounded uniformly with respect to the parameter $mu$. Here, ${ U(t,s: t, sinmathbb{R}}$ is the evolution family generated by ${A(t}_{tin mathbb{R}}$, $mu$ is a real number and q is a fixed positive number.
Cauchy problem for multiscale conservation laws: Application to structured cell populations
Shang, Peipei
2010-01-01
In this paper, we study a vector conservation law that models the growth and selection of ovarian follicles. During each ovarian cycle, only a definite number of follicles ovulate, while the others undergo a degeneration process called atresia. This work is motivated by a multiscale mathematical model starting on the cellular scale, where ovulation or atresia result from a hormonally controlled selection process. A two-dimensional conservation law describes the age and maturity structuration of the follicular cell populations. The densities intersect through a coupled hyperbolic system between different follicles and cell phases, which results in a vector conservation law and coupling boundary conditions. The maturity velocity functions possess both a local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with bounded initial and boundary data.
Mutational analysis a joint framework for Cauchy problems in and beyond vector spaces
Lorenz, Thomas
2010-01-01
Ordinary differential equations play a central role in science and have been extended to evolution equations in Banach spaces. For many applications, however, it is difficult to specify a suitable normed vector space. Shapes without a priori restrictions, for example, do not have an obvious linear structure. This book generalizes ordinary differential equations beyond the borders of vector spaces with a focus on the well-posed Cauchy problem in finite time intervals. Here are some of the examples: - Feedback evolutions of compact subsets of the Euclidean space - Birth-and-growth processes of random sets (not necessarily convex) - Semilinear evolution equations - Nonlocal parabolic differential equations - Nonlinear transport equations for Radon measures - A structured population model - Stochastic differential equations with nonlocal sample dependence and how they can be coupled in systems immediately - due to the joint framework of Mutational Analysis. Finally, the book offers new tools for modelling.
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Milena Netka
2009-01-01
Full Text Available The paper is concerned with weak solutions of a generalized Cauchy problem for a nonlinear system of first order differential functional equations. A theorem on the uniqueness of a solution is proved. Nonlinear estimates of the Perron type are assumed. A method of integral functional inequalities is used.
Ill-posedness of the Cauchy problem for totally degenerate system of conservation laws
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Wladimir Neves
2005-11-01
Full Text Available In this paper we answer some open questions concerning totally degenerate systems of conservation laws. We study the augmented Born-Infeld system, which is the Born-Infeld model augmented by two additional conservations laws. This system is a nice example of totally degenerate system of conservation laws and, global smooth solutions are conjectured to exist when the initial-data is smooth. We show that this conjecture is false, for the more natural and general condition of initial-data. In fact, first we show that does not exist global smooth solution for any 2X2 totally degenerated system of conservation laws, which the characteristics speeds do not have singular points. Moreover, we sharpen the conjecture in Majda [20]. Under the same hypothesis of initial-data, we show that the Riemann Problem is not well-posed, which follows for weak solutions of the Cauchy Problem. In the end, we prove some results on the direction of well-posedness for the less physically initial-data.
Institute of Scientific and Technical Information of China (English)
Deng Yinbin; Yang Fen
2008-01-01
This article is contributed to the Cauchy problem u/t = △u + K(ㄧxㄧ)up in Rn x (0,T), u(x,0) =(ψ)(x) in Rn; with initial function(ψ)≠0. The stability of positive radial steady state, which are positive solutions of △u + K(ㄧxㄧ)up =0, is obtained when p is critical for general K(ㄧxㄧ).
Institute of Scientific and Technical Information of China (English)
冯玉瑚; 朱凡昌
2004-01-01
Solutions of fuzzy differential equations provide a noteworthy example of time-dependent fuzzy sets. The purpose of this paper is to introduce functions of a suitable Lyapunov-like type and to show the existence and uniqueness theorem for the Cauchy problem of fuzzy differential equations under non-Lipschitz conditions. The comparison principles and the existence and uniqueness theorems of this paper generalize many well-known results up to now.
WELL-POSEDNESS FOR THE CAUCHY PROBLEM TO THE HIROTA EQUATION IN SOBOLEV SPACES OF NEGATIVE INDICES
Institute of Scientific and Technical Information of China (English)
HUOZHAOHUI; JIA YUELING
2005-01-01
The local well-posedness of the Cauchy problem for the Hirota equation is established for low regularity data in Sobolev spaces Hs(s ≥ -1-4). Moreover, the global well-posedness for L2 data follows from the local well-posedness and the conserved quantity. For data in Hs(s ＞ 0), the global well-posedness is also proved. The main idea is to use the generalized trilinear estimates, associated with the Fourier restriction norm method.
Diffuse interface methods for inverse problems: case study for an elliptic Cauchy problem
Burger, Martin; Løseth Elvetun, Ole; Schlottbom, Matthias
2015-12-01
Many inverse problems have to deal with complex, evolving and often not exactly known geometries, e.g. as domains of forward problems modeled by partial differential equations. This makes it desirable to use methods which are robust with respect to perturbed or not well resolved domains, and which allow for efficient discretizations not resolving any fine detail of those geometries. For forward problems in partial differential equations methods based on diffuse interface representations have gained strong attention in the last years, but so far they have not been considered systematically for inverse problems. In this work we introduce a diffuse domain method as a tool for the solution of variational inverse problems. As a particular example we study ECG inversion in further detail. ECG inversion is a linear inverse source problem with boundary measurements governed by an anisotropic diffusion equation, which naturally cries for solutions under changing geometries, namely the beating heart. We formulate a regularization strategy using Tikhonov regularization and, using standard source conditions, we prove convergence rates. A special property of our approach is that not only operator perturbations are introduced by the diffuse domain method, but more important we have to deal with topologies which depend on a parameter \\varepsilon in the diffuse domain method, i.e. we have to deal with \\varepsilon -dependent forward operators and \\varepsilon -dependent norms. In particular the appropriate function spaces for the unknown and the data depend on \\varepsilon . This prevents the application of some standard convergence techniques for inverse problems, in particular interpreting the perturbations as data errors in the original problem does not yield suitable results. We consequently develop a novel approach based on saddle-point problems. The numerical solution of the problem is discussed as well and results for several computational experiments are reported. In
Energy Technology Data Exchange (ETDEWEB)
Addona, Davide, E-mail: d.addona@campus.unimib.it [Università degli Studi di Milano Bicocca, (MILANO BICOCCA) Dipartimento di Matematica (Italy)
2015-08-15
We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton–Jacobi–Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified.
Solutions of fractional diffusion problems
Directory of Open Access Journals (Sweden)
Rabha W. Ibrahim
2010-10-01
Full Text Available Using the concept of majorant functions, we prove the existence and uniqueness of holomorphic solutions to nonlinear fractional diffusion problems. The analytic continuation of these solutions is studied and the singularity for two cases are posed.
Institute of Scientific and Technical Information of China (English)
苗长兴
2003-01-01
In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data in the homogeneous spaces. We give a method which can be used to construct local mild solutionsof the abstract Cauchy problem in Cσ,s,p and Lq([O, T);Hs,p) by introducing the concept of both admissiblequintuplet and compatible space and establishing time-space estimates for solutions to the linear parabolic typeequations. For the small data, we prove that these results can be extended globally in time. We also study theregularity of the solution to the abstract Cauchy problem for nonlinear parabolic type equations in Cσ,s,p. Asan application, we obtain the same result for Navier-Stokes equations with weak initial data in homogeneousSobolev spaces.
Angular processes related to Cauchy random walks
Cammarota, Valemtina
2011-01-01
We study the angular process related to random walks in the Euclidean and in the non-Euclidean space where steps are Cauchy distributed. This leads to different types of non-linear transformations of Cauchy random variables which preserve the Cauchy density. We give the explicit form of these distributions for all combinations of the scale and the location parameters. Continued fractions involving Cauchy random variables are analyzed. It is shown that the $n$-stage random variables are still Cauchy distributed with parameters related to Fibonacci numbers. This permits us to show the convergence in distribution of the sequence to the golden ratio.
Well-posedness of Cauchy problems for Korteweg-de Vries-Benjamin-Ono equation and Hirota equation
Institute of Scientific and Technical Information of China (English)
HUO Zhaohui; GUO Boling
2004-01-01
The well-posedness of the Cauchy problems to the Korteweg-de Vries-Benjamin-Ono equation and Hirota equation is considered. For the Korteweg-de Vries-Benjamin-Ono equation, local result is established for data in Hs(R)(s≥-1/8). Moreover, the global well-posedness for data in L2(R) can be obtained. For Hirota equation, local result is established for initial data in Hs(s≥1/4) .In addition, the local solution is proved to be global in Hs (s≥1) if the initial data are in Hs (s≥1) by energy inequality and the generalization of the trilinear estimates associated with the Fourier restriction norm method.
Institute of Scientific and Technical Information of China (English)
2007-01-01
In this paper, we consider the Cauchy problem with initial data given on a semi-bounded axis for inhomogeneous quasilinear hyperbolic systems. Under the assumption that the rightmost (resp. leftmost) eigenvalue is weakly linearly degenerate and the inhomogeneous term satisfies the corresponding matching condition, we obtain the global existence and uniqueness of C1 solution with small and decaying initial data.
Thieme, Horst R.; Vosseler, Hauke
2002-01-01
A convolution of Stieltjes type is introduced for operator families of bounded strong variation and vector valued $L_1$-functions. Using this tool, perturbation theorems for integrated semigroups of bounded strong variation are derived, and improved results on $L_p$-solutions to the inhomogeneous Cauchy problem are obtained.
Fractional Moments on Bandit Problems
B, Ananda Narayanan
2012-01-01
Reinforcement learning addresses the dilemma between exploration to find profitable actions and exploitation to act according to the best observations already made. Bandit problems are one such class of problems in stateless environments that represent this explore/exploit situation. We propose a learning algorithm for bandit problems based on fractional expectation of rewards acquired. The algorithm is theoretically shown to converge on an eta-optimal arm and achieve O(n) sample complexity. Experimental results show the algorithm incurs substantially lower regrets than parameter-optimized eta-greedy and SoftMax approaches and other low sample complexity state-of-the-art techniques.
Cauchy problem of the generalized Zakharov type system in [Formula: see text].
You, Shujun; Ning, Xiaoqi
2017-01-01
In this paper, we consider the initial value problem for a two-dimensional generalized Zakharov system with quantum effects. We prove the existence and uniqueness of global smooth solutions to the initial value problem in the Sobolev space through making a priori integral estimates and the Galerkin method.
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Marzena Pytel-Kudela
2006-01-01
Full Text Available The analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations in Hilbert spaces are studied using theory of bi-linear forms in respectively rigged Hilbert spaces triples. Theorems specifying the existence of a dissolving operator for a class of adiabatically perturbed nonautonomous partial differential equations are stated. Some applications of the results obtained are discussed.
Symmetry Reduction and Cauchy Problems for a Class of Fourth-Order Evolution Equations
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LI Ji-Na; ZHANG Shun-Li
2008-01-01
We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations to Cauehy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evolution equations which admit certain higher-order generalized conditional symmetries (GCSs) and give some examples to show the main reduction procedure.These reductions cannot be derived within the framework of the standard Lie approach,which hints that the technique presented here is something essential for the dimensional reduction of evolution equations.
Cauchy problems for fifth-order KdV equations in weighted Sobolev spaces
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Eddye Bustamante
2015-05-01
Full Text Available In this work we study the initial-value problem for the fifth-order Korteweg-de Vries equation $$ \\partial_{t}u+\\partial_{x}^{5}u+u^k\\partial_{x}u=0, \\quad x,t\\in \\mathbb{R}, \\; k=1,2, $$ in weighted Sobolev spaces $H^s(\\mathbb{R}\\cap L^2(\\langle x \\rangle^{2r}dx$. We prove local and global results. For the case $k=2$ we point out the relationship between decay and regularity of solutions of the initial-value problem.
THE CAUCHY PROBLEM OF NONLINEAR SCHR(O)DINGER-BOUSSINESQ EQUATIONS IN Hs(Rd)
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Han Yongqian
2005-01-01
In this paper, the local well posedness and global well posedness of solutions for the initial value problem (IVP) of nonlinear Schrodinger-Boussinesq equations is considered in Hs(Rd) by resorting Besov spaces, where real number s ≥ 0.
On analytic continuability of the missing Cauchy datum for Helmholtz boundary problems
DEFF Research Database (Denmark)
Karamehmedovic, Mirza
2015-01-01
We relate the domains of analytic continuation of Dirichlet and Neumann boundary data for Helmholtz problems in two or more independent variables. The domains are related à priori, locally and explicitly in terms of complex polyrectangular neighbourhoods of planar pieces of the boundary...
Asymptotic property of solutions to the random cauchy problem for wave equations
Dôku, Isamu
1986-01-01
Based on the existence and uniqueness theorem for random wave equations, we consider asymptotic behaviors of solutions to the random initial value problem. We describe conditions for the equipartition of stochastic energy (or SE for short) by making use of the random spectral theory and, according to Goldstein's semigroup method, we prove the asymptotically equipartitioned SE theorem and the so-called virial theorem of classical mechanics, and also study the probabilistic characterization of the conditions for equipartition. In addition, we show at last the equipartition of SE from a finite time onwards.
The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations
Choi, Young-Pil; Kwon, Bongsuk
2016-07-01
We present a new hydrodynamic model consisting of the pressureless Euler equations and the isentropic compressible Navier-Stokes equations where the coupling of two systems is through the drag force. This coupled system can be derived, in the hydrodynamic limit, from the particle-fluid equations that are frequently used to study the medical sprays, aerosols and sedimentation problems. For the proposed system, we first construct the local-in-time classical solutions in an appropriate L2 Sobolev space. We also establish the a priori large-time behavior estimate by constructing a Lyapunov functional measuring the fluctuation of momentum and mass from the averaged quantities, and using this together with the bootstrapping argument, we obtain the global classical solution. The large-time behavior estimate asserts that the velocity functions of the pressureless Euler and the compressible Navier-Stokes equations are aligned exponentially fast as time tends to infinity.
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
[1]Oleinik, O. A., Samokhin,V. N., Mathematical Models in Boundary Layer Theorem, Boca Raton; Chapman and Hall/CRC, 1999.[2]Volpert, A.I., Hudjaev, S.I., On the problem for quasilinear degenerate parabolic equations of second order(Russian), Mat. Sb., 1967, 3: 374-396.[3]Zhao, J., Uniqueness of solutions of quasilinear degenerate parabolic equations, Northeastern Math. J., 1985,1(2): 153-165.[4]Wu, Z., Yin, J., Some properties of functions in BVx and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Northeastern Math. J., 1989,5(4): 395-422.[5]Brezis, H., Crandall, M.G., Uniqueness of solutions of the initial value problem for ut- △ψ(u) = 0, J.Math. Pures et Appl., 1979, 58: 153-163.[6]Kruzkov, S.N., First order quasilinear equations in several independent varaiables, Math. USSR-Sb., 1970, 10:217-243.[7]Cockburn, B., Gripenberg, G., Continious dependence on the nonlinearities of solutions of degenerate parabolic equations, J. Diff. Equatiaons, 1999, 151: 231-251.[8]Volpert, A.I., BV space and quasilinear equations, Mat. Sb., 1967, 73: 255-302.[9]Volpert, A.I., Hudjave, S.I., Analysis of class of discontinuous functions and the equations of mathematical physics (Russian), Izda. Nauka Moskwa, 1975.[10]Evans, L.C., Weak convergence methods for nonlinear partial differential equations, Conference Board of the Mathematical Sciences, Regional Conferences Series in Mathematics Number 74, 1998.[11]Wu, Z., Zhao, J., Yin, J., et al., Nonlinear Diffusion Equations, Singapore: Word Scientific, 2001.
On the Cauchy problem of a 2 times 2 system of nonstrictly hyperbolic conservation laws
Energy Technology Data Exchange (ETDEWEB)
Kan, P.T.
1989-01-01
Global existence for a 2 {times} 2 system of nonstrictly hyperbolic conservation law is established for data of arbitrary bounded variation. This result is obtained by proving a convergence theorem for the method of artificial viscosity applied to this system of conservation law. For this purpose, the method of compensated compactness and an analysis of the entropy functions are used. This system under consideration is a special case of a canonical class of 2 {times} 2 systems of conservation laws with quadratic flux functions possessing an isolated umbilic point (point of coinciding wave speeds where strict hyperbolicity fails) at the origin of the state space. These systems arise as model equations to equations used in oil reservoir simulations. Their wave curves and Riemann problem solutions are known to exhibit complexities not seen in any strictly hyperbolic systems. In this thesis, besides establishing global existence for a special system in the canonical class, general properties of a subclass are also investigated. The geometry of rarefaction wave curves are analytically studied and Riemann invariants are constructed. An L{sup {infinity}} bound (independent of the viscosity) for the solutions of the corresponding viscous systems are obtained. Also studied is the monotonicity properties of the wave speeds in the Riemann invariant plane.
Institute of Scientific and Technical Information of China (English)
Yan ZHANG
2007-01-01
When a target manifold is complete with a bounded curvature, we prove that there exists a unique global solution which satisfies the Euler-lagrange equation of (τ)(φ,(φ))=fR1+1{(|dφ|2+}dtdx for the given Cauchy data.
Energy Technology Data Exchange (ETDEWEB)
Panov, E Yu [Novgorod State University, Novgorod (Russian Federation)
2002-12-31
We construct a theory of locally summable generalized entropy solutions (g.e. solutions) of the Cauchy problem for a first-order non-homogeneous quasilinear equation with continuous flux vector satisfying a linear restriction on its growth. We prove the existence of greatest and least g.e. solutions, suggest sufficient conditions for uniqueness of g.e. solutions, prove several versions of the comparison principle, and obtain estimates for the L{sup p}-norms of solution with respect to the space variables. We establish the uniqueness of g.e. solutions in the case when the input data are periodic functions of the space variables.
Morassi, Antonino; Vessella, Sergio
2010-01-01
We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. The paper is self-contained and the Carleman estimate, from which the sharp three sphere inequality is derived, is proved in an elementary and direct way based on standard integration by parts.
Nonlocal Problems for Fractional Differential Equations via Resolvent Operators
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Zhenbin Fan
2013-01-01
Full Text Available We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchy operator by means of the analytic resolvent method. Based on this result, we derive the existence of mild solutions for nonlocal fractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively. An example is also given to illustrate our theory.
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D. C. Kent
1985-01-01
Full Text Available This paper is concerned with the notion of ordered Cauchy space which is given a simple internal characterization in Section 2. It gives a discription of the category of ordered Cauchy spaces which have ordered completions, and a construction of the fine completion functor on this category. Sections 4 through 6 deals with certain classes of ordered Cauchy spaces which have ordered completions; examples are given which show that the fine completion does not preserve such properties as uniformizability, regularity, or total boundedness. From these results, it is evident that a further study of ordered Cauchy completions is needed.
Sekerzh-Zen'kovich, S. Ya.
2016-10-01
The problem of Cauchy-Poisson type is considered in the framework of potential tsunami model with impulse cylindical compactly supported source discontinuous with respect to the radial horizontal coordinate under the assumption that the depth of the liquid is constant. An analytical solution of the problem for a special choice of values of the parameters of the source is given. The connection the solution thus obtained with nonstandard characteristics introduced by Maslov is discussed. The graphs of the time history of the free surface elevation are compared with the marigrams, received for the 2011 tsunami from the DART stations 21418, 21413, and 51407, and also with graphs calculated for the well-known model with a "simple" source.
Fractional Calculus in Wave Propagation Problems
Mainardi, Francesco
2012-01-01
Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this lecture we devote our attention to wave propagation problems in linear viscoelastic media. Our purpose is to outline the role of fractional calculus in providing simplest evolution processes which are intermediate between diffusion and wave propagation. The present treatment mainly reflects the research activity and style of the author in the related scientific areas during the last decades.
Institute of Scientific and Technical Information of China (English)
刘德朋
2001-01-01
本文就形如 utt＝a2(utt+(2)/(x)ux) 的 Cauchy 问题进行了讨论,得到其相容性条件和古典解的存在性定理,进而对于更一般的一类形如 autt+2butt+cutt+(2)/(x)(aux+bux)=0(a、b、c是常量,且b2-ac＞0)的方程得到类似结果.
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El-Bachir Yallaoui
2012-06-01
Full Text Available In this article, we introduce a new class of analytic functions of the unit disc $mathbf{D}$ namely the Exponential Cauchy Transforms $mathbf{{K}_{e}}$ defined by f(z= {displaystyleint_{mathbf{T}}} expleft[ Kleft( xzight ight] dmu(x where $Kleft( zight =left( 1-zight ^{-1}$ is classical Cauchy kernel and $mu(x$ is a complex Borel measures and $x$ belongs to the unit circle $mathbf{T}$ . We use Laguerre polynomials to explore the coefficients of the Taylor expansions of the kernel and Peron's formula to study the asymptotic behavior of the Taylor coefficients. Finally we investigate relationships between our new class $mathbf{{K}_{e}}$, the classical Cauchy space $mathbf{K}$ and the Hardy spaces $H^{p}$.
Superlinear singular fractional boundary-value problems
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Imed Bachar
2016-04-01
Full Text Available In this article, we study the superlinear fractional boundary-value problem $$\\displaylines{ D^{\\alpha }u(x =u(xg(x,u(x,\\quad 00$. The function $g(x,u\\in C((0,1\\times [ 0,\\infty ,[0,\\infty$ that may be singular at x=0 and x=1 is required to satisfy convenient hypotheses to be stated later.
Dafermos, M
2002-01-01
This talk describes some recent results [16] regarding the problem of uniqueness in the large (also known as strong cosmic censorship) for the initial value problem in general relativity. In order to isolate the essential analytic features of the problem from the complicated setting of gravitational collapse in which it arises, some familiarity with conformal properties of certain celebrated special solutions of the theory of relativity will have to be developed. This talk is an attempt to present precisely these features to an audience of non-specialists, in a way which will hopefully fully motivate a certain characteristic initial value problem for the spherically-symmetric Einstein-Maxwell-Scalar Field system. The considerations outlined here leading to this particular initial value problem are well known in the physics relativity community, where the problem of uniqueness has been studied heuristically [1, 22] and numerically [2, 3]. In [16], the global behavior of solutions to this IVP, in particular, th...
Approximate (m, n)-Cauchy-Jensen Additive Mappings in C*-algebras
Institute of Scientific and Technical Information of China (English)
John Michael RASSIAS; Kil Woung JUN; Hark-Mahn KIM
2011-01-01
Concerning the stability problem of functional equations,we introduce a general (m,n)-Cauchy-Jensen functional equation and establish new theorems about the Hyers-Ulam stability of general (m,n)-Cauchy-Jensen additive mappings in C*-algebras,which generalize the results obtained for Cauchy-Jensen type additive mappings.
Institute of Scientific and Technical Information of China (English)
钱爱林; 毛建峰; 桂咏新
2011-01-01
Helmholtz方程Cauchy问题是严重不适定问题,本文我们在一个带形区域上考虑了一类Helmholtz方程Cauchy问题:已知Cauchy数据u(0,y)=g(y),在区间0＜x＜1上求解.我们用半离散的中心差分方法得到了这一问题的正则化解,给出了正则化参数的选取规则,得到了误差估计.%The Cauchy problem of Helmholtz equation is severely ill-posed problem. In this paper, we consider the Cauchy problem for the Helmholtz equation where the Cauchy data is given at x = 0 and the solution is sought in the interval 0 ＜ x ＜ 1. A semi-discrete difference schemes together with a choice of regularization parameter is presented and error estimate is obtained.
DEFF Research Database (Denmark)
Ghorbani, Mohammad
2013-01-01
In this paper we introduce an instance of the well-know Neyman–Scott cluster process model with clusters having a long tail behaviour. In our model the offspring points are distributed around the parent points according to a circular Cauchy distribution. Using a modified Cramér-von Misses test st...... statistic and the simulated pointwise envelopes it is shown that this model fits better than the Thomas process to the frequently analyzed long-leaf pine data-set....
From Students' Problem-Solving Strategies to Connections in Fractions
Flores, Alfinio; Klein, Erika
2005-01-01
Strategies that children used to solve a fraction problem are presented, and an insight into how students think about divisions and fractions is described. Teachers can use these strategies to help students establish connections related to fractions.
Santucci, F.; Santini, P. M.
2016-10-01
We study the generalization of the dispersionless Kadomtsev-Petviashvili (dKP) equation in n+1 dimensions and with nonlinearity of degree m+1, a model equation describing the propagation of weakly nonlinear, quasi one-dimensional waves in the absence of dispersion and dissipation, and arising in several physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear optics. In 2 + 1 dimensions and with quadratic nonlinearity, this equation is integrable through a novel inverse scattering transform, and it has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. In higher dimensions and with higher nonlinearity, the generalized dKP equations are not integrable, but their invariance under motions on the paraboloid allows one to construct in this paper a family of exact solutions describing waves constant on their paraboloidal wave front and breaking simultaneously in all points of it, developing after breaking either multivaluedness or single-valued discontinuous profiles (shocks). Then such exact solutions are used to build the longtime behavior of the solutions of the Cauchy problem, for small and localized initial data, showing that wave breaking of small initial data takes place in the longtime regime if and only if m(n-1)≤slant 2. Lastly, the analytic aspects of such wave breaking are investigated in detail in terms of the small initial data, in both cases in which the solution becomes multivalued after breaking or it develops a shock. These results, contained in the 2012 master’s thesis of one of the authors (FS) [1], generalize those obtained in [2] for the dKP equation in n+1 dimensions with quadratic nonlinearity, and are obtained following the same strategy.
Existence of Solution for Fractional Differential Problem with a Parameter
Institute of Scientific and Technical Information of China (English)
Shi Ai-ling; Zhang Shu-qin
2014-01-01
We apply the method of lower and upper solutions combined with mono-tone iterations to fractional differential problem with a parameter. The existence of minimal and maximal solutions is proved for the fractional differential problem with a parameter.
Directory of Open Access Journals (Sweden)
Jordan Hristov
2016-01-01
Full Text Available The article addresses a reappraisal of the famous Ward–Tordai equation describing the equilibrium of surfactants at air/liquid interfaces under diffusion control. The new derivation is entirely developed in the light of fractional calculus. The unified approach demonstrates that this equation can be clearly reformulated as a nonlinear ordinary time-fractional equation of order 1/2. The work formulates versions with different isotherms. A simple solution of the case with the Henry’s isotherm and a discussion of a Cauchy problem involving the Freundlich isotherm are provided.
Singular and degenerate cauchy problems
Carroll, R.W
1976-01-01
In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank mat
THE CAUCHY-KOVALEVSKAYA THEOREM-OLD AND NEW
Institute of Scientific and Technical Information of China (English)
W. Tutschke
2005-01-01
The paper surveys interactions between complex and functional-analytic methods in the Cauchy-Kovalevskaya theory. For instance, the behaviour of the derivative of a bounded holomorphic function led to abstract versions of the Cauchy-Kovalevskaya Theorem.Recent trends in the Cauchy-Kovalevskaya theory are based on the concept of associated differential operators. Since an evolution operator may posses several associated operators, initial data may be decomposed into components belonging to different associated spaces. This technique makes it also possible to solve ill-posed initial value problems.
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.
Controllability Problem of Fractional Neutral Systems: A Survey
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Artur Babiarz
2017-01-01
Full Text Available The following article presents recent results of controllability problem of dynamical systems in infinite-dimensional space. Generally speaking, we describe selected controllability problems of fractional order systems, including approximate controllability of fractional impulsive partial neutral integrodifferential inclusions with infinite delay in Hilbert spaces, controllability of nonlinear neutral fractional impulsive differential inclusions in Banach space, controllability for a class of fractional neutral integrodifferential equations with unbounded delay, controllability of neutral fractional functional equations with impulses and infinite delay, and controllability for a class of fractional order neutral evolution control systems.
p-topological Cauchy completions
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J. Wig
1999-01-01
Full Text Available The duality between “regular” and “topological” as convergence space properties extends in a natural way to the more general properties “p-regular” and “p-topological.” Since earlier papers have investigated regular, p-regular, and topological Cauchy completions, we hereby initiate a study of p-topological Cauchy completions. A p-topological Cauchy space has a p-topological completion if and only if it is “cushioned,” meaning that each equivalence class of nonconvergent Cauchy filters contains a smallest filter. For a Cauchy space allowing a p-topological completion, it is shown that a certain class of Reed completions preserve the p-topological property, including the Wyler and Kowalsky completions, which are, respectively, the finest and the coarsest p-topological completions. However, not all p-topological completions are Reed completions. Several extension theorems for p-topological completions are obtained. The most interesting of these states that any Cauchy-continuous map between Cauchy spaces allowing p-topological and p′-topological completions, respectively, can always be extended to a θ-continuous map between any p-topological completion of the first space and any p′-topological completion of the second.
Recognizing Similarities between Fraction Word Problems.
Hardiman, Pamela Thibodeau
Deciding how to approach a word problem for solution is a critical stage of problem solving, and is the stage which frequently presents considerable difficulty for novices. Do novices use the same information that experts do in deciding that two problems would be solved similarly? This set of four studies indicates that novices rely more on…
Institute of Scientific and Technical Information of China (English)
蒋毅; 成和平; 孟宪良; 蒲成林
2006-01-01
For the Cauchy problem for the nonlinear Klein-Gordon equation with potential,we define new stable and unstable sets for the initial data.We prove that if during the evolution enters into the unstable set,the solution blows up in finite time.If during the evolution enters into the stable set,the solution is global.By using scaling argument,we also answer the question of how small the initial data are the global solution of the Cauchy problem exists.%对带势的非线性Klein-Gordon方程柯西问题,我们定义了新的对于初值的稳定和不稳定集.我们证明了如果发展进入了不稳定集,解在有限时间内爆破;如果发展进入了稳定集,解整体存在.运用势并讨论,我们回答了当初值为多少时,柯西问题的整体解存在.
On Fractional Variational Problems which Admit Local Transformations
Malinowska, Agnieszka B
2012-01-01
We extend the second Noether theorem to fractional variational problems which are invariant under infinitesimal transformations that depend upon $r$ arbitrary functions and their fractional derivatives in the sense of Caputo. Our main result is illustrated using the fractional Lagrangian density of the electromagnetic field.
Demonstration of Cauchy: Understanding Algebraic
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T.L. Costa
2012-11-01
Full Text Available ABSTRACT: In this study we present some considerations about the End of Course Work undergraduate Full Degree in Mathematics / UFMT, drafted in 2011, and by taking title "A story about Cauchy and Euler's theorem on polyhedra" that gave birth to our research project Master of Education, begun in 2012, on the approaches of Euler's theorem on polyhedra in mathematics textbooks. At work in 2011 presented some considerations about the history of Euler's theorem for polyhedra which focus the demonstration presented by Cauchy (1789-1857, who tries to generalize it, relying on assumptions not observable in Euclidean geometry. Therefore, we seek the accessible literature on the history of mathematics; relate some aspects of the demonstration Cauchy with historical events on the development of mathematics in the nineteenth century, which allowed the acceptance of such a demonstration by mathematicians of his time.Keywords: History of Mathematics. Euler's Theorem on Polyhedra. Demonstration of Cauchy.
From Sturm-Liouville problems to fractional and anomalous diffusions
D'Ovidio, Mirko
2010-01-01
Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those diffusions, the explicit laws of certain stable processes turn out to be fundamental. This paper directs one's efforts towards the explicit representation of solutions to fractional and anomalous diffusions related to Sturm-Liouville problems of fractional order associated to fractional power function spaces. Furthermore, we study a new version of the Bochner's subordination rule and we establish some connections between subordination and space-fractional operator
Carlini, A.; Novikov, I. D.
We consider the action principle to derive the classical, relativistic motion of a selfinteracting particle in a 4D Lorentzian spacetime containing a wormhole and which allows the existence of closed time-like curves. In particular, we study the case of a pointlike particle subject to a “hard-sphere” self-interaction potential and which can traverse the wormhole an arbitrary number of times, and show that the only possible trajectories for which the classical action is stationary are those which are globally self-consistent. Generically, the multiplicity of these trajectories (defined as the number of self-consistent solutions to the equations of motion beginning with given Cauchy data) is finite, and it becomes infinite if certain constraints on the same initial data are satisfied. This confirms the previous conclusions (for a nonrelativistic model) by Echeverria, Klinkhammer and Thorne that the Cauchy initial value problem in the presence of a wormhole “time machine” is classically “ill-posed” (far too many solutions). Our results further extend the recent claim by Novikov et al. that the “principle of self-consistency” is a natural consequence of the “principle of minimal action.”
Carlini, A
1996-01-01
We consider the action principle to derive the classical, relativistic motion of a self-interacting particle in a 4-D Lorentzian spacetime containing a wormhole and which allows the existence of closed time-like curves. In particular, we study the case of a pointlike particle subject to a `hard-sphere' self-interaction potential and which can traverse the wormhole an arbitrary number of times, and show that the only possible trajectories for which the classical action is stationary are those which are globally self-consistent. Generically, the multiplicity of these trajectories (defined as the number of self-consistent solutions to the equations of motion beginning with given Cauchy data) is finite, and it becomes infinite if certain constraints on the same initial data are satisfied. This confirms the previous conclusions (for a non-relativistic model) by Echeverria, Klinkhammer and Thorne that the Cauchy initial value problem in the presence of a wormhole `time machine' is classically `ill-posed' (far too m...
Existence of solutions of abstract fractional impulsive semilinear evolution equations
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K. Balachandran
2010-01-01
Full Text Available In this paper we prove the existence of solutions of fractional impulsive semilinear evolution equations in Banach spaces. A nonlocal Cauchy problem is discussed for the evolution equations. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the theory.
ISOPERIMETRIC PROBLEMS OF THE CALCULUS OF VARIATIONS WITH FRACTIONAL DERIVATIVES
Institute of Scientific and Technical Information of China (English)
Ricardo Almeida; Rui A. C. Ferreira; Delfim F. M. Torres
2012-01-01
In this article,we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with the lower bound of the fractional derivatives are considered.
Solvability for fractional order boundary value problems at resonance
Hu Zhigang; Liu Wenbin
2011-01-01
Abstract In this paper, by using the coincidence degree theory, we consider the following boundary value problem for fractional differential equation D 0 + α x ( t ) = f ( t , x ( t ) , x ′ ( t ) , x ″ ( t ) ) , t ∈ [ 0 , 1 ] , x ( 0 ) = x ( 1 ) , x ′ ( 0 ) = x ″ ( 0 ) = 0 , where D 0 + α denotes the Caputo fractional differential o...
On nonlocal problems for fractional differential equations in Banach spaces
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XiWang Dong
2011-01-01
Full Text Available In this paper, we study the existence and uniqueness of solutions to the nonlocal problems for the fractional differential equation in Banach spaces. New sufficient conditions for the existence and uniqueness of solutions are established by means of fractional calculus and fixed point method under some suitable conditions. Two examples are given to illustrate the results.
An approach for solving linear fractional programming problems ...
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An approach for solving linear fractional programming problems. ... Journal of the Nigerian Association of Mathematical Physics. Journal Home · ABOUT · Advanced Search ... Open Access DOWNLOAD FULL TEXT Subscription or Fee Access ...
On Alternative Optimal Solutions to Linear Fractional Optimization Problems
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ShengjiaXue
2004-01-01
The structure of the optimal solution set is derived for linear fractional optimization problems with the representation theorem of polyhedral sets．And the computational procedure in determining all optimal solutions is also given．
Monotone method for initial value problem for fractional diffusion equation
Institute of Scientific and Technical Information of China (English)
ZHANG Shuqin
2006-01-01
Using the method of upper and lower solutions and its associated monotone iterative, consider the existence and uniqueness of solution of an initial value problem for the nonlinear fractional diffusion equation.
An inverse Sturm–Liouville problem with a fractional derivative
Jin, Bangti
2012-05-01
In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm-Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and numerical reconstructions of the potential with a Newton method from finite spectral data are presented. Surprisingly, it allows very satisfactory reconstructions for both smooth and discontinuous potentials, provided that the order . α∈. (1,. 2) of fractional derivative is sufficiently away from 2. © 2012 Elsevier Inc.
Solvability for fractional order boundary value problems at resonance
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Hu Zhigang
2011-01-01
Full Text Available Abstract In this paper, by using the coincidence degree theory, we consider the following boundary value problem for fractional differential equation D 0 + α x ( t = f ( t , x ( t , x ′ ( t , x ″ ( t , t ∈ [ 0 , 1 ] , x ( 0 = x ( 1 , x ′ ( 0 = x ″ ( 0 = 0 , where D 0 + α denotes the Caputo fractional differential operator of order α, 2 < α ≤ 3. A new result on the existence of solutions for above fractional boundary value problem is obtained. Mathematics Subject Classification (2000: 34A08, 34B15.
On Bimodal Offset Cauchy Distribution
Directory of Open Access Journals (Sweden)
Girija S.V.S.
2013-05-01
Full Text Available The bivariate Cauchy distribution has received applications in many areas, including biological analyses, clinical trials, stochastic modeling of decreasing failure rate life components, study of labour turnover, queuing theory and reliability (Nayak (1987 and Lee and Gross (1991. In the study of biological analyses, clinical trials and reliability circular distributions will yield suitable results. Circular data arises in a number of different areas such as geological, meteorological, biological and industrial sciences. It is not suggestive to use standard statistical techniques to model circular data, due to the circular geometry of the sample space (p.2 Jammalamadaka and Sen Gupta (2001. It is possible to construct a circular model by transforming a bivariate linear random variate to just its directional component and the resultant model is called ‘offset distribution’. In the literature most of the available circular models were constructed by wrapping a linear model. In recent years some wrapped models were constructed by Dattatreya Rao et al (2007. Here an attempt is made to exploit method of offsetting on Bivariate Cauchy distribution to construct a circular model named by us “OFFSET CAUCHY DISTRIBUTION (OC”. The characteristic function of the Offset Cauchy model is derived and its characteristics are discussed.
Chebyshev Finite Difference Method for Fractional Boundary Value Problems
Directory of Open Access Journals (Sweden)
Boundary
2015-09-01
Full Text Available This paper presents a numerical method for fractional differential equations using Chebyshev finite difference method. The fractional derivatives are described in the Caputo sense. Numerical results show that this method is of high accuracy and is more convenient and efficient for solving boundary value problems involving fractional ordinary differential equations. AMS Subject Classification: 34A08 Keywords and Phrases: Chebyshev polynomials, Gauss-Lobatto points, fractional differential equation, finite difference 1. Introduction The idea of a derivative which interpolates between the familiar integer order derivatives was introduced many years ago and has gained increasing importance only in recent years due to the development of mathematical models of a certain situations in engineering, materials science, control theory, polymer modelling etc. For example see [20, 22, 25, 26]. Most fractional order differential equations describing real life situations, in general do not have exact analytical solutions. Several numerical and approximate analytical methods for ordinary differential equation Received: December 2014; Accepted: March 2015 57 Journal of Mathematical Extension Vol. 9, No. 3, (2015, 57-71 ISSN: 1735-8299 URL: http://www.ijmex.com Chebyshev Finite Difference Method for Fractional Boundary Value Problems H. Azizi Taft Branch, Islamic Azad University Abstract. This paper presents a numerical method for fractional differential equations using Chebyshev finite difference method. The fractional derivative
Control of Cauchy System for an Elliptic Operator
Institute of Scientific and Technical Information of China (English)
G.MASSENGO MOPHOU; O.NAKOULIMA
2009-01-01
The control of a Cauchy system for an elliptic operator seems to be globally an open problem. In this paper, we analyze this problem using a regularization method which consists in viewing a singular problem as a limit of a family of well-posed problems. Following this analysis and assuming that the interior of considered convex is non-empty, we obtain a singular optimality system (S.O.S.) for the considered control problem.
A note on stability and Cauchy time functions
Sánchez, Miguel
2013-01-01
Since the solution of the so-called folk problems of smoothability, there has been a special interest in the properties of classical time and volume functions of spacetimes, among them the recent references arXiv:1108.5120v3 and arXiv:1301.2909v1 . Here we supply some information that completes the one provided in these two papers and discuss some results. As an illustration, we show that any Cauchy temporal function for a globally hyperbolic spacetime remains Cauchy temporal for close metrics - which, in particular, implies stability for global hyperbolicity.
Variational approach for restoring blurred images with cauchy noise
DEFF Research Database (Denmark)
Sciacchitano, Federica; Dong, Yiqiu; Zeng, Tieyong
2015-01-01
model, we add a quadratic penalty term, which guarantees the uniqueness of the solution. Due to the convexity of our model, the primal dual algorithm is employed to solve the minimization problem. Experimental results show the effectiveness of the proposed method for simultaneously deblurring...... and denoising images corrupted by Cauchy noise. Comparison with other existing and well-known methods is provided as well....
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...
Fractional and multivariable calculus model building and optimization problems
Mathai, A M
2017-01-01
This textbook presents a rigorous approach to multivariable calculus in the context of model building and optimization problems. This comprehensive overview is based on lectures given at five SERC Schools from 2008 to 2012 and covers a broad range of topics that will enable readers to understand and create deterministic and nondeterministic models. Researchers, advanced undergraduate, and graduate students in mathematics, statistics, physics, engineering, and biological sciences will find this book to be a valuable resource for finding appropriate models to describe real-life situations. The first chapter begins with an introduction to fractional calculus moving on to discuss fractional integrals, fractional derivatives, fractional differential equations and their solutions. Multivariable calculus is covered in the second chapter and introduces the fundamentals of multivariable calculus (multivariable functions, limits and continuity, differentiability, directional derivatives and expansions of multivariable ...
Fractional evolution equation nonlocal problems with noncompact semigroups
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Xuping Zhang
2016-01-01
Full Text Available This paper is concerned with the existence results of mild solutions to the nonlocal problem of fractional semilinear integro-differential evolution equations. New existence theorems are obtained by means of the fixed point theorem for condensing maps. The results extend and improve some related results in this direction.
Fractional Extensions of some Boundary Value Problems in Oil Strata
Indian Academy of Sciences (India)
Mridula Garg; Alka Rao
2007-05-01
In the present paper, we solve three boundary value problems related to the temperature field in oil strata - the fractional extensions of the incomplete lumped formulation and lumped formulation in the linear case and the fractional generalization of the incomplete lumped formulation in the radial case. By using the Caputo differintegral operator and the Laplace transform, the solutions are obtained in integral forms where the integrand is expressed in terms of the convolution of some auxiliary functions of Wright function type. A generalization of the Laplace transform convolution theorem, known as Efros’ theorem is widely used.
Institute of Scientific and Technical Information of China (English)
曾有栋; 陈祖墀
2002-01-01
本文研究奇异半线性抛物方程ut-Δu+V1(x)u=V2(x)up,x∈Rn＼{0},t＞0的Cauchy问题解的存在性.这里,V1(x),V2(x)可以在原点具有奇性.利用Kato类函数和Green tight函数及不动点定理证明了问题存在正的奇异解,它在原点具有奇性.%In this paper we consider the Cauchy problem for the singular semilinear parabolic equation ut-Δu+V1(x)u=V2(x)up,x∈Rn＼{0},t＞0, where V1(x),V2(x) may have singularities at the origin. Using functions of the Kato class and the Green tight functions we got the existence of the positive solution being singular at the origin.
Complex integration and Cauchy's theorem
Watson, GN
2012-01-01
This brief monograph by one of the great mathematicians of the early twentieth century offers a single-volume compilation of propositions employed in proofs of Cauchy's theorem. Developing an arithmetical basis that avoids geometrical intuitions, Watson also provides a brief account of the various applications of the theorem to the evaluation of definite integrals.Author G. N. Watson begins by reviewing various propositions of Poincaré's Analysis Situs, upon which proof of the theorem's most general form depends. Subsequent chapters examine the calculus of residues, calculus optimization, the
An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems
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Mohammad Maleki
2012-01-01
Full Text Available An adaptive pseudospectral method is presented for solving a class of multiterm fractional boundary value problems (FBVP which involve Caputo-type fractional derivatives. The multiterm FBVP is first converted into a singular Volterra integrodifferential equation (SVIDE. By dividing the interval of the problem to subintervals, the unknown function is approximated using a piecewise interpolation polynomial with unknown coefficients which is based on shifted Legendre-Gauss (ShLG collocation points. Then the problem is reduced to a system of algebraic equations, thus greatly simplifying the problem. Further, some additional conditions are considered to maintain the continuity of the approximate solution and its derivatives at the interface of subintervals. In order to convert the singular integrals of SVIDE into nonsingular ones, integration by parts is utilized. In the method developed in this paper, the accuracy can be improved either by increasing the number of subintervals or by increasing the degree of the polynomial on each subinterval. Using several examples including Bagley-Torvik equation the proposed method is shown to be efficient and accurate.
A Mathematical Programming Approach to the Fractionation Problem in Chemoradiotherapy
Salari, Ehsan; Bortfeld, Thomas
2013-01-01
In concurrent chemoradiotherapy, chemotherapeutic agents are administered during the course of radiotherapy to enhance the primary tumor control. However, that often comes at the expense of increased risk of normal-tissue complications. The additional biological damage is mainly attributed to two mechanisms of action, which are the independent cytotoxic activity of chemotherapeutic agents and their interactive cooperation with radiation. The goal of this study is to develop a mathematical framework to obtain drug and radiation administration schedules that maximize the therapeutic gain for concurrent chemoradiotherapy. In particular, we analyze the impact of incorporating these two mechanisms into the radiation fractionation problem. Considering each mechanism individually, we first derive closed-form expressions for the optimal radiation fractionation regimen and the corresponding drug administration schedule. We next study the case in which both mechanisms are simultaneously present and develop a dynamic pr...
THREE POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Mujeeb ur Rehman; Rahmat Ali Khan; Naseer Ahmad Asif
2011-01-01
In this paper,we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type cDδ0+u(t) =f(t,u(t),cDσ0+u(t)),t ∈[0,T],u(0) =αu(η),u(T) =βu(η),where1 ＜δ＜2,0＜σ＜ 1,α,β∈R,η∈(0,T),αη(1-β)+(1-α)(T-βη) ≠0 and cDoδ+,cDσ0+ are the Caputo fractional derivatives.We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results.Examples are also included to show the applicability of our results.
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 the 1970's, we demonstrate how they imply regularity properties in full scales of -Sobolev spaces and Hölder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of s are derived by use of recent results on -calculus. We also include an overview...
Mitrinović, Dragoslav S
1993-01-01
Volume 1, i. e. the monograph The Cauchy Method of Residues - Theory and Applications published by D. Reidel Publishing Company in 1984 is the only book that covers all known applications of the calculus of residues. They range from the theory of equations, theory of numbers, matrix analysis, evaluation of real definite integrals, summation of finite and infinite series, expansions of functions into infinite series and products, ordinary and partial differential equations, mathematical and theoretical physics, to the calculus of finite differences and difference equations. The appearance of Volume 1 was acknowledged by the mathematical community. Favourable reviews and many private communications encouraged the authors to continue their work, the result being the present book, Volume 2, a sequel to Volume 1. We mention that Volume 1 is a revised, extended and updated translation of the book Cauchyjev raeun ostataka sa primenama published in Serbian by Nau~na knjiga, Belgrade in 1978, whereas the greater part ...
Directory of Open Access Journals (Sweden)
Jeffrey W. Lyons
2017-01-01
Full Text Available For \\(\\alpha\\in(1,2]\\, the singular fractional boundary value problem \\[D^{\\alpha}_{0^+}x+f\\left(t,x,D^{\\mu}_{0^+}x\\right=0,\\quad 0\\lt t\\lt 1,\\] satisfying the boundary conditions \\(x(0=D^{\\beta}_{0^+}x(1=0\\, where \\(\\beta\\in(0,\\alpha-1]\\, \\(\\mu\\in(0,\\alpha-1]\\, and \\(D^{\\alpha}_{0^+}\\, \\(D^{\\beta}_{0^+}\\ and \\(D^{\\mu}_{0^+}\\ are Riemann-Liouville derivatives of order \\(\\alpha\\, \\(\\beta\\ and \\(\\mu\\ respectively, is considered. Here \\(f\\ satisfies a local Carathéodory condition, and \\(f(t,x,y\\ may be singular at the value 0 in its space variable \\(x\\. Using regularization and sequential techniques and Krasnosel'skii's fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given.
The origins of Cauchy's rigorous calculus
Grabiner, Judith V
2005-01-01
This text examines the reinterpretation of calculus by Augustin-Louis Cauchy and his peers in the 19th century. These intellectuals created a collection of well-defined theorems about limits, continuity, series, derivatives, and integrals. 1981 edition.
Cauchy problem for derivors in finite dimension
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Jean-Francois Couchouron
2001-05-01
Full Text Available In this paper we study the uniqueness of solutions to ordinary differential equations which fail to satisfy both accretivity condition and the uniqueness condition of Nagumo, Osgood and Kamke. The evolution systems considered here are governed by a continuous operators $A$ defined on $mathbb{R}^N$ such that $A$ is a derivor; i.e., $-A$ is quasi-monotone with respect to $(mathbb{R}^{+}^N$.
Institute of Scientific and Technical Information of China (English)
刘转转
2011-01-01
在概率论中,求解形如E[φ(X)] -1/(√2πσ)∫-∞+∞φ(x)e-(x-μ)2/(2σ2)ds的积分是很重要的.但即使φ(x)是初等函数如xn,ems,sinmx等,用常规的分部积分法也不易处理.而1维热传导方程初值问题有形如前面的积分解和含有微分算子的级数解,由解的唯一性将把这类期望的积分运算转化为含有微分的级数运算.通过举例说明了该方法在求解数字特征、特征函数等方面的简便实用性,并以公式形式给出了xn,ems,sinmx等解析函数的期望.最后作为补充,给出了n维类似的结论.%In probability theory, the integral such asE [ψ(X) ] =1 -Γ2πσ∫+∞-∞ ψ(x)e-(x-n)2-2σ2 dx is very important,but it is not easy to solve it by using integration by parts, though ψ( x) is the elementary function such asxn, emx , smmx,etc. The solution of the Cauchy problem of one-dimension heat equation has two forms: the integral as E[ψ> (X) ] which was mentioned and the progression with differential operator. Because of the unique of the solution, integral oper-ation can be changed into differential operation. The formulas will be easily given in the form of examples about n-umerical characteristic and characteristic function, etc, when ψ( x) is the elementary function such as xn, emx, sinmx, and the product of them. Finally, the theory of re-dimension heat equation will be given.
On the solutions of fractional order of evolution equations
Morales-Delgado, V. F.; Taneco-Hernández, M. A.; Gómez-Aguilar, J. F.
2017-01-01
In this paper we present a discussion of generalized Cauchy problems in a diffusion wave process, we consider bi-fractional-order evolution equations in the Riemann-Liouville, Liouville-Caputo, and Caputo-Fabrizio sense. Through Fourier transforms and Laplace transform we derive closed-form solutions to the Cauchy problems mentioned above. Similarly, we establish fundamental solutions. Finally, we give an application of the above results to the determination of decompositions of Dirac type for bi-fractional-order equations and write a formula for the moments for the fractional vibration of a beam equation. This type of decomposition allows us to speak of internal degrees of freedom in the vibration of a beam equation.
Analytic capacity, rectifiability, menger curvature and the Cauchy integral
Pajot, Hervé
2002-01-01
Based on a graduate course given by the author at Yale University this book deals with complex analysis (analytic capacity), geometric measure theory (rectifiable and uniformly rectifiable sets) and harmonic analysis (boundedness of singular integral operators on Ahlfors-regular sets). In particular, these notes contain a description of Peter Jones' geometric traveling salesman theorem, the proof of the equivalence between uniform rectifiability and boundedness of the Cauchy operator on Ahlfors-regular sets, the complete proofs of the Denjoy conjecture and the Vitushkin conjecture (for the latter, only the Ahlfors-regular case) and a discussion of X. Tolsa's solution of the Painlevé problem.
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 regular...
Directory of Open Access Journals (Sweden)
Firat Evirgen
2016-04-01
Full Text Available In this paper, a class of Nonlinear Programming problem is modeled with gradient based system of fractional order differential equations in Caputo's sense. To see the overlap between the equilibrium point of the fractional order dynamic system and theoptimal solution of the NLP problem in a longer timespan the Multistage Variational İteration Method isapplied. The comparisons among the multistage variational iteration method, the variationaliteration method and the fourth order Runge-Kutta method in fractional and integer order showthat fractional order model and techniques can be seen as an effective and reliable tool for finding optimal solutions of Nonlinear Programming problems.
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…
An inverse problem for a one-dimensional time-fractional diffusion problem
Jin, Bangti
2012-06-26
We study an inverse problem of recovering a spatially varying potential term in a one-dimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources. The unique identifiability of the potential is shown for two cases, i.e. the flux at one end and the net flux, provided that the set of input sources forms a complete basis in L 2(0, 1). An algorithm of the quasi-Newton type is proposed for the efficient and accurate reconstruction of the coefficient from finite data, and the injectivity of the Jacobian is discussed. Numerical results for both exact and noisy data are presented. © 2012 IOP Publishing Ltd.
Existence of solutions to boundary value problem of fractional differential equations with impulsive
Directory of Open Access Journals (Sweden)
Weihua JIANG
2016-12-01
Full Text Available In order to solve the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line, the existence of solutions to the boundary problem is specifically studied. By defining suitable Banach spaces, norms and operators, using the properties of fractional calculus and applying the contraction mapping principle and Krasnoselskii's fixed point theorem, the existence of solutions for the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line is proved, and examples are given to illustrate the existence of solutions to this kind of equation boundary value problems.
An FPTAS for the fractional group Steiner tree problem
Directory of Open Access Journals (Sweden)
Slobodan Jelić
2015-10-01
Full Text Available This paper considers a linear relaxation of the cut-based integer programming formulation for the group Steiner tree problem (FGST. We combine the approach of Koufogiannakis and Young (2013 with the nearly-linear time approximation scheme for the minimum cut problem of Christiano et. al (2011 in order to develop a fully polynomial time approximation scheme for FGST problem. Our algorithm returns the solution to FGST where the objective function value is a maximum of 1+6ε times the optimal, for ε ∈〈0;1/6] in Õ(mk(m+n^4/3 ε^–16/3/ε^2 time, where n, m and k are the numbers of vertices, edges and groups in the group Steiner tree instance, respectively. This algorithm has a better worst-case running time than algorithm by Garg and Khandekar (2002 where the number of groups is sufficiently large.
External Field QED on Cauchy Surfaces
Deckert, D -A
2015-01-01
The Shale-Stinespring Theorem (1965) together with Ruijsenaar's criterion (1977) provide a necessary and sufficient condition for the implementability of the evolution of external field quantum electrodynamics between constant-time hyperplanes on standard Fock space. The assertion states that an implementation is possible if and only if the spacial components of the external electromagnetic four-vector potential $A_\\mu$ are zero. We generalize this result to smooth, space-like Cauchy surfaces and, for general $A_\\mu$, show how the second-quantized Dirac evolution can always be implemented as a map between varying Fock spaces. Furthermore, we give equivalence classes of polarizations, including an explicit representative, that give rise to those admissible Fock spaces. We prove that the polarization classes only depend on the tangential components of $A_\\mu$ w.r.t. the particular Cauchy surface, and show that they behave naturally under Lorentz and gauge transformations.
Directory of Open Access Journals (Sweden)
Ebrahim Youness
1993-11-01
Full Text Available The problem dual to a multi-objective fractional programming problems is defined by using the concept of dual space of the objective space and using the concept of subgradient. Some assumptions considered in recent works are relaxed in our proposed approach.
Moving-boundary problems for the time-fractional diffusion equation
Directory of Open Access Journals (Sweden)
Sabrina D. Roscani
2017-02-01
Full Text Available We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation. The time-fractional derivative of order $\\alpha\\in (0,1$ is taken in the sense of Caputo. We study the asymptotic behaivor, as t tends to infinity, of a general solution by using a fractional weak maximum principle. Also, we give some particular exact solutions in terms of Wright functions.
Local fractional Euler’s method for the steady heat-conduction problem
Directory of Open Access Journals (Sweden)
Gao Feng
2016-01-01
Full Text Available In this paper, the local fractional Euler’s method is proposed to consider the steady heat-conduction problem for the first time. The numerical solution for the local fractional heat-relaxation equation is presented. The comparison between numerical and exact solutions is discussed.
Three-Point Boundary Value Problems for Conformable Fractional Differential Equations
Directory of Open Access Journals (Sweden)
H. Batarfi
2015-01-01
Full Text Available We study a fractional differential equation using a recent novel concept of fractional derivative with initial and three-point boundary conditions. We first obtain Green's function for the linear problem and then we study the nonlinear differential equation.
Problem Solving as a Professional Development Strategy for Teachers: A Case Study with Fractions
Perdomo-Díaz, Josefa; Felmer, Patricio; Randolph, Valeria; González, Guillermo
2017-01-01
In this paper we present a professional development course designed to impact on teachers' mathematical knowledge for teaching fractions. The main features of the course are the use of i) problem solving activities related with mathematical knowledge for teaching fractions ii) peer discussions and iii) monitor's interventions focused on answering…
SOLVABILITY FOR FRACTIONAL-ORDER TWO-POINT BOUNDARY VALUE PROBLEM AT RESONANCE
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
In this paper, we are concerned with the existence of solution to a boundary value problem of nonlinear fractional differential equation at resonance. By means of the coincidence degree theory, the existence of solution is obtained.
Solution of Semi-Boundless Mixed Problem for Time-fractional Telegraph Equation
Institute of Scientific and Technical Information of China (English)
2007-01-01
In this paper, we study the semi-boundless mixed problem for time-fractional telegraph equation. We are able to use the integral transform method (the Fourier sin and cos transforms) to obtain the solution.
Darboux problem for implicit impulsive partial hyperbolic fractional order differential equations
Directory of Open Access Journals (Sweden)
Said Abbas
2011-11-01
Full Text Available In this article we investigate the existence and uniqueness of solutions for the initial value problems, for a class of hyperbolic impulsive fractional order differential equations by using some fixed point theorems.
Institute of Scientific and Technical Information of China (English)
Liu YANG; Zongmin QIAO
2012-01-01
In this paper,the existence and multiplicity of positive solutions for Robin type boundary value problem of differential equation involving the Riemann-Liouville fractional order derivative are established.
A goal programming procedure for solving fuzzy multiobjective fractional linear programming problems
Tunjo Perić; Zoran Babić; Sead Rešić
2014-01-01
This paper presents a modification of Pal, Moitra and Maulik's goal programming procedure for fuzzy multiobjective linear fractional programming problem solving. The proposed modification of the method allows simpler solving of economic multiple objective fractional linear programming (MOFLP) problems, enabling the obtained solutions to express the preferences of the decision maker defined by the objective function weights. The proposed method is tested on the production planning example.
Institute of Scientific and Technical Information of China (English)
SU XIN-WEI
2011-01-01
This paper is devoted to study the existence and uniqueness of solutions to a boundary value problem of nonlinear fractional differential equation with impulsive effects. The arguments are based upon Schauder and Banach fixed-point theorems. We improve and generalize the results presented in [B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Analysis: Hybrid Systems, 3(2009), 251258].
Higher-order Laplace equations and hyper-Cauchy distributions
Orsingher, Enzo
2012-01-01
In this paper we introduce new distributions which are solutions of higher-order Laplace equations. It is proved that their densities can be obtained by folding and symmetrizing Cauchy distributions. Another class of probability laws related to higher-order Laplace equations is obtained by composing pseudo-processes with positively-skewed Cauchy distributions which produce asymmetric Cauchy densities in the odd-order case. A special attention is devoted to the third-order Laplace equation where the connection between the Cauchy distribution and the Airy functions is obtained and analyzed.
Direct and inverse source problems for a space fractional advection dispersion equation
Aldoghaither, Abeer
2016-05-15
In this paper, direct and inverse problems for a space fractional advection dispersion equation on a finite domain are studied. The inverse problem consists in determining the source term from final observations. We first derive the analytic solution to the direct problem which we use to prove the uniqueness and the unstability of the inverse source problem using final measurements. Finally, we illustrate the results with a numerical example.
Khader, M. M.
2015-10-01
In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method. The algorithm is based on a combination of the useful properties of Chebyshev polynomial approximation and finite difference method. We implement this technique to solve numerically the non-linear programming problem which are governed by fractional differential equations (FDEs). The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the Caputo fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The application of the method to the generated FDEs leads to algebraic systems which can be solved by an appropriate method. Two numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method. A comparison with the fourth-order Runge-Kutta method is given.
Cauchy-Schwartz Inequality and Time Entanglement Witness%Cauchy-Schwartz不等式与时间纠缠判据
Institute of Scientific and Technical Information of China (English)
颜辉
2011-01-01
在量子光学中,利用Cauchy-Schwartz不等式可以定义不同性质的光场,从经典光场到量子光场,甚至单光子场;但是Cauchy-Schwartz不等式却不能描述时间纠缠光子对,也就是说并不是违反Cauchy-Sehwartz不等式的双光子态都是时间纠缠态.本文系统介绍Cauchy-Schwartz不等式、时间纠缠判据,以及两者的关系.
Homogenization of a Cauchy continuum towards a micromorphic continuum
Hütter, Geralf
2017-02-01
The micromorphic theory of Eringen and Mindlin, including special cases like strain gradient theory or Cosserat theory, is widely used to model size effects and localization phenomena. The heuristic construction of such theories based on thermodynamic considerations is well-established. However, the identification of corresponding constitutive laws and of the large number of respective constitutive parameters limits the practical application of such theories. In the present contribution, a closed procedure for the homogenization of a Cauchy continuum at the microscale towards a fully micromorphic continuum is derived including explicit definitions of all involved generalized macroscopic stress and deformation measures. The boundary value problem to be solved on the microscale is formulated either for using static or kinematic boundary conditions. The procedure is demonstrated with an example.
An intermediate distribution between Gaussian and Cauchy distributions
Liu, Tong; Dai, Wu-Sheng; Xie, Mi
2012-01-01
In this paper, we construct an intermediate distribution linking the Gaussian and the Cauchy distribution. We provide the probability density function and the corresponding characteristic function of the intermediate distribution. Because many kinds of distributions have no moment, we introduce weighted moments. Specifically, we consider weighted moments under two types of weighted functions: the cut-off function and the exponential function. Through these two types of weighted functions, we can obtain weighted moments for almost all distributions. We consider an application of the probability density function of the intermediate distribution on the spectral line broadening in laser theory. Moreover, we utilize the intermediate distribution to the problem of the stock market return in quantitative finance.
Proof of linear instability of the Reissner-Nordstr\\"om Cauchy horizon under scalar perturbations
Luk, Jonathan
2015-01-01
It has long been suggested that solutions to linear scalar wave equation $$\\Box_g\\phi=0$$ on a fixed subextremal Reissner-Nordstr\\"om spacetime with non-vanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to $W^{1,2}_{loc}$. This instability is related to the celebrated blue shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein-Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner-Nordstr\\"om spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price's law decay is generically sharp along the event horiz...
Developing Pre-Service Teachers Understanding of Fractions through Problem Posing
Toluk-Ucar, Zulbiye
2009-01-01
This study investigated the effect of problem posing on the pre-service primary teachers' understanding of fraction concepts enrolled in two different versions of a methods course at a university in Turkey. In the experimental version, problem posing was used as a teaching strategy. At the beginning of the study, the pre-service teachers'…
Yakubova, Gulnoza; Hughes, Elizabeth M.; Hornberger, Erin
2015-01-01
The purpose of this study was to determine the effectiveness of a point-of-view video modeling intervention to teach mathematics problem-solving when working on word problems involving subtracting mixed fractions with uncommon denominators. Using a multiple-probe across students design of single-case methodology, three high school students with…
Sharp, Emily; Shih Dennis, Minyi
2017-01-01
This study used a multiple probe across participants design to examine the effects of a model drawing strategy (MDS) intervention package on fraction comparing and ordering word problem-solving performance of three Grade 4 students. MDS is a form of cognitive strategy instruction for teaching word problem solving that includes explicit instruction…
THE COLLOCATION METHODS FOR SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNELS
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
This paper applies the singular integral operators,singular quadrature operators and discretization matrices associated withsingular integral equations with Cauchy kernels, which are established in [1],to give a unified framework for various collocation methods of numericalsolutions of singular integral equations with Cauchy kernels. Under theframework, the coincidence of the direct quadrature method and the indirectquadrature method is very simple and obvious.
A New Generalization on Cauchy-Schwarz Inequality
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Songting Yin
2017-01-01
Full Text Available We extend the well-known Cauchy-Schwarz inequality involving any number of real or complex functions and also give a necessary and sufficient condition for the equality. This is another generalized version of the Cauchy-Schwarz inequality.
A generalized Cauchy process and its application to relaxation phenomena
Energy Technology Data Exchange (ETDEWEB)
Lim, S C [Faculty of Engineering, Multimedia University, 63100 Cyberjaya, Selangor (Malaysia); Li Ming [School of Information Science and Technology, East China Normal University, Shanghai 200026 (China)
2006-03-24
We study some of the basic properties of a generalized Cauchy process indexed by two parameters. The application of the Lamperti transformation to the generalized Cauchy process leads to a self-similar process which preserves the long-range dependence. The asymptotic properties of spectral density of the process are derived. Possible application of this process to model relaxation phenomena is considered.
Institute of Scientific and Technical Information of China (English)
SONG Li-mei; WENG Pei-xuan
2012-01-01
In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α ∈ (3,4],where the fractional derivative D0α+ is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.
Application of geometric algebra to electromagnetic scattering the Clifford-Cauchy-Dirac technique
Seagar, Andrew
2016-01-01
This work presents the Clifford-Cauchy-Dirac (CCD) technique for solving problems involving the scattering of electromagnetic radiation from materials of all kinds. It allows anyone who is interested to master techniques that lead to simpler and more efficient solutions to problems of electromagnetic scattering than are currently in use. The technique is formulated in terms of the Cauchy kernel, single integrals, Clifford algebra and a whole-field approach. This is in contrast to many conventional techniques that are formulated in terms of Green's functions, double integrals, vector calculus and the combined field integral equation (CFIE). Whereas these conventional techniques lead to an implementation using the method of moments (MoM), the CCD technique is implemented as alternating projections onto convex sets in a Banach space. The ultimate outcome is an integral formulation that lends itself to a more direct and efficient solution than conventionally is the case, and applies without exception to all types...
Self-accelerating Massive Gravity: Superluminality, Cauchy Surfaces and Strong Coupling
Motloch, Pavel; Joyce, Austin; Motohashi, Hayato
2015-01-01
Self-accelerating solutions in massive gravity provide explicit, calculable examples that exhibit the general interplay between superluminality, the well-posedness of the Cauchy problem, and strong coupling. For three particular classes of vacuum solutions, one of which is new to this work, we construct the conformal diagram for the characteristic surfaces on which isotropic stress-energy perturbations propagate. With one exception, all solutions necessarily possess spacelike characteristics, indicating perturbative superluminality. Foliating the spacetime with these surfaces gives a pathological frame where kinetic terms of the perturbations vanish, confusing the Hamiltonian counting of degrees of freedom. This frame dependence distinguishes the vanishing of kinetic terms from strong coupling of perturbations or an ill-posed Cauchy problem. We give examples where spacelike characteristics do and do not originate from a point where perturbation theory breaks down and where spacelike surfaces do or do not inte...
Fractional Calculus and -Valently Starlike Functions
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Özkan Öznur
2009-01-01
Full Text Available Abstract In this investigation, the authors prove coefficient bounds, distortion inequalities for fractional calculus of a family of multivalent functions with negative coefficients, which is defined by means of a certain nonhomogenous Cauchy-Euler differential equation.
Cauchy-Rassias Stability of Cauchy-Jensen Additive Mappings in Banach Spaces
Institute of Scientific and Technical Information of China (English)
Choonkil BAAK
2006-01-01
Let X, Y be vector spaces. It is shown that if a mapping f: X → Y satisfiesf(x+y/2+z)+f(x-y/2+z)=f(x)+2f(z), (0.1)f(x+y/2+z)-f(x-y/2+z)= f(y), (0.2)or2f(x+y/2+z)=f(x)+f(y)+2f(z) (0.3)for all x, y, z ∈ X, then the mapping f: X → Y is Cauchy additive.Furthermore, we prove the Cauchy-Rassias stability of the functional equations (0.1), (0.2) and (0.3) in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebra.
The Bayesian Formulation and Well-Posedness of Fractional Elliptic Inverse Problems
Trillos, Nicolas Garcia
2016-01-01
We study the inverse problem of recovering the order and the diffusion coefficient of an elliptic fractional partial differential equation from a finite number of noisy observations of the solution. We work in a Bayesian framework and show conditions under which the posterior distribution is given by a change of measure from the prior. Moreover, we show well-posedness of the inverse problem, in the sense that small perturbations of the observed solution lead to small Hellinger perturbations of the associated posterior measures. We thus provide a mathematical foundation to the Bayesian learning of the order ---and other inputs--- of fractional models.
The Modified Adomian Decomposition Method for Nonlinear Fractional Boundary Value Problems
Institute of Scientific and Technical Information of China (English)
WANG Jie
2012-01-01
We use the modified Adomian decomposition method(ADM) for solving the nonlinear fractional boundary value problem Dα0+u(x)=f(x,u(x)), 0＜x＜1, 3＜α≤4u(0) =α0, u″(0) =α2 (1)u(1) =β0, u″(1) =β2where Dα0+u is Caputo fractional derivative and α0,α2,β0,β2 is not zero at all,and f:[0,1] x R → R is continuous.The calculated numerical results show reliability and efficiency of the algorithm given.The numerical procedure is tested on linear and nonlinear problems.
Ortiz, Néstor; Sarbach, Olivier
2014-04-01
A spherical dust cloud which is initially at rest and which has a monotonously decaying density profile collapses and forms a shell-focusing singularity. Provided the density profile is not too flat, meaning that its second radial derivative is negative at the centre, this singularity is visible to local, and sometimes even to global observers. According to the strong cosmic censorship conjecture, such naked singularities should be unstable under generic, non-spherical perturbations of the initial data or when more realistic matter models are considered. In an attempt to gain further insight into this stability issue, in this work we initiate the analysis of a simpler but related problem. We discuss the stability of test fields propagating in the vicinity of the Cauchy horizon associated to the naked central singularity. We first study the high-frequency limit and show that the fields undergo a blueshift as they approach the Cauchy horizon. However, in contrast to what occurs at inner horizons of black holes, we show that the blueshift is uniformly bounded along incoming and outgoing null rays. Motivated by this boundedness result, we take a step beyond the geometric optics approximation and consider the Cauchy evolution of spherically symmetric test scalar fields. We prove that under reasonable conditions on the initial data a suitable rescaled field can be continuously extended to the Cauchy horizon. In particular, this result implies that the physical field is everywhere finite on the Cauchy horizon away from the central singularity.
Dixon, Juli K.; Andreasen, Janet B.; Avila, Cheryl L.; Bawatneh, Zyad; Deichert, Deana L.; Howse, Tashana D.; Turner, Mercedes Sotillo
2014-01-01
A goal of this study was to examine elementary preservice teachers' (PSTs) ability to contextualize and decontextualize fraction subtraction by asking them to write word problems to represent fraction subtraction expressions and to choose prewritten word problems to support given fraction subtraction expressions. Three themes emerged from the…
Kassa, Semu Mitiku; Tsegay, Teklay Hailay
2017-08-01
Tri-level optimization problems are optimization problems with three nested hierarchical structures, where in most cases conflicting objectives are set at each level of hierarchy. Such problems are common in management, engineering designs and in decision making situations in general, and are known to be strongly NP-hard. Existing solution methods lack universality in solving these types of problems. In this paper, we investigate a tri-level programming problem with quadratic fractional objective functions at each of the three levels. A solution algorithm has been proposed by applying fuzzy goal programming approach and by reformulating the fractional constraints to equivalent but non-fractional non-linear constraints. Based on the transformed formulation, an iterative procedure is developed that can yield a satisfactory solution to the tri-level problem. The numerical results on various illustrative examples demonstrated that the proposed algorithm is very much promising and it can also be used to solve larger-sized as well as n-level problems of similar structure.
Directory of Open Access Journals (Sweden)
Nurdan Cetin
2014-01-01
Full Text Available We consider a multiobjective linear fractional transportation problem (MLFTP with several fractional criteria, such as, the maximization of the transport profitability like profit/cost or profit/time, and its two properties are source and destination. Our aim is to introduce MLFTP which has not been studied in literature before and to provide a fuzzy approach which obtain a compromise Pareto-optimal solution for this problem. To do this, first, we present a theorem which shows that MLFTP is always solvable. And then, reducing MLFTP to the Zimmermann’s “min” operator model which is the max-min problem, we construct Generalized Dinkelbach’s Algorithm for solving the obtained problem. Furthermore, we provide an illustrative numerical example to explain this fuzzy approach.
Existence of Weak Solutions for Nonlinear Time-Fractional p-Laplace Problems
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Meilan Qiu
2014-01-01
Full Text Available The existence of weak solution for p-Laplace problem is studied in the paper. By exploiting the relationship between the Nehari manifold and fibering maps and combining the compact imbedding theorem and the behavior of Palais-Smale sequences in the Nehari manifold, the existence of weak solutions is established. By means of the Arzela-Ascoli fixed point theorem, some existence results of the corresponding time-fractional equations of the p-Laplace problem are obtained.
Study on the Nonsingular Problem of Fractional-Order Terminal Sliding Mode Control
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Kening Li
2013-01-01
Full Text Available An improved type of control strategy combining the fractional calculus with nonsingular terminal sliding mode control named non-singular fractional terminal sliding mode control (NFOTSM is proposed for the nonlinear tire-road friction control system of vehicle in this paper. A fractional-order switching manifold is proposed, and the corresponding control law is formulated based on the Lyapunov stability theory to guarantee the sliding condition. The proposed controller ensures the finite time stability of the closed-loop system. Then, a terminal attractor is introduced to solve the singularity problem of fractional terminal sliding mode control (FOTSM. Finally, the performance of the NFOTSM is fully investigated compared with other related algorithms to find the effectiveness for the tire-road friction system. The results show that the NFOTSM has better performance than other related algorithms.
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Mohamed Jleli
2014-01-01
Full Text Available A class of nonlinear multipoint boundary value problems for singular fractional differential equations is considered. By means of a coupled fixed point theorem on ordered sets, some results on the existence and uniqueness of positive solutions are obtained.
Characteristic equation method for fractal heat-transfer problem via local fractional calculus
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Liu Geng-Yuan
2016-01-01
Full Text Available In this paper the fractal heat-transfer problem described by the theory of local fractional calculus is considered. The non-differentiable-type solution of the heat-transfer equation is obtained. The characteristic equation method is proposed as a powerful technology to illustrate the analytical solution of the partial differential equation in fractal heat transfer.
Isik, Cemalettin; Kar, Tugrul
2012-01-01
The present study aimed to make an error analysis in the problems posed by pre-service elementary mathematics teachers about fractional division operation. It was carried out with 64 pre-service teachers studying in their final year in the Department of Mathematics Teaching in an eastern university during the spring semester of academic year…
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Archana Chauhan
2012-12-01
Full Text Available In this article, we establish a general framework for finding solutions for impulsive fractional integral boundary-value problems. Then, we prove the existence and uniqueness of solutions by applying well known fixed point theorems. The obtained results are illustrated with an example for their feasibility.
Positive solutions of multi-point boundary value problem of fractional differential equation
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De-xiang Ma
2015-07-01
Full Text Available By means of two fixed-point theorems on a cone in Banach spaces, some existence and multiplicity results of positive solutions of a nonlinear fractional differential equation boundary value problem are obtained. The proofs are based upon some properties of Green’s function, which are also the key of the paper.
Preservice Teachers' Algebraic Reasoning and Symbol Use on a Multistep Fraction Word Problem
Cullen, Amanda L.; Tobias, Jennifer M.; Safak, Elif; Kirwan, J. Vince; Wessman-Enzinger, Nicole M.; Wickstrom, Megan H.; Baek, Jae M.
2017-01-01
Previous research on preservice teachers' understanding of fractions and algebra has focused on one or the other. To extend this research, we examined 85 undergraduate elementary education majors and middle school mathematics education majors' solutions and solution paths (i.e., the ways or methods in which preservice teachers solve word problems)…
Existence of three solutions for impulsive nonlinear fractional boundary value problems
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Shapour Heidarkhani
2017-01-01
Full Text Available In this work we present new criteria on the existence of three solutions for a class of impulsive nonlinear fractional boundary-value problems depending on two parameters. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results.
On resolution to Wu's conjecture on Cauchy function's exterior singularities
Institute of Scientific and Technical Information of China (English)
Theodore Yaotsu Wu
2011-01-01
This is a series of studies on Wu's conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy's function f(z) (z = x + iy) and its integral J[∫(z)] = (2πi)-1 φ f(t)(t - z)-1dt taken along the unit circle as contour C, inside which (the open domain D+)f(z) is regular but has singularities distributed in open domain Doutside C. Resolution is given to the inverse problem that the singularities of f(z) can be determined in analytical form in terms of the values f(t) of f(z) numerically prescribed on C (｜t｜ = 1), as so enunciated by Wu's conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle, for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction, and essential singularities aretreated with employing the generalized Hilbert transforms.These new methods are applicable to relevant problems in mathematics, engineering and technology in analogy with resolving the inverse problem presented here.
Abstract Description of Internet Traffic of Generalized Cauchy Type
Directory of Open Access Journals (Sweden)
Ming Li
2012-01-01
. Therefore, there is a limitation for fGn to accurately model traffic. Recently, the generalized Cauchy (GC process was introduced to model traffic with the flexibility to separately measure the fractal dimension DGC and the Hurst parameter HGC of traffic. However, there is a fundamental problem whether or not there exists the generality that the GC model is more conformable with real traffic than single parameter models, such as fGn, irrelevant of traffic traces used in experimental verification. The solution to that problem remains unknown but is desired for model evaluation in traffic theory or for model selection against specific issues, such as queuing analysis relating to the autocorrelation function (ACF of arrival traffic. The key contribution of this paper is our solution to that fundamental problem (see Theorem 3.17 with the following features in analysis. (i Set-valued analysis of the traffic of the fGn type. (ii Set-valued analysis of the traffic of the GC type. (iii Revealing the generality previously mentioned by comparing metrics of the traffic of the fGn type to that of the GC type.
Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation
Ye, Zhuan; Xu, Xiaojing
2016-04-01
As a continuation of the previous work [48], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [13]. Consequently, this result significantly improves the recent works [13,45,48].
Spatially fractional-order viscoelasticity, non-locality and a new kind of anisotropy
Hanyga, Andrzej
2011-01-01
Spatial non-locality of space-fractional viscoelastic equations of motion is studied. Relaxation effects are accounted for by replacing second-order time derivatives by lower-order fractional derivatives and their generalizations. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind anisotropy, associated with angular dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Explicit fundamental solutions of the Cauchy problem are constructed for some cases isotropic and anisotropic non-locality.
Radiative transfer in plane-parallel media and Cauchy integral equations III. The finite case
Rutily, Bernard; Chevallier, Loïc
2006-01-01
We come back to the Cauchy integral equations occurring in radiative transfer problems posed in finite, plane-parallel media with light scattering taken as monochromatic and isotropic. Their solution is calculated following the classical scheme where a Cauchy integral equation is reduced to a couple of Fredholm integral equations. It is expressed in terms of two auxiliary functions $\\zeta_+$ and $\\zeta_-$ we introduce in this paper. These functions show remarkable analytical properties in the complex plane. They satisfy a simple algebraic relation which generalizes the factorization relation of semi-infinite media. They are regular in the domain of the Fredholm integral equations they satisfy, and thus can be computed accurately. As an illustration, the X- and Y-functions are calculated in the whole complex plane, together with the extension in this plane of the so-called Sobouti's functions.
Li, Dongfang; Zhang, Jiwei
2016-10-01
Anomalous diffusion behavior in many practical problems can be described by the nonlinear time-fractional parabolic problems on unbounded domain. The numerical simulation is a challenging problem due to the dependence of global information from time fractional operators, the nonlinearity of the problem and the unboundedness of the spacial domain. To overcome the unboundedness, conventional computational methods lead to extremely expensive costs, especially in high dimensions with a simple treatment of boundary conditions by making the computational domain large enough. In this paper, based on unified approach proposed in [25], we derive the efficient nonlinear absorbing boundary conditions (ABCs), which reformulates the problem on unbounded domain to an initial boundary value problem on bounded domain. To overcome nonlinearity, we construct a linearized finite difference scheme to solve the reduced nonlinear problem such that iterative methods become dispensable. And the stability and convergence of our linearized scheme are proved. Most important, we prove that the numerical solutions are bounded by the initial values with a constant coefficient, i.e., the constant coefficient is independent of the time. Overall, the computational cost can be significantly reduced comparing with the usual implicit schemes and a simple treatment of boundary conditions. Finally, numerical examples are given to demonstrate the efficiency of the artificial boundary conditions and theoretical results of the schemes.
Cho, Yonggeun
2016-05-04
This paper is devoted to the mathematical analysis of a class of nonlinear fractional Schrödinger equations with a general Hartree-type integrand. We show the well-posedness of the associated Cauchy problem and prove the existence and stability of standing waves under suitable assumptions on the nonlinearity. Our proofs rely on a contraction argument in mixed functional spaces and the concentration-compactness method. © 2015 World Scientific Publishing Company
Construction of Involution Cauchy-Hadamard Type MDS Matrices%对合Cauchy-Hadamard型MDS矩阵的构造
Institute of Scientific and Technical Information of China (English)
崔霆; 金晨辉
2010-01-01
MDS矩阵和对合MDS矩阵在分组密码中有广泛应用.该文将考察同时是Hadamard矩阵和Cauchy矩阵的那些MDS矩阵,给出了这类矩阵的结构、构造方法和个数,从而得到了MDS矩阵一种新的构造方法.该文还证明了Cauchy-Hadamard型MDS矩阵都等效于对合的Cauchy-Hadamard型MDS矩阵,并给出了由Cauchy-Hadamard型MDS矩阵构造对合的Cauchy-Hadamard型MDS矩阵的方法.
Robinson manifolds and Cauchy-Riemann spaces
Trautman, A
2002-01-01
A Robinson manifold is defined as a Lorentz manifold (M, g) of dimension 2n >= 4 with a bundle N subset of C centre dot TM such that the fibres of N are maximal totally null and there holds the integrability condition [Sec N, Sec N] subset of Sec N. The real part of N intersection N-bar is a bundle of null directions tangent to a congruence of null geodesics. This generalizes the notion of a shear-free congruence of null geodesics (SNG) in dimension 4. Under a natural regularity assumption, the set M of all these geodesics has the structure of a Cauchy-Riemann manifold of dimension 2n - 1. Conversely, every such CR manifold lifts to many Robinson manifolds. Three definitions of a CR manifold are described here in considerable detail; they are equivalent under the assumption of real analyticity, but not in the smooth category. The distinctions between these definitions have a bearing on the validity of the Robinson theorem on the existence of null Maxwell fields associated with SNGs. This paper is largely a re...
Boundary value problem for one-dimensional fractional differential advection-dispersion equation
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Khasambiev Mokhammad Vakhaevich
2014-07-01
Full Text Available An equation commonly used to describe solute transport in aquifers has attracted more attention in recent years. After a formal study of some aspects of the advection-diffusion equation, basically from the mathematical point of view with the solution of a differential equation with fractional derivative, the main interest to this problem shifted onto physical aspects of the dynamical system, such as the total energy and the dynamical response. In this regard it should be pointed out that the interaction with environment is expressed in terms of stochastic arrow of time. This allows one also to reach a progress in one more issue. Formerly the equation of advection-diffusion was not obtained from any physical principles. However, mainly the success concerns linear fractional systems. In fact, there are many cases in which linear treatments are not sufficient. The more general systems described by nonlinear fractional differential equations have not been studied enough. The ordinary calculus brings out clearly that essentially new phenomena occur in nonlinear systems, which generally cannot occur in linear systems. Due to vast range of application of the fractional advection-dispersion equation, a lot of work has been done to find numerical solution and fundamental solution of this equation. The research on the analytical solution of initial-boundary problem for space-fractional advection-dispersion equation is relatively new and is still at an early stage of development. In this paper, we will take use of the method of variable separation to solve space-fractional advection-dispersion equation with initial boundary data.
Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems
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D. Baleanu
2013-01-01
Full Text Available We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs on semi-infinite interval, using generalized Laguerre polynomials. We derive the operational matrix of Caputo fractional derivative of the generalized Laguerre polynomials which is applied together with generalized Laguerre tau approximation for implementing a spectral solution of linear multiterm FDEs on semi-infinite interval subject to initial conditions. The generalized Laguerre pseudo-spectral approximation based on the generalized Laguerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem. Through several numerical examples, we confirm the accuracy and performance of the proposed spectral algorithms. Indeed, the methods yield accurate results, and the exact solutions are achieved for some tested problems.
On existence and uniqueness of positive solutions to a class of fractional boundary value problems
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Caballero J
2011-01-01
Full Text Available Abstract The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fractional boundary value problem D 0 + α u ( t + f ( t , u ( t = 0 , 0 < t < 1 , u ( 0 = u ( 1 = u ′ ( 0 = 0 , where 2 < α ≤ 3 and D 0 + α is the Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem in partially ordered metric spaces. The autonomous case of this problem was studied in the paper [Zhao et al., Abs. Appl. Anal., to appear], but in Zhao et al. (to appear, the question of uniqueness of the solution is not treated. We also present some examples where we compare our results with the ones obtained in Zhao et al. (to appear. 2010 Mathematics Subject Classification: 34B15
Institute of Scientific and Technical Information of China (English)
袁益让
1999-01-01
For compressible two-phase displacement problem, a kind of characteristic finite difference fractional steps schemes is put forward and thick and thin grids are used to form a complete set. Some techniques, such as piecewise biquadratic interpolation, of calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in L~2 norm are derived to determine the error in the approximate solution.
POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEM OF FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In this paper,we establish sufficient conditions for the existence of positive solutions to a general class of integral boundary value problem(BVP) of nonlinear fractional functional differential equation.A differential operator is taken in the RiemannLiouville sense.Our analysis relies on the Krasnosel'skii fixed-point theorem in cones.We also give examples to illustrate the applicability of our results.
The Fractional Fourier Transform and Its Application to Energy Localization Problems
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ter Morsche Hennie G
2003-01-01
Full Text Available Applying the fractional Fourier transform (FRFT and the Wigner distribution on a signal in a cascade fashion is equivalent to a rotation of the time and frequency parameters of the Wigner distribution. We presented in ter Morsche and Oonincx, 2002, an integral representation formula that yields affine transformations on the spatial and frequency parameters of the -dimensional Wigner distribution if it is applied on a signal with the Wigner distribution as for the FRFT. In this paper, we show how this representation formula can be used to solve certain energy localization problems in phase space. Examples of such problems are given by means of some classical results. Although the results on localization problems are classical, the application of generalized Fourier transform enlarges the class of problems that can be solved with traditional techniques.
Fractional Goal Programming for Fuzzy Solid Transportation Problem with Interval Cost
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B. Radhakrishnan
2014-09-01
Full Text Available In this paper, we study a solid transportation problem with interval cost using fractional goal programming approach (FGP. In real life applications of the FGP problem with multiple objectives, it is difficult for the decision-maker(s to determine the goal value of each objective precisely as the goal values are imprecise, vague, or uncertain. Therefore, a fuzzy goal programming model is developed for this purpose. The proposed model presents an application of fuzzy goal programming to the solid transportation problem. Also, we use a special type of non-linear (hyperbolic membership functions to solve multi-objective transportation problem. It gives an optimal compromise solution. The proposed model is illustrated by using an example.
Directory of Open Access Journals (Sweden)
Somayeh Nemati
2016-11-01
Full Text Available In this paper, we consider the second-kind Chebyshev polynomials (SKCPs for the numerical solution of the fractional optimal control problems (FOCPs. Firstly, an introduction of the fractional calculus and properties of the shifted SKCPs are given and then operational matrix of fractional integration is introduced. Next, these properties are used together with the Legendre-Gauss quadrature formula to reduce the fractional optimal control problem to solving a system of nonlinear algebraic equations that greatly simplifies the problem. Finally, some examples are included to confirm the efficiency and accuracy of the proposed method.
Methods and Algorithms for Solving Inverse Problems for Fractional Advection-Dispersion Equations
Aldoghaither, Abeer
2015-11-12
Fractional calculus has been introduced as an e cient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous di↵usion processes such as contaminant transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particle transport such as particles with velocity variations and long-rest periods. Mathematical modeling of physical phenomena requires the identification of pa- rameters and variables from available measurements. This is referred to as an inverse problem. In this work, we are interested in studying theoretically and numerically inverse problems for space Fractional Advection-Dispersion Equation (FADE), which is used to model solute transport in porous media. Identifying parameters for such an equa- tion is important to understand how chemical or biological contaminants are trans- ported throughout surface aquifer systems. For instance, an estimate of the di↵eren- tiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the medium. Our main contribution is to propose a novel e cient algorithm based on modulat-ing functions to estimate the coe cients and the di↵erentiation order for space FADE, which can be extended to general fractional Partial Di↵erential Equation (PDE). We also show how the method can be applied to the source inverse problem. This work is divided into two parts: In part I, the proposed method is described and studied through an extensive numerical analysis. The local convergence of the proposed two-stage algorithm is proven for 1D space FADE. The properties of this method are studied along with its limitations. Then, the algorithm is generalized to the 2D FADE. In part II, we analyze direct and inverse source problems for a space FADE. The problem consists of recovering the source term using final
Fast multi-swarm optimization based-on Cauchy mutation and crossover operation
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
The standard particle swarm optimization (PSO) algorithm is a novel evolutionary algorithm in which each particle studies its own previous best solution and the group's previous best solutions to optimization problems. One problem in PSO is its tendency of trapping into local optima. In this paper, a multi-swarm technique based on fast particle swarm optimization (FPSO) algorithm is proposed by introducing crossover operation. FPSO is global search algorithm which can prevent PSO from trapping into local optima in light of Cauchy mutation. Though it can get high optimizing precision, the convergence rate is not satisfactory. FMSO can not only find satisfying solutions, but also speed up the search.
Fuzzy linear fractional bi-level multi-objective programming problems
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nemat safaei
2012-08-01
Full Text Available The Kuhn-Tuker condition has become nowadays an important tool in the hands of investigation for checking the optimality in optimization literature. In the present paper with use of a Taylor series and Kuhn-Tucker conditions approach, we solve a fuzzy linear fractional bilevel multi-objective programming (FLFBL-MOP problem. The Taylor series is an expansion of a series that represents a function. In the proposed approach, membership functions associated with each level(s ofthe objective(s of FLFBL-MOP problems are transformed and unied by using a Taylor series approach. By using the Kuhn-Tucker conditions, the problem is reduced to a single objective and nally, numericalexample is given to illustrates the efficiency and superiority of the proposed approach.
Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem
Zheng, G. H.; Wei, T.
2010-11-01
In this paper, a backward diffusion problem for a space-fractional diffusion equation (SFDE) in a strip is investigated. Such a problem is obtained from the classical diffusion equation in which the second-order space derivative is replaced with a Riesz-Feller derivative of order β in (0, 2]. We show that such a problem is severely ill-posed and further propose a new regularization method and apply a spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical methods are effective.
Existence of solutions to fractional boundary-value problems with a parameter
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Ya-Ning Li
2013-06-01
Full Text Available This article concerns the existence of solutions to the fractional boundary-value problem $$displaylines{ -frac{d}{dt} ig(frac{1}{2} {}_0D_t^{-eta}+ frac{1}{2}{}_tD_{T}^{-eta}igu'(t=lambda u(t+abla F(t,u(t,quad hbox{a.e. } tin[0,T], cr u(0=0,quad u(T=0. }$$ First for the eigenvalue problem associated with it, we prove that there is a sequence of positive and increasing real eigenvalues; a characterization of the first eigenvalue is also given. Then under different assumptions on the nonlinearity F(t,u, we show the existence of weak solutions of the problem when $lambda$ lies in various intervals. Our main tools are variational methods and critical point theorems.
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Paras Bhatnagar
2012-10-01
Full Text Available Kaul and Kaur [7] obtained necessary optimality conditions for a non-linear programming problem by taking the objective and constraint functions to be semilocally convex and their right differentials at a point to be lower semi-continuous. Suneja and Gupta [12] established the necessary optimality conditions without assuming the semilocal convexity of the objective and constraint functions but their right differentials at the optimal point to be convex. Suneja and Gupta [13] established necessary optimality conditions for an efficient solution of a multiobjective non-linear programming problem by taking the right differentials of the objective functions and constraintfunctions at the efficient point to be convex. In this paper we obtain some results for a properly efficient solution of a multiobjective non-linear fractional programming problem involving semilocally convex and related functions by assuming generalized Slater type constraint qualification.
Minimax fractional programming problem involving nonsmooth generalized α-univex functions
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Anurag JAYSWAL
2013-01-01
Full Text Available In this paper, we introduce a new class of generalized α-univex functions where the involved functions are locally Lipschitz. We extend the concept of α-type I invex [S. K. Mishra, J. S. Rautela, On nondifferentiable minimax fractional programming under generalized α-type I invexity, J. Appl. Math. Comput. 31 (2009 317-334] to α-univexity and an example is provided to show that there exist functions that are α-univex but not α-type I invex. Furthermore, Karush-Kuhn-Tucker-type sufficient optimality conditions and duality results for three different types of dual models are obtained for nondifferentiable minimax fractional programming problem involving generalized α-univex functions. The results in this paper extend some known results in the literature.
Institute of Scientific and Technical Information of China (English)
Yi-rang Yuan
2004-01-01
For compressible two-phase displacement problem,the modified upwind finite difference fractional steps schemes are put forward.Some techniques,such as calculus of variations,commutative law of multiplication of difference operators,decomposition of high order difference operators,the theory of prior estimates and techniques are used.Optimal order estimates in L 2 norm are derived for the error in the approximate solution.This method has already been applied to the numerical simulation of seawater intrusion and migration-accumulation of oil resources.
ALE Fractional Step Finite Element Method for Fluid-Structure Nonlinear Interaction Problem
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
A computational procedure is developed to solve the problems of coupled motion of a structure and a viscous incompressible fluid. In order to incorporate the effect of the moving surface of the structure as well as the free surface motion, the arbitrary Lagrangian-Eulerian formulation is employed as the basis of the finite element spatial discretization. For numerical integration in time, the fraction step method is used. This method is useful because one can use the same linear interpolation function for both velocity and pressure. The method is applied to the nonlinear interaction of a structure and a tuned liquid damper. All computations are performed with a personal computer.
Institute of Scientific and Technical Information of China (English)
GAO Ying; RONG Wei-dong
2008-01-01
This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators are the difference of differentiable function and convex function. Under the assumption of Calmness Constraint Qualification the Kuhn-Tucker type necessary conditions for efficient solution are given, and the Kuhn-Tucker type sufficient conditions for efficient solution are presented under the assumptions of (F, α, ρ, d)-V-convexity.Subsequently, the optimality conditions for two kinds of duality models are formulated and duality theorems are proved.
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Z. Denton
2017-01-01
Full Text Available In this work we investigate integro-differential initial value problems with Riemann Liouville fractional derivatives where the forcing function is a sum of an increasing function and a decreasing function. We will apply the method of lower and upper solutions and develop two monotone iterative techniques by constructing two sequences that converge uniformly and monotonically to minimal and maximal solutions. In the first theorem we will construct two natural sequences and in the second theorem we will construct two intertwined sequences. Finally, we illustrate our results with an example.
Area theorem and smoothness of compact Cauchy horizons
Minguzzi, E
2014-01-01
We obtain an improved version of the area theorem for not necessarily differentiable horizons which, in conjunction with a recent result on the completeness of generators, allows us to prove that under the null energy condition every compactly generated Cauchy horizon is smooth and compact. We explore the consequences of this result for time machines, topology change, black holes and cosmic censorship. For instance, it is shown that compact Cauchy horizons cannot form in a non-empty spacetime which satisfies the stable dominant energy condition wherever there is some source content.
UNCONDITIONAL CAUCHY SERIES AND UNIFORM CONVERGENCE ON MATRICES
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A. AIZPURU; A. GUTIERREZ-DAVILA
2004-01-01
The authors obtain new characterizations of unconditional Cauchy series in terms of separation properties of subfamilies of p(N), and a generalization of the Orlicz-Pettis Theorem is also obtained. New results on the uniform convergence on matrices and a new version of the Hahn-Schur summation theorem are proved. For matrices whose rows define unconditional Cauchy series, a better sufficient condition for the basic Matrix Theorem of Antosik and Swartz, new necessary conditions and a new proof of that theorem are given.
Can a particle detector cross a Cauchy horizon?
Juárez-Aubry, Benito A
2015-01-01
Cauchy horizons are well known to exhibit instabilities in classical spacetime dynamics and singularities in quantum field theory. We analyse the response of an Unruh-DeWitt particle detector that falls towards a Cauchy horizon, in terms of the specifics of the horizon, the choice of the quantum state and the specifics of the detector's trajectory. As a prototype, we study in detail the case for the $1+1$ Reissner-Nordstr\\"om black hole with a scalar field in the Hartle-Hawking state. Comparisons are made with the response of a detector that falls into a Schwarzschild-like singularity.
ON CAUCHY-POMPEIU FORMULA FOR FUNCTIONS WITH VALUES IN A UNIVERSAL CLIFFORD ALGEBRA
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无
2003-01-01
This paper obtains the Cauchy-Pompeiu formula on certain distinguishedboundary for functions with values in a universal Clifford algebra. This formula is just anextension of the Cauchy's integral formula obtained in [11].
Fractional Stefan problems exhibiting lumped and distributed latent-heat memory effects
Voller, Vaughan R.; Falcini, Federico; Garra, Roberto
2013-04-01
We consider fractional Stefan melting problems which involve a memory of the latent-heat accumulation. We show that the manner in which the memory of the latent-heat accumulation is recorded depends on the assumed nature of the transition between the liquid and the solid phases. When a sharp interface between the liquid and the solid phases is assumed, the memory of the accumulation of the latent heat is “lumped” in the history of the speed of the interface. In contrast, when a diffuse interface is assumed, the memory of the accumulation is “distributed” throughout the liquid phase. By use of an example problem, we demonstrate that the equivalence of the sharp- and diffuse-interface models can only occur when there is no memory in the system.
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Zhiqiang Zhou
2017-01-01
Full Text Available We study the pricing of the American options with fractal transmission system under two-state regime switching models. This pricing problem can be formulated as a free boundary problem of time-fractional partial differential equation (FPDE system. Firstly, applying Laplace transform to the governing FPDEs with respect to the time variable results in second-order ordinary differential equations (ODEs with two free boundaries. Then, the solutions of ODEs are expressed in an explicit form. Consequently the early exercise boundaries and the values for the American option are recovered using the Gaver-Stehfest formula. Numerical comparisons of the methods with the finite difference methods are carried out to verify the efficiency of the methods.
Divisor problems and the pair correlation for the fractional parts of $n^2\\alpha$
Truelsen, Jimi Lee
2009-01-01
Z. Rudnick and P. Sarnak have proved that the pair correlation for the fractional parts of $n^2 \\alpha$ is Poissonian for almost all $\\alpha$. However, they were not able to find a specific $\\alpha$ for which it holds. We show that the problem is related to the problem of determining the number of $(a,b,r) \\in \\N^3$ such that $a \\le M$, $b \\le N$, $r \\le K$ and $p ab \\equiv r (q)$ for $p$ and $q$ coprime. With suitable assumptions on the relative size of $K$, $M$, $N$ and $q$ one should expect there to be $KMN/q$ such triples asymptotically and we will show that this holds on average.
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A.S. BERDYSHEV; A. CABADA; B.Kh. TURMETOV
2014-01-01
This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.
Institute of Scientific and Technical Information of China (English)
G.J. Zalmai; Qing-hong Zhang
2007-01-01
A semi-infinite programming problem is a mathematical programming problem with a finite number of variables and infinitely many constraints. Duality theories and generalized convexity concepts are important research topics in mathematical programming. In this paper, we discuss a fairly large number of parametric duality results under various generalized (η, p)-invexity assumptions for a semi-infinite minmax fractional programming problem.
Lectures on Cauchy's problem in linear partial differential equations
Hadamard, Jacques
2003-01-01
Would well repay study by most theoretical physicists."" - Physics Today""An overwhelming influence on subsequent work on the wave equation."" - Science Progress""One of the classical treatises on hyperbolic equations."" - Royal Naval Scientific ServiceDelivered at Columbia University and the Universities of Rome and Zürich, these lectures represent a pioneering investigation. Jacques Hadamard based his research on prior studies by Riemann, Kirchhoff, and Volterra. He extended and improved Volterra's work, applying its theories relating to spherical and cylindrical waves to all normal hyperbol
Parameters Identification for an Abstract Cauchy Problem by Quasilinearization
1989-10-01
if in addition ’ is of bounded variation on (-r,01, then DF(q) satisfies (1111). -12- Proof: Let A = max lakI and IhI = max IhkI Then we obtain the...zero as q 4 q* and (H8) holds. If ’ is of bounded variation on I-r,0, then y and ; are of bounded variation on {-r,TI. By 115, Theorem 2.1.7(b)] this...4.5) satisfies (1110). Moreover, if in addition u is of bounded variation on [0,T], then DG(q) satisfies (H13). Proof: Using (4.5) in place of (4.4
Nonlocal Cauchy problem for nonlinear mixed integrodifferential equations
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H.L. Tidke
2010-12-01
Full Text Available The present paper investigates the existence and uniqueness of mild and strong solutions of a nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition. The results obtained by using Schauder fixed point theorem and the theory of semigroups.
Cauchy problem as a two-surface based `geometrodynamics'
Rácz, István
2014-01-01
Four-dimensional spacetimes foliated by a two-parameter family of homologous two-surfaces are considered in Einstein's theory of gravity. By combining a 1+(1+2) decomposition, the canonical form of the spacetime metric and a suitable specification of the conformal structure of the foliating two-surfaces a gauge fixing is introduced. It is shown that, in terms of the chosen geometrically distinguished variables, the 1+3 Hamiltonian and momentum constraints can be recast into the form of a parabolic equation and a first order symmetric hyperbolic system, respectively. Initial data to this system can be given on one of the two-surfaces foliating the three-dimensional initial data surface. The 1+3 reduced Einstein's equations are also determined. By combining the 1+3 momentum constraint with the reduced system of the secondary 1+2 decomposition a mixed hyperbolic-hyperbolic system is formed. It is shown that solutions to this mixed hyperbolic-hyperbolic system are also solutions to the full set of Einstein's equa...
Fuchs, Lynn S.; Schumacher, Robin F.; Long, Jessica; Namkung, Jessica; Malone, Amelia S.; Wang, Amber; Hamlett, Carol L.; Jordan, Nancy C.; Siegler, Robert S.; Changas, Paul
2016-01-01
The purposes of this study were to (a) investigate the efficacy of a core fraction intervention program on understanding and calculation skill and (b) isolate the effects of different forms of fraction word-problem (WP) intervention delivered as part of the larger program. At-risk 4th graders (n = 213) were randomly assigned at the individual…
Nested iterations and strengthened Cauchy-Bunyakowski-Schwarz inequalities.
J.H. Brandts; S. Korotov; M. Krizek
2004-01-01
ABSTRACT: It is known that in one, two, and three spatial dimensions, the optimal constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality for the Laplacian for red-refined linear finite element spaces, takes values zero, $\\half\\sqrt{2}$ and $\\half\\sqrt{3}$, respectively. In this pap
Institute of Scientific and Technical Information of China (English)
刘柏鑫; 任婷婷
2016-01-01
研究三维不可压双流体MHD方程Cauchy问题，给出该问题在小初值条件下解的整体存在性结果。%This paper deals with the Cauchy pronlems of 3D two-fluid MHD model. The global well-posedness is obtained under the two-fluid MHD model.
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D. Goos
2015-01-01
Full Text Available We consider the time-fractional derivative in the Caputo sense of order α∈(0, 1. Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in R+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when α↗1 of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems when α = 1, and the fractional diffusion equation becomes the heat equation.
Metcalfe, Arron W S; Ashkenazi, Sarit; Rosenberg-Lee, Miriam; Menon, Vinod
2013-10-01
Baddeley and Hitch's multi-component working memory (WM) model has played an enduring and influential role in our understanding of cognitive abilities. Very little is known, however, about the neural basis of this multi-component WM model and the differential role each component plays in mediating arithmetic problem solving abilities in children. Here, we investigate the neural basis of the central executive (CE), phonological (PL) and visuo-spatial (VS) components of WM during a demanding mental arithmetic task in 7-9 year old children (N=74). The VS component was the strongest predictor of math ability in children and was associated with increased arithmetic complexity-related responses in left dorsolateral and right ventrolateral prefrontal cortices as well as bilateral intra-parietal sulcus and supramarginal gyrus in posterior parietal cortex. Critically, VS, CE and PL abilities were associated with largely distinct patterns of brain response. Overlap between VS and CE components was observed in left supramarginal gyrus and no overlap was observed between VS and PL components. Our findings point to a central role of visuo-spatial WM during arithmetic problem-solving in young grade-school children and highlight the usefulness of the multi-component Baddeley and Hitch WM model in fractionating the neural correlates of arithmetic problem solving during development.
Jia, Junxiong; Peng, Jigen; Yang, Jiaqing
2017-04-01
In this paper, we focus on a space-time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result of an inverse source problem on the determination of the temporal component of the inhomogeneous term, which seems to be the first theoretical result of the inverse problem for such a general fractional diffusion model.
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Dongyuan Liu
2015-01-01
Full Text Available We consider the following state dependent boundary-value problem D0+αy(t-pD0+βg(t,y(σ(t+f(t,y(τ(t=0, 0
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Lingju Kong
2013-04-01
Full Text Available We study the existence of multiple solutions to the boundary value problem $$displaylines{ frac{d}{dt}Big(frac12{}_0D_t^{-eta}(u'(t+frac12{}_tD_T^{-eta}(u'(t Big+lambda abla F(t,u(t=0,quad tin [0,T],cr u(0=u(T=0, }$$ where $T>0$, $lambda>0$ is a parameter, $0leqeta<1$, ${}_0D_t^{-eta}$ and ${}_tD_T^{-eta}$ are, respectively, the left and right Riemann-Liouville fractional integrals of order $eta$, $F: [0,T]imesmathbb{R}^Nomathbb{R}$ is a given function. Our interest in the above system arises from studying the steady fractional advection dispersion equation. By applying variational methods, we obtain sufficient conditions under which the above equation has at least three solutions. Our results are new even for the special case when $eta=0$. Examples are provided to illustrate the applicability of our results.
Zhai, Chengbo; Hao, Mengru
2014-01-01
By using Krasnoselskii's fixed point theorem, we study the existence of at least one or two positive solutions to a system of fractional boundary value problems given by -D(0+)(ν1)y1(t) = λ1a1(t)f(y1(t), y2(t)), - D(0+)(ν2)y2(t) = λ2a2(t)g(y1(t), y2(t)), where D(0+)(ν) is the standard Riemann-Liouville fractional derivative, ν1, ν2 ∈ (n - 1, n] for n > 3 and n ∈ N, subject to the boundary conditions y1((i))(0) = 0 = y ((i))(0), for 0 ≤ i ≤ n - 2, and [D(0+)(α)y1(t)] t=1 = 0 = [D(0+ (α)y2(t)] t=1, for 1 ≤ α ≤ n - 2, or y1((i))(0) = 0 = y ((i))(0), for 0 ≤ i ≤ n - 2, and [D(0+)(α)y1(t)] t=1 = ϕ1(y1), [D(0+)(α)y2(t)] t=1 = ϕ2(y2), for 1 ≤ α ≤ n - 2, ϕ1, ϕ2 ∈ C([0,1], R). Our results are new and complement previously known results. As an application, we also give an example to demonstrate our result.
Divisor problems and the pair correlation for the fractional parts of n^{2}α
DEFF Research Database (Denmark)
Truelsen, Jimi Lee
2010-01-01
Z. Rudnick and P. Sarnak have proved that the pair correlation for the fractional parts of n2α is Poissonian for almost all α. However, they were not able to find a specific α for which it holds. We show that the problem is related to the problem of determining the number of (a, b, r) ∈ N3 such t...
Modified Mixed Lagrangian-Eulerian Method Based on Numerical Framework of MT3DMS on Cauchy Boundary.
Suk, Heejun
2016-07-01
MT3DMS, a modular three-dimensional multispecies transport model, has long been a popular model in the groundwater field for simulating solute transport in the saturated zone. However, the method of characteristics (MOC), modified MOC (MMOC), and hybrid MOC (HMOC) included in MT3DMS did not treat Cauchy boundary conditions in a straightforward or rigorous manner, from a mathematical point of view. The MOC, MMOC, and HMOC regard the Cauchy boundary as a source condition. For the source, MOC, MMOC, and HMOC calculate the Lagrangian concentration by setting it equal to the cell concentration at an old time level. However, the above calculation is an approximate method because it does not involve backward tracking in MMOC and HMOC or allow performing forward tracking at the source cell in MOC. To circumvent this problem, a new scheme is proposed that avoids direct calculation of the Lagrangian concentration on the Cauchy boundary. The proposed method combines the numerical formulations of two different schemes, the finite element method (FEM) and the Eulerian-Lagrangian method (ELM), into one global matrix equation. This study demonstrates the limitation of all MT3DMS schemes, including MOC, MMOC, HMOC, and a third-order total-variation-diminishing (TVD) scheme under Cauchy boundary conditions. By contrast, the proposed method always shows good agreement with the exact solution, regardless of the flow conditions. Finally, the successful application of the proposed method sheds light on the possible flexibility and capability of the MT3DMS to deal with the mass transport problems of all flow regimes.
Alisultanov, Z. Z.; Meilanov, R. P.
2012-10-01
We consider the problem of the effective interaction potential in a quantum many-particle system leading to the fractional-power dispersion law. We show that passing to fractional-order derivatives is equivalent to introducing a pair interparticle potential. We consider the case of a degenerate electron gas. Using the van der Waals equation, we study the equation of state for systems with a fractional-power spectrum. We obtain a relation between the van der Waals constant and the phenomenological parameter α, the fractional-derivative order. We obtain a relation between energy, pressure, and volume for such systems: the coefficient of the thermal energy is a simple function of α. We consider Bose—Einstein condensation in a system with a fractional-power spectrum. The critical condensation temperature for 1 ideal system, where α = 2.
Positive Solutions to Fractional Boundary Value Problems with Nonlinear Boundary Conditions
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Nemat Nyamoradi
2013-01-01
Full Text Available We consider a system of boundary value problems for fractional differential equation given by D0+βϕp(D0+αu(t=λ1a1(tf1(u(t,v(t, t∈(0,1, D0+βϕp(D0+αv(t=λ2a2(tf2(u(t,v(t, t∈(0,1, where 1<α, β≤2, 2<α+β≤4, λ1, λ2 are eigenvalues, subject either to the boundary conditions D0+αu(0=D0+αu(1=0, u(0=0, D0+β1u(1-Σi=1m-2a1i D0+β1u(ξ1i=0, D0+αv(0=D0+αv(1=0, v(0=0, D0+β1v(1-Σi=1m-2a2i D0+β1v(ξ2i=0 or D0+αu(0=D0+αu(1=0, u(0=0, D0+β1u(1-Σi=1m-2a1i D0+β1u(ξ1i=ψ1(u, D0+αv(0=D0+αv(1=0, v(0=0, D0+β1v(1-Σi=1m-2a2i D0+β1v(ξ2i=ψ2(v, where 0<β1<1, α-β1-1≥0 and ψ1, ψ2:C([0,1]→[0, ∞ are continuous functions. The Krasnoselskiis fixed point theorem is applied to prove the existence of at least one positive solution for both fractional boundary value problems. As an application, an example is given to demonstrate some of main results.
Duan, Beiping; Zheng, Zhoushun; Cao, Wen
2016-08-01
In this paper, we revisit two spectral approximations, including truncated approximation and interpolation for Caputo fractional derivative. The two approaches have been studied to approximate Riemann-Liouville (R-L) fractional derivative by Chen et al. and Zayernouri et al. respectively in their most recent work. For truncated approximation the reconsideration partly arises from the difference between fractional derivative in R-L sense and Caputo sense: Caputo fractional derivative requires higher regularity of the unknown than R-L version. Another reason for the reconsideration is that we distinguish the differential order of the unknown with the index of Jacobi polynomials, which is not presented in the previous work. Also we provide a way to choose the index when facing multi-order problems. By using generalized Hardy's inequality, the gap between the weighted Sobolev space involving Caputo fractional derivative and the classical weighted space is bridged, then the optimal projection error is derived in the non-uniformly Jacobi-weighted Sobolev space and the maximum absolute error is presented as well. For the interpolation, analysis of interpolation error was not given in their work. In this paper we build the interpolation error in non-uniformly Jacobi-weighted Sobolev space by constructing fractional inverse inequality. With combining collocation method, the approximation technique is applied to solve fractional initial-value problems (FIVPs). Numerical examples are also provided to illustrate the effectiveness of this algorithm.
Time machines with the compactly determined Cauchy horizon
Krasnikov, S
2014-01-01
The building of a time machine, if possible at all, requires the relevant regions of spacetime to be compact (that is, physically speaking, free from sources of unpredictability such as infinities and singularities). Motivated by this argument we consider the spacetimes with the compactly determined Cauchy horizons (CDCHs), the defining property of which is the compactness of $\\overline{J^-(\\EuScript U)}\\cap J^+(\\EuScript S_0)$, where $\\EuScript U$ is an open subset of the Cauchy horizon and $\\EuScript S_0$ is a Cauchy surface of the initial globally hyperbolic region $\\ingh$. The following two facts are established: 1) $\\ingh$ has no globally hyperbolic maximal extension. This means that by shaping appropriately a precompact portion of a globally hyperbolic region one can \\emph{force} the Universe to produce either a closed causal curve, or a quasiregular singularity, whichever it abhors less; 2) Before a CDCH is formed a null geodesic appears which infinitely approaches the horizon returning again and again...
Cauchy-horizon singularity inside perturbed Kerr black holes
Burko, Lior M; Zenginoǧlu, Anıl
2016-01-01
The Cauchy horizon inside a perturbed Kerr black hole develops an instability that transforms it into a curvature singularity. We solve for the linearized Weyl scalars $\\psi_0$ and $\\psi_4$ and for the curvature scalar $R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}$ along outgoing null rays approaching the Cauchy horizon in the interior of perturbed Kerr black holes using the Teukolsky equation, and compare our results with those found in perturbation analysis. Our results corroborate the previous perturbation analysis result that at its early parts the Cauchy horizon evolves into a deformationally-weak, null, scalar-curvature singularity. We find excellent agreement for $\\psi_0(u={\\rm const},v)$, where $u,v$ are advanced and retarded times, respectively. We do find, however, that the exponential growth rate of $R_{\\alpha\\beta\\gamma\\delta}R^{\\alpha\\beta\\gamma\\delta}(u={\\rm const},v)$ approaching the singularity is dramatically slower than that found in perturbation analysis, and that the angular freq...
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Min Jia
2012-01-01
Full Text Available We study a model arising from porous media, electromagnetic, and signal processing of wireless communication system -tαx(t=f(t,x(t,x'(t,x”(t,…,x(n-2(t, 0
Sri Ranga, A.
1984-01-01
Just as the denominator polynomials of a J-fraction are orthogonal polynomials with respect to some moment functional, the denominator polynomials of an M-fraction are shown to satisfy a skew orthogonality relation with respect to a stronger moment functional. Many of the properties of the numerators and denominators of an M- fraction are also studied using this pseudo orthogonality relation of the denominator polynomials. Properties of the zeros of the denominator polyno...
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Ruihui Huang
2013-01-01
Full Text Available We study the existence and uniqueness of solutions for a class of antiperiodic boundary value problems of the fractional differential equation with a p-Laplacian operator. Based on the Leray-Schauder nonlinear alternative, several sufficient conditions of the existence and uniqueness of solution of the above problem are established. Our results improve and complement the recent work of Chen and Liu, 2012.
Luchko, Yuri; Mainardi, Francesco
2013-06-01
In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations behave quite differently regarding their response to a localized disturbance. Whereas the diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.
Deliyianni, Eleni; Gagatsis, Athanasios; Elia, Iliada; Panaoura, Areti
2016-01-01
The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1,701 primary…
Deliyianni, Eleni; Gagatsis, Athanasios; Elia, Iliada; Panaoura, Areti
2016-01-01
The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1,701 primary…
Luchko, Yuri; Povstenko, Yuriy
2012-01-01
In this paper, the one-dimensional time-fractional diffusion-wave equation with the fractional derivative of order $1 \\le \\alpha \\le 2$ is revisited. This equation interpolates between the diffusion and the wave equations that behave quite differently regarding their response to a localized disturbance: whereas the diffusion equation describes a process, where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. For the time fractional diffusion-wave equation, the propagation speed of a disturbance is infinite, but its fundamental solution possesses a maximum that disperses with a finite speed. In this paper, the fundamental solution of the Cauchy problem for the time-fractional diffusion-wave equation, its maximum location, maximum value, and other important characteristics are investigated in detail. To illustrate analytical formulas, results of numerical calculations and plots are presented. Numerical algorithms and programs used to produce pl...
Kim, Sun A.; Wang, Peishi; Michaels, Craig A.
2015-01-01
This article investigates the effects of fraction word problem-solving instruction involving explicit teaching of the concrete-representational-abstract sequence with culturally relevant teaching examples for 3 low-performing Asian immigrant English learners who spoke a language other than English at home. We used a multiple probe design across…
Directory of Open Access Journals (Sweden)
Obidjon Kh. Abdullaev
2016-06-01
Full Text Available In this work, we study the existence and uniqueness of solutions to non-local boundary value problems with integral gluing condition. Mixed type equations (parabolic-hyperbolic involving the Caputo fractional derivative have loaded parts in Riemann-Liouville integrals. Thus we use the method of integral energy to prove uniqueness, and the method of integral equations to prove existence.
Prestack AVA inversion of exact Zoeppritz equations based on modified Trivariate Cauchy distribution
Zhou, Lin; Li, Jingye; Chen, Xiaohong; Liu, Xingye; Chen, Li
2017-03-01
Obtaining interlayer weak reflection information that helps identify properties and accurate density information from complex and elusive reservoirs is particularly important for reservoir characterization and detection. However, conventional prestack amplitude variation with incidence angle inversion method is strongly influenced by the accuracy of the approximate Zoeppritz equations, which suppresses weak reflections coming from the commonly used prior distribution. In this paper, we address these problems by using exact Zoeppritz equations. First, the objective function of the inverse problem was constructed and the modified Cauchy distribution was introduced as the prior information by utilizing Bayes' theorem. In the complicated objective function, the forward operators are the sophisticated and nonlinear Zoeppritz equations with respect to estimate parameters. We then combined the idea of generalized linear inversion with iterative reweighed least-squares algorithm in order to solve the problem. Generalized linear inversion was used to solve the objective function, from which a nonlinear solution of the model parameters' perturbations can be calculated. The iterative reweighed least-squares algorithm was applied to solve the nonlinear expression in an attempt to obtain an updated iterative formula of the model parameters. Therefore the prestack amplitude variation with incidence angle inversion was able to be performed in order to better characterize a reservoir. Both synthetic and field data examples show that the new method can not only directly inverse P-wave velocity, S-wave velocity and density, but also provides accurate estimation results, particularly for density. The introduction of the modified Trivariate Cauchy prior constraints effectively estimated and inverted elastic parameters of weak reflections. Both examples demonstrated the feasibility and effectiveness of the proposed method.
On the generalized Cauchy function and new Conjecture on its exterior singularities
Institute of Scientific and Technical Information of China (English)
Theodore Yaotsu Wu
2011-01-01
This article studies on Cauchy's function f(z) and its integral, (2πi)J[f(z)] ≡ Φc f(t)dt/(t - z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D+ bounded by C and the open domain D- outside C. ( 1 ) With f(z) assumed to be Cn (n ＜ ∞-times continuously differentiable) (Λ)z ∈ D+ and in a neighborhood of C, f(z) and its derivatives f(n)(z) are proved uniformly continuous in the closed domain D+ = [D+ + C].(2) Cauchy's integral formulas and their derivatives (V)z ∈ D+(or (V)z ∈ D-) are proved to converge uniformly in D+ (or in D- = [D- + C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f(z) and J[f(z)]) are shown extended to hold for the complement function F(z), defined to be Cn(V)z ∈ D- and about C. (4) The uniform convergence theorems for f(z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four generalized Hilbert-type integral transforms are proved. (5) Further,the singularity distribution of f(z) in D- is elucidated by considering the direct problem exemplified with several typical singularities prescribed in D-. (6) A comparative study is made between generalized integral formulas and Plemelj's formulas on their differing basic properties. (7) Physical significances of these formulas are illustrated with applications to nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f(z) in domain D-, based on the continuous numerical value of f(z)(V)z ∈ D+ = [D+ + C], is presented for resolution as a conjecture.
Directory of Open Access Journals (Sweden)
Guotao Wang
2012-01-01
Full Text Available We study nonlinear impulsive differential equations of fractional order with irregular boundary conditions. Some existence and uniqueness results are obtained by applying standard fixed-point theorems. For illustration of the results, some examples are discussed.
Eigenvalue Problem for Nonlinear Fractional Differential Equations with Integral Boundary Conditions
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Guotao Wang
2014-01-01
Full Text Available By employing known Guo-Krasnoselskii fixed point theorem, we investigate the eigenvalue interval for the existence and nonexistence of at least one positive solution of nonlinear fractional differential equation with integral boundary conditions.
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Johnny Henderson
2016-01-01
Full Text Available We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with two parameters, subject to coupled integral boundary conditions.
Heart Failure with Preserved Ejection Fraction in the Elderly: Scope of the Problem
Upadhya, Bharathi; Taffet, George E.; Cheng, Che Ping; Kitzman, Dalane W.
2015-01-01
Heart failure with preserved ejection fraction (HFpEF) is the most common form of heart failure (HF) in older adults, particularly women, and is increasing in prevalence as the population ages. With morbidity and mortality on par with HF with reduced ejection fraction, it remains a most challenging clinical syndrome for the practicing clinician and basic research scientist. Originally considered to be predominantly caused by diastolic dysfunction, more recent insights indicate that HFpEF in o...
Cauchy-perturbative matching and outer boundary conditions computational studies
Rezzolla, L; Matzner, R A; Rupright, M E; Shapiro, S L; Rezzolla, Luciano; Abrahams, Andrew M; Matzner, Richard A.; Rupright, Mark E.; Shapiro, Stuart L.
1999-01-01
We present results from a new technique which allows extraction of gravitational radiation information from a generic three-dimensional numerical relativity code and provides stable outer boundary conditions. In our approach we match the solution of a Cauchy evolution of the nonlinear Einstein field equations to a set of one-dimensional linear equations obtained through perturbation techniques over a curved background. We discuss the validity of this approach in the case of linear and mildly nonlinear gravitational waves and show how a numerical module developed for this purpose is able to provide an accurate and numerically convergent description of the gravitational wave propagation and a stable numerical evolution.
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Imed Bachar
2014-01-01
Full Text Available We are interested in the following fractional boundary value problem: Dαu(t+atuσ=0, t∈(0,∞, limt→0t2-αu(t=0, limt→∞t1-αu(t=0, where 1<α<2, σ∈(-1,1, Dα is the standard Riemann-Liouville fractional derivative, and a is a nonnegative continuous function on (0,∞ satisfying some appropriate assumptions related to Karamata regular variation theory. Using the Schauder fixed point theorem, we prove the existence and the uniqueness of a positive solution. We also give a global behavior of such solution.
α-times Integrated Regularized Cosine Functions and Second Order Abstract Cauchy Problens
Institute of Scientific and Technical Information of China (English)
张寄洲; 陶有山
2001-01-01
In this paper, α -times integrated C-regularized cosine functions and mild α-times integrated C-existence families of second order are introduced. Equivalences are proved among α -times integrated C-regularized cosine function for a linear operator A, C-wellposed of (α + 1)-times abstract Cauchy problem and mild α -times integrated C-existence family of second order for A when the commutable condition is satisfied. In addition, if A = C-1AC, they are also equivalent to A generating the α -times integrated C-regularized cosine finction.The characterization of an exponentially botnded mild α -times integrated C-existence family of second order is given out in terms of a Laplace transform.
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Syed Tauseef Mohyud-Din
2015-01-01
Full Text Available This paper witnesses the coupling of an analytical series expansion method which is called reduced differential transform with fractional complex transform. The proposed technique is applied on three mathematical models, namely, fractional Kaup-Kupershmidt equation, generalized fractional Drinfeld-Sokolov equations, and system of coupled fractional Sine-Gordon equations subject to the appropriate initial conditions which arise frequently in mathematical physics. The derivatives are defined in Jumarie’s sense. The accuracy, efficiency, and convergence of the proposed technique are demonstrated through the numerical examples. It is observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation of fractional differential equations. The proposed technique is independent of complexities arising in the calculation of Lagrange multipliers, Adomian’s polynomials, linearization, discretization, perturbation, and unrealistic assumptions and hence gives the solution in the form of convergent power series with elegantly computed components. All the examples show that the proposed combination is a powerful mathematical tool to solve other nonlinear equations also.
A Priori Estimates for Solutions of Boundary Value Problems for Fractional-Order Equations
Alikhanov, A A
2011-01-01
We consider boundary value problems of the first and third kind for the diffusionwave equation. By using the method of energy inequalities, we find a priori estimates for the solutions of these boundary value problems.
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Soheil Salahshour
2015-02-01
Full Text Available In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville or a solution with increasing length of their support (Hukuhara difference. Then, in order to solve the FFDE analytically, we introduce the fuzzy Laplace transform of the Caputo H-derivative. To the best of our knowledge, there is limited research devoted to the analytical methods to solve the FFDE under the fuzzy Caputo fractional differentiability. An analytical solution is presented to confirm the capability of the proposed method.
Cauchy pdf modelling and its application to SAR image despeckling
Institute of Scientific and Technical Information of China (English)
Chen Guozhong; Liu Xingzhao
2008-01-01
Synthetic aperture radar(SAR)imagery is a kind of coherent system that produces a random pattern,named speckle,which degrades the merit of SAR images and affects their further application seriously.Therefore,how to restore SAR image from the speckle has become a necessary step in post-processing of image.A new despeckling method is putforth on the basis of wavelet.First.a new approach on the basis of"second kind statistics"is used to estimate the dispersion parameter of the Cauchy distribution.Then,this Cauchy prior is applied to model the distribution of the wavelet coefficients for the log-transformed reflectance of SAR image.Based on the above ideas,a new homomorphic wavelet-based maximum a posterior(MAP)despeckling method is proposed.Finally,the simulated speckled image and the real SAR image are used to verify our proposed method and the results show that it outperforms the other methods in terms of the speckle reduction and the feature retention.
Cauchy-perturbative matching revisited: tests in spherical symmetry
Zink, B L; Pazos, E; Tiglio, M; Diener, Peter; Pazos, Enrique; Tiglio, Manuel; Zink, Burkhard
2006-01-01
During the last few years progress has been made on several fronts making it possible to revisit Cauchy-perturbative matching (CPM) in numerical relativity in a more robust and accurate way. This paper is the first in a series where we plan to analyze CPM in the light of these new results. Here we start by testing high-order summation-by-parts operators, penalty boundaries and contraint-preserving boundary conditions applied to CPM in a setting that is simple enough to study all the ingredients in great detail: Einstein's equations in spherical symmetry, describing a black hole coupled to a massless scalar field. We show that with the techniques described above, the errors introduced by Cauchy-perturbative matching are very small, and that very long term and accurate CPM evolutions can be achieved. Our tests include the accretion and ring-down phase of a Schwarzschild black hole with CPM, where we find that the discrete evolution introduces, with a low spatial resolution of \\Delta r = M/10, an error of 0.3% a...
MATCHING PURSUITS AMONG SHIFTED CAUCHY KERNELS IN HIGHER-DIMENSIONAL SPACES
Institute of Scientific and Technical Information of China (English)
钱涛; 王晋勋; 杨燕
2014-01-01
Appealing to the Clifford analysis and matching pursuits, we study the adaptive decompositions of functions of several variables of finite energy under the dictionaries con-sisting of shifted Cauchy kernels. This is a realization of matching pursuits among shifted Cauchy kernels in higher-dimensional spaces. It offers a method to process signals in arbitrary dimensions.
Hyperelastic bodies under homogeneous Cauchy stress induced by non-homogeneous finite deformations
Mihai, L Angela
2016-01-01
We discuss whether homogeneous Cauchy stress implies homogeneous strain in isotropic nonlinear elasticity. While for linear elasticity the positive answer is clear, we exhibit, through detailed calculations, an example with inhomogeneous continuous deformation but constant Cauchy stress. The example is derived from a non rank-one convex elastic energy. Connections to conforming and non-conforming finite element implementations are drawn.
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Chatnugrob Sangsawang
2016-06-01
Full Text Available This paper addresses a problem of the two-stage flexible flow shop with reentrant and blocking constraints in Hard Disk Drive Manufacturing. This problem can be formulated as a deterministic FFS|stage=2,rcrc, block|Cmax problem. In this study, adaptive Hybrid Particle Swarm Optimization with Cauchy distribution (HPSO was developed to solve the problem. The objective of this research is to find the sequences in order to minimize the makespan. To show their performances, computational experiments were performed on a number of test problems and the results are reported. Experimental results show that the proposed algorithms give better solutions than the classical Particle Swarm Optimization (PSO for all test problems. Additionally, the relative improvement (RI of the makespan solutions obtained by the proposed algorithms with respect to those of the current practice is performed in order to measure the quality of the makespan solutions generated by the proposed algorithms. The RI results show that the HPSO algorithm can improve the makespan solution by averages of 14.78%.
Newton, Kristie J.; Willard, Catherine; Teufel, Christopher
2014-01-01
The purpose of this study was to better understand how students with learning disabilities, including those who struggle specifically with mathematics, engage with fraction computation. In particular, we examined error patterns, the influence of like and unlike denominators on these patterns, and correct solution methods. Although skill-related…
1989-09-30
The Theory of Linear Operators, The Principia Press (1936). 13. W. Guttinger, "Generalized Functions and Dispersion Relations in Physics...Monagine," Acta Mathematica , Vol 29 (1905), pg 101. 17, K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, Orlando, (1974). 18. Y. N
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Fenghua Miao
2013-07-01
Full Text Available In this article, we study the fractional q-difference boundary-value problems with p-Laplacian operator $$displaylines{ D_{q}^{gamma}(phi_p(D_{q}^{alpha}u(t + f(t,u(t=0, quad 0 1. By using a fixed-point theorem in partially ordered sets, we obtain sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions.
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Gülcan Özkum
2013-01-01
Full Text Available The study in this paper mainly concerns the inverse problem of determining an unknown source function in the linear fractional differential equation with variable coefficient using Adomian decomposition method (ADM. We apply ADM to determine the continuous right hand side functions fx and ft in the heat-like diffusion equations Dtαux,t=hxuxxx,t+fx and Dtαux,t=hxuxxx,t+ft, respectively. The results reveal that ADM is very effective and simple for the inverse problem of determining the source function.
Some aspects of fractional diffusion equations of single and distributed order
Mainardi, Francesco; Gorenflo, Rudolf
2007-01-01
The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $\\beta \\in (0,1)$. The fundamental solution for the Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process related to a phenomenon of sub-diffusion (the variance grows in time sub-linearly). A further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one. Then the fundamental solution is still a probability density of a non-Markovian process that, however, is no longer self-similar but exhibits a corresponding distribution of time-scales.
Solvability of fractional multi-point boundary-value problems with p-Laplacian operator at resonance
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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.
Signaling, Entanglement, and Quantum Evolution Beyond Cauchy Horizons
Yurtsever, U; Yurtsever, Ulvi; Hockney, George
2005-01-01
Consider a bipartite entangled system half of which falls through the event horizon of an evaporating black hole, while the other half remains coherently accessible to experiments in the exterior region. Beyond complete evaporation, the evolution of the quantum state past the Cauchy horizon cannot remain unitary, raising the questions: How can this evolution be described as a quantum map, and how is causality preserved? What are the possible effects of such nonstandard quantum evolution maps on the behavior of the entangled laboratory partner? More generally, the laws of quantum evolution under extreme conditions in remote regions (not just in evaporating black-hole interiors, but possibly near other naked singularities and regions of extreme spacetime structure) remain untested by observation, and might conceivably be non-unitary or even nonlinear, raising the same questions about the evolution of entangled states. The answers to these questions are subtle, and are linked in unexpected ways to the fundamenta...
Differential geometric formulation of the Cauchy Navier equations
Schadt, Frank
2011-01-01
The paper presents a reformulation of some of the most basic entities and equations of linear elasticity - the stress and strain tensor, the Cauchy Navier equilibrium equations, material equations for linear isotropic bodies - in a modern differential geometric language using differential forms and lie derivatives. Similar steps have been done successfully in general relativity, quantum physics and electrodynamics and are of great use in those fields. In Elasticity Theory, however, such a modern differential geometric approach is much less common. Furthermore, existing reformulations demand a vast knowledge of differential geometry, including nonstandard entities such as vector valued differential forms and the like. This paper presents a less general but more easily accessible approach to using modern differential geometry in elasticity theory than those published up to now.
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Mansour Saraj
2012-06-01
Full Text Available In this paper we propose a fuzzy goal programming method for obtaining a satisfactory solution to a bi-level multi-objective absolute-value fractional programming (BLMO-A-FP problems. In the proposed approach, the membership functions for the de ned fuzzy goals of all objective functions at the two levels as well as the membership functions for vector of fuzzy goals of the decision variables controlled by upper level decision maker (ULDM are developed in the model formulation of the problem. Then fuzzy goal programming technique is used for achieving highest degree of each of the membership goals by minimizing negative and positive deviational variables. The method of variable change on the under- and over-deviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem eciently by using linear goal programming methodology. Theoretical results is illustrated with the help of a numerical.
Existence and comparison of smallest eigenvalues for a fractional boundary-value problem
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Paul W. Eloe
2014-02-01
Full Text Available The theory of $u_0$-positive operators with respect to a cone in a Banach space is applied to the fractional linear differential equations $$ D_{0+}^{\\alpha} u+\\lambda_1p(tu=0\\quad\\text{and}\\quad D_{0+}^{\\alpha} u+\\lambda_2q(tu=0, $$ $0< t< 1$, with each satisfying the boundary conditions $u(0=u(1=0$. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained.
Foundation of fractional Langevin equation: harmonization of a many-body problem.
Lizana, Ludvig; Ambjörnsson, Tobias; Taloni, Alessandro; Barkai, Eli; Lomholt, Michael A
2010-05-01
In this study we derive a single-particle equation of motion, from first principles, starting out with a microscopic description of a tracer particle in a one-dimensional many-particle system with a general two-body interaction potential. Using a harmonization technique, we show that the resulting dynamical equation belongs to the class of fractional Langevin equations, a stochastic framework which has been proposed in a large body of works as a means of describing anomalous dynamics. Our work sheds light on the fundamental assumptions of these phenomenological models and a relation derived by Kollmann.
Institute of Scientific and Technical Information of China (English)
姚磊华
1997-01-01
A generalized upwind scheme with fractional steps for 3-D mathematical models of convection dominating groundwater quality is suggested. The mass transport equation is split into a convection equation and a dispersive equation. The generalized upwind scheme is used to solve the convection equation and the finite element method is used to compute the dispersive equation. These procedures which not only overcome the phenomenon of the negative concentration and numerical dispersion appear frequently with normal FEM or FDM to solve models of convection dominating groundwater transport but also avoid the step for computing each node velocity give a more suitable method to calculate the concentrations of the well points.
Heart failure with preserved ejection fraction in the elderly: scope of the problem.
Upadhya, Bharathi; Taffet, George E; Cheng, Che Ping; Kitzman, Dalane W
2015-06-01
Heart failure with preserved ejection fraction (HFpEF) is the most common form of heart failure (HF) in older adults, particularly women, and is increasing in prevalence as the population ages. With morbidity and mortality on par with HF with reduced ejection fraction, it remains a most challenging clinical syndrome for the practicing clinician and basic research scientist. Originally considered to be predominantly caused by diastolic dysfunction, more recent insights indicate that HFpEF in older persons is typified by a broad range of cardiac and non-cardiac abnormalities and reduced reserve capacity in multiple organ systems. The globally reduced reserve capacity is driven by: 1) inherent age-related changes; 2) multiple, concomitant co-morbidities; 3) HFpEF itself, which is likely a systemic disorder. These insights help explain why: 1) co-morbidities are among the strongest predictors of outcomes; 2) approximately 50% of clinical events in HFpEF patients are non-cardiovascular; 3) clinical drug trials in HFpEF have been negative on their primary outcomes. Embracing HFpEF as a true geriatric syndrome, with complex, multi-factorial pathophysiology and clinical heterogeneity could provide new mechanistic insights and opportunities for progress in management. This article is part of a Special Issue entitled CV Aging. Copyright © 2015 Elsevier Ltd. All rights reserved.
Deswal, Sunita; Kalkal, Kapil Kumar; Sheoran, Sandeep Singh
2016-09-01
A mathematical model of fractional order two-temperature generalized thermoelasticity with diffusion and initial stress is proposed to analyze the transient wave phenomenon in an infinite thermoelastic half-space. The governing equations are derived in cylindrical coordinates for a two dimensional axi-symmetric problem. The analytical solution is procured by employing the Laplace and Hankel transforms for time and space variables respectively. The solutions are investigated in detail for a time dependent heat source. By using numerical inversion method of integral transforms, we obtain the solutions for displacement, stress, temperature and diffusion fields in physical domain. Computations are carried out for copper material and displayed graphically. The effect of fractional order parameter, two-temperature parameter, diffusion, initial stress and time on the different thermoelastic and diffusion fields is analyzed on the basis of analytical and numerical results. Some special cases have also been deduced from the present investigation.
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I. J. Cabrera
2012-01-01
Full Text Available We are concerned with the existence and uniqueness of a positive and nondecreasing solution for the following nonlinear fractional m-point boundary value problem: D0+αu(t+f(t,u(t=0, 0
Kraisler, Eli
2015-01-01
Many approximations within density-functional theory spuriously predict that a many-electron system can dissociate into fractionally charged fragments. Here, we revisit the case of dissociated diatomic molecules, known to exhibit this problem when studied within standard approaches, including the local spin-density approximation (LSDA). By employing our recently proposed [E. Kraisler and L. Kronik, Phys. Rev. Lett. 110, 126403 (2013)] ensemble-generalization we find that asymptotic fractional dissociation is eliminated in all systems examined, even if the underlying exchange-correlation (xc) is still the LSDA. Furthermore, as a result of the ensemble generalization procedure, the Kohn-Sham potential develops a spatial step between the dissociated atoms, reflecting the emergence of the derivative discontinuity in the xc energy functional. This step, predicted in the past for the exact Kohn-Sham potential and observed in some of its more advanced approximate forms, is a desired feature that prevents any fractio...
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.
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.
Energy Technology Data Exchange (ETDEWEB)
Zhuang, Qiao, E-mail: zhqsdu@163.com; Yu, Bo; Jiang, Xiaoyun, E-mail: wqjxyf@sdu.edu.cn
2015-01-01
In this paper, a time-fractional heat conduction problem is mathematically proposed for an experimental heat conduction process in a 3-layer composite medium. A numerical solution to the direct problem is obtained with finite difference method. In regard to the inverse problem, the optimal order of Caputo fractional derivative is estimated with Levenberg–Marquardt method. Comparing with the carbon–carbon experimental data, the results show that the time-fractional heat conduction model provides an effective and accurate simulation of the experimental data. The rationality of the proposed time-fractional model and validity of Levenberg–Marquardt method in solving the time-fractional inverse heat conduction problem are also manifested according to the results. By conducting the sensitivity analysis, the feasibility of the parameter estimation is further discussed.
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A. H. Bhrawy
2013-01-01
Full Text Available We extend the application of the Galerkin method for treating the multiterm fractional differential equations (FDEs subject to initial conditions. A new shifted Legendre-Galerkin basis is constructed which satisfies exactly the homogeneous initial conditions by expanding the unknown variable using a new polynomial basis of functions which is built upon the shifted Legendre polynomials. A new spectral collocation approximation based on the Gauss-Lobatto quadrature nodes of shifted Legendre polynomials is investigated for solving the nonlinear multiterm FDEs. The main advantage of this approximation is that the solution is expanding by a truncated series of Legendre-Galerkin basis functions. Illustrative examples are presented to ensure the high accuracy and effectiveness of the proposed algorithms are discussed.
Institute of Scientific and Technical Information of China (English)
Chang-feng Ma
2004-01-01
This paper provides an convergence analysis of a fractional-step projection method for the controlled-source electromagnetic induction problems in heterogenous electrically conducting media by means of finite element approximations. Error estimates in finite time are given. And it is verified that provided the time step τ is sufficiently small, the proposed algorithm yields for finite time T an error of (O)(hs + τ) in the L2-norm for the magnetic field H, where h is the mesh size and 1/2 ＜ s ≤ 1.
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Tatar Nasser-eddine
2011-01-01
Full Text Available Abstract A second-order abstract problem of neutral type with derivatives of non-integer order in the nonlinearity as well as in the nonlocal conditions is investigated. This model covers many of the existing models in the literature. It extends the integer order case to the fractional case in the sense of Caputo. A fixed point theorem is used to prove existence of mild solutions. AMS Subject Classification 26A33, 34K40, 35L90, 35L70, 35L15, 35L07
Exercise intolerance in heart failure with preserved ejection fraction:more than a heart problem
Institute of Scientific and Technical Information of China (English)
Bharathi Upadhya; Mark J Haykowsky; Joel Eggebeen; Dalane W Kitzman
2015-01-01
Heart failure (HF) with preserved ejection fraction (HFpEF) is the most common form of HF in older adults, and is increasing in preva-lence as the population ages. Furthermore, HFpEF is increasing out of proportion to HF with reduced EF (HFrEF), and its prognosis is worsening while that of HFrEF is improving. Despite the importance of HFpEF, our understanding of its pathophysiology is incomplete, and optimal treatment remains largely undefined. A cardinal feature of HFpEF is reduced exercise tolerance, which correlates with symptoms as well as reduced quality of life. The traditional concepts of exercise limitations have focused on central dysfunction related to poor cardiac pump function. However, the mechanisms are not exclusive to the heart and lungs, and the understanding of the pathophysiology of this dis-ease has evolved. Substantial attention has focused on defining the central versus peripheral mechanisms underlying the reduced functional capacity and exercise tolerance among patients with HF. In fact, physical training can improve exercise tolerance via peripheral adaptive mechanisms even in the absence of favorable central hemodynamic function. In addition, the drug trials performed to date in HFpEF that have focused on influencing cardiovascular function have not improved exercise capacity. This suggests that peripheral limitations may play a significant role in HF limiting exercise tolerance, a hallmark feature of HFpEF.
Discrete random walk models for space-time fractional diffusion
Energy Technology Data Exchange (ETDEWEB)
Gorenflo, Rudolf; Mainardi, Francesco; Moretti, Daniele; Pagnini, Gianni; Paradisi, Paolo
2002-11-01
A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By space-time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order {alpha} is part of (0,2] and skewness {theta} (module{theta}{<=}{l_brace}{alpha},2-{alpha}{r_brace}), and the first-order time derivative with a Caputo derivative of order {beta} is part of (0,1]. Such evolution equation implies for the flux a fractional Fick's law which accounts for spatial and temporal non-locality. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.
On the origins of generalized fractional calculus
Kiryakova, Virginia
2015-11-01
In Fractional Calculus (FC), as in the (classical) Calculus, the notions of derivatives and integrals (of first, second, etc. or arbitrary, incl. non-integer order) are basic and co-related. One of the most frequent approach in FC is to define first the Riemann-Liouville (R-L) integral of fractional order, and then by means of suitable integer-order differentiation operation applied over it (or under its sign) a fractional derivative is defined - in the R-L sense (or in Caputo sense). The first mentioned (R-L type) is closer to the theoretical studies in analysis, but has some shortages - from the point of view of interpretation of the initial conditions for Cauchy problems for fractional differential equations (stated also by means of fractional order derivatives/ integrals), and also for the analysts' confusion that such a derivative of a constant is not zero in general. The Caputo (C-) derivative, arising first in geophysical studies, helps to overcome these problems and to describe models of applied problems with physically consistent initial conditions. The operators of the Generalized Fractional Calculus - GFC (integrals and derivatives) are based on commuting m-tuple (m = 1, 2, 3, …) compositions of operators of the classical FC with power weights (the so-called Erdélyi-Kober operators), but represented in compact and explicit form by means of integral, integro-differential (R-L type) or differential-integral (C-type) operators, where the kernels are special functions of most general hypergeometric kind. The foundations of this theory are given in Kiryakova 18. In this survey we present the genesis of the definitions of the GFC - the generalized fractional integrals and derivatives (of fractional multi-order) of R-L type and Caputo type, analyze their properties and applications. Their special cases are all the known operators of classical FC, their generalizations introduced by other authors, the hyper-Bessel differential operators of higher integer
Spectral Cauchy Characteristic Extraction of strain, news and gravitational radiation flux
Handmer, Casey J; Winicour, Jeffrey
2016-01-01
We present a new approach for the Cauchy-characteristic extraction of gravitational radiation strain, news function, and the flux of the energy-momentum, supermomentum and angular momentum associated with the Bondi-Metzner-Sachs asymptotic symmetries. In Cauchy-characteristic extraction, a characteristic evolution code propagates numerical data on an inner worldtube supplied by a Cauchy evolution code to obtain the space-time metric in a neighborhood of null infinity. The metric is first determined in a scrambled form in terms of coordinates determined by the Cauchy formalism. In prior treatments, the waveform is first extracted from this metric and then transformed into an asymptotic inertial coordinate system. This procedure provides the physically proper description of the waveform and the radiated energy but it does not generalize to determine the flux of angular momentum or supermomentum. Here we formulate and implement a new approach which transforms the full metric into an asymptotic inertial frame and...
Analysis of linear waves near the Cauchy horizon of cosmological black holes
Hintz, Peter
2015-01-01
We show that linear scalar waves are bounded and continuous up to the Cauchy horizon of Reissner-Nordstr\\"om-de Sitter and Kerr-de Sitter spacetimes, and in fact decay exponentially fast to a constant along the Cauchy horizon. We obtain our results by modifying the spacetime beyond the Cauchy horizon in a suitable manner, which puts the wave equation into a framework in which a number of standard as well as more recent microlocal regularity and scattering theory results apply. In particular, the conormal regularity of waves at the Cauchy horizon - which yields the boundedness statement - is a consequence of radial point estimates, which are microlocal manifestations of the blue-shift and red-shift effects.
Energy Technology Data Exchange (ETDEWEB)
Philipp, M; Vergnat, C; Mueller, U; Sanctuary, R; Baller, J; Krueger, J K [Laboratoire de Physique des Materiaux, Universite du Luxembourg, 162A, avenue de la Faiencerie, L-1511 Luxembourg (Luxembourg); Possart, W [Fachbereich Werkstoffwissenschaften, Universitaet des Saarlandes, D-66123 Saarbruecken (Germany); Alnot, P [LPMI, Universite Nancy (France)], E-mail: martine.philipp@uni.lu
2009-01-21
The non-equilibrium process of polymerization of reactive polymers can be accompanied by transition phenomena like gelation or the chemical glass transition. The sensitivity of the mechanical properties at hypersonic frequencies-including the generalized Cauchy relation-to these transition phenomena is studied for three different polyurethanes using Brillouin spectroscopy. As for epoxies, the generalized Cauchy relation surprisingly holds true for the non-equilibrium polymerization process and for the temperature dependence of polyurethanes. Neither the sol-gel transition nor the chemical and thermal glass transitions are visible in the representation of the generalized Cauchy relation. Taking into account the new results and combining them with general considerations about the elastic properties of the isotropic state, an improved physical foundation of the generalized Cauchy relation is proposed.
Philipp, M; Vergnat, C; Müller, U; Sanctuary, R; Baller, J; Possart, W; Alnot, P; Krüger, J K
2009-01-21
The non-equilibrium process of polymerization of reactive polymers can be accompanied by transition phenomena like gelation or the chemical glass transition. The sensitivity of the mechanical properties at hypersonic frequencies-including the generalized Cauchy relation-to these transition phenomena is studied for three different polyurethanes using Brillouin spectroscopy. As for epoxies, the generalized Cauchy relation surprisingly holds true for the non-equilibrium polymerization process and for the temperature dependence of polyurethanes. Neither the sol-gel transition nor the chemical and thermal glass transitions are visible in the representation of the generalized Cauchy relation. Taking into account the new results and combining them with general considerations about the elastic properties of the isotropic state, an improved physical foundation of the generalized Cauchy relation is proposed.
Ortiz, Néstor
2013-01-01
A spherical dust cloud which is initially at rest and which has a monotonously decaying density profile collapses and forms a shell-focussing singularity. Provided the density profile is not too flat, meaning that its second radial derivative is negative at the center, this singularity is visible to local, and sometimes even to global observers. According to the strong cosmic censorship conjecture, such naked singularities should be unstable under generic, nonspherical perturbations of the initial data or when more realistic matter models are considered. In an attempt to gain some understanding about this stability issue, in this work we initiate the analysis of a simpler but related problem. We discuss the stability of test fields propagating in the vicinity of the Cauchy horizon associated to the naked central singularity. We first study the high-frequency limit and show that the fields undergo a blueshift as they approach the Cauchy horizon. However, in contrast to what occurs at inner horizons of black ho...
Tapiero, Charles S.; Vallois, Pierre
2016-11-01
The premise of this paper is that a fractional probability distribution is based on fractional operators and the fractional (Hurst) index used that alters the classical setting of random variables. For example, a random variable defined by its density function might not have a fractional density function defined in its conventional sense. Practically, it implies that a distribution's granularity defined by a fractional kernel may have properties that differ due to the fractional index used and the fractional calculus applied to define it. The purpose of this paper is to consider an application of fractional calculus to define the fractional density function of a random variable. In addition, we provide and prove a number of results, defining the functional forms of these distributions as well as their existence. In particular, we define fractional probability distributions for increasing and decreasing functions that are right continuous. Examples are used to motivate the usefulness of a statistical approach to fractional calculus and its application to economic and financial problems. In conclusion, this paper is a preliminary attempt to construct statistical fractional models. Due to the breadth and the extent of such problems, this paper may be considered as an initial attempt to do so.
Abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm
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Wang Rong-Nian
2011-01-01
Full Text Available Abstract In the present paper, we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm, where the operator A in the linear part is the generator of a compact analytic semigroup. New criterions, ensuring the existence of mild solutions, are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by A, Krasnoselkii's fixed point theorem and Schauder's fixed point theorem. An application to a fractional partial integro-differential equation with nonlocal initial condition is also considered. Mathematics subject classification (2000 26A33, 34G10, 34G20
Directory of Open Access Journals (Sweden)
Zhenlai Han
2012-11-01
Full Text Available In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with $p$-Laplacian operator $$displaylines{ D_{0+}^eta(phi_p(D_{0+}^alpha u(t+a(tf(u=0, quad 0
Bachher, M.; Sarkar, N.
2016-11-01
An electromagneto-thermoelastic coupled problem for a homogeneous, isotropic, thermally and electrically conducting half-space solid whose surface is subjected to a thermal shock is considered in two-dimensional space. The equations of the theory of generalized electromagneto-thermoelasticity with fractional derivative heat transfer allowing the second sound effects are considered. An initial magnetic field acts parallel to the plane boundary of the half-space. The normal mode analysis and the eigenvalue approach techniques are used to solve the resulting nondimensional coupled field equations for the three theories. Numerical results for the temperature, displacements and thermal stresses distributions are presented graphically and discussed. A comparison is made with the results obtained in the presence and absence of the magnetic field.
Institute of Scientific and Technical Information of China (English)
Xiaohong HAO; Zongfu ZHOU
2013-01-01
In this paper,we consider the following multi-point boundary value problem of fractional differential equation Dα0+u(t) =f(t,u(t),Dα-10+ u(t),Dα+20u(t),Dα-30u(t)),t ∈ (0,1),I4-α0+u(0) =0,Dα-10+u(0) =nΣi=1αi Dα-10+u(ξi),Dα-20+u(1) =nΣj=1 βjDα-20+ u(ηj),Dα-30+u(1)-Dα-30+u(0) =Dα-20+2u(1/2),where 3 ＜ α ≤ 4 is a real number.By applying Mawhin coincidence degree theory and constructing suitable operators,some existence results of solutions can be established.
Chaotic and Chaos-Like Behavior in Continued Fractions
Shuji, OBATA; Shigeru, OHKURO; Toshiaki, MAEDA; Physics Laboratory, Faculty of Science and Engineering, Tokyo Denki University; Laboratory of Information aud System Engineering, Hachinohe Institute of Technology; DEPARTMENT OF MATHEMATICAL SCIENCES, TOKYO DENKI UNIVERSITY
1999-01-01
Chaotic and chaos-like behavior in continued fractions is studied with respect to several types of maps, including a logistic map. Various numerical phenomena in the continued fractions are investigated, where the fractions correspond to fractal structures. Cyclic terms in the Cauchy distribution areas are introduced, including the chaos-like behavior. It is indicated that such mixed states of distributions and cycles are common in the chaotic and chaos-like behavior.
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 ...
Fundamental solutions to time-fractional heat conduction equations in two joint half-lines
Povstenko, Yuriy
2013-10-01
Heat conduction in two joint half-lines is considered under the condition of perfect contact, i.e. when the temperatures at the contact point and the heat fluxes through the contact point are the same for both regions. The heat conduction in one half-line is described by the equation with the Caputo time-fractional derivative of order α, whereas heat conduction in another half-line is described by the equation with the time derivative of order β. The fundamental solutions to the first and second Cauchy problems as well as to the source problem are obtained using the Laplace transform with respect to time and the cos-Fourier transform with respect to the spatial coordinate. The fundamental solutions are expressed in terms of the Mittag-Leffler function and the Mainardi function.
Who Gave you the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus
Borovik, Alexandre; 10.1007/s10699-011-9235-x
2011-01-01
Cauchy's contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Bjorling, an infinitesimal definition of the criterion of uniform convergence. Cauchy's foundational stance is hereby reconsidered.
NSCT Domain Additive Watermark Detection Using RAO Hypothesis Test and Cauchy Distribution
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Hongbo Bi
2016-01-01
Full Text Available We presented a RAO hypothesis detector by modeling Cauchy distribution for the Nonsubsampled Contourlet Transform (NSCT subband coefficients in the field of additive spread spectrum image watermarking. Firstly, the NSCT subband coefficients were modeled following the Cauchy distributions, and the Fit of Goodness shows that Cauchy distribution fits the NSCT subband coefficients more accurately than the Generalized Gaussian Distribution (GGD commonly used. Secondly, a blind RAO test watermark detector was derived in the NSCT domain, which does not need the knowledge of embedding strength at the receiving end. Finally, compared to the other three state-of-art detectors, the robustness of the proposed watermarking scheme was evaluated when the watermarked images were attacked by JPEG compression, random noise, low pass filtering, and median filtering. Experimental results show that, compared with the other three detectors, the proposed RAO detector guarantees the lower probability of miss under the given probability of false alarm.
Institute of Scientific and Technical Information of China (English)
ZHAO Lan-hao; LI Tong-chun; WANG Ling; HERREROS M. I.; PASTOR M.
2006-01-01
A two-step Taylor-Galerkin fractional-step finite element method, which is of second order accuracy in space and time, was proposed for the three-dimensional free surface problem. With this method, the intermediate velocity was explicitly obtained by neglecting pressure gradient term, and then the velocity was corrected by adding the effects of pressure once the pressure field had been obtained from the pressure Poisson equation. The level set approach was applied to track implicitly the free surface. In order to track the free surface, the transport equation of the level set function was solved at each time step and the level set function is reinitialized through iteration to maintain it as a distance function. The governing equations of the system were discretized by the two- step Taylor-Galerkin method, which is of high-order accuracy and easy to be used. The validity and reliability of this method in this article were proved by two numerical examples.
Institute of Scientific and Technical Information of China (English)
PING Yun-Xia; LIU Lie; ZHANG Chao-Min; CHENG Ze
2011-01-01
Violations of Bell inequality, Cauchy-Schwarz inequality and entanglement in a two-mode three-level atomic system are investigated. It is shown that there are some states, which are entangled but do not violate Bell inequality in this system. Moreover, the relations of violations of Bell inequality, Cauchy-Schwarz inequality, and entanglement are discussed in detail.
Zhang, Huifeng; Lei, Xiaohui; Wang, Chao; Yue, Dong; Xie, Xiangpeng
2017-01-01
Since wind power is integrated into the thermal power operation system, dynamic economic emission dispatch (DEED) has become a new challenge due to its uncertain characteristics. This paper proposes an adaptive grid based multi-objective Cauchy differential evolution (AGB-MOCDE) for solving stochastic DEED with wind power uncertainty. To properly deal with wind power uncertainty, some scenarios are generated to simulate those possible situations by dividing the uncertainty domain into different intervals, the probability of each interval can be calculated using the cumulative distribution function, and a stochastic DEED model can be formulated under different scenarios. For enhancing the optimization efficiency, Cauchy mutation operation is utilized to improve differential evolution by adjusting the population diversity during the population evolution process, and an adaptive grid is constructed for retaining diversity distribution of Pareto front. With consideration of large number of generated scenarios, the reduction mechanism is carried out to decrease the scenarios number with covariance relationships, which can greatly decrease the computational complexity. Moreover, the constraint-handling technique is also utilized to deal with the system load balance while considering transmission loss among thermal units and wind farms, all the constraint limits can be satisfied under the permitted accuracy. After the proposed method is simulated on three test systems, the obtained results reveal that in comparison with other alternatives, the proposed AGB-MOCDE can optimize the DEED problem while handling all constraint limits, and the optimal scheme of stochastic DEED can decrease the conservation of interval optimization, which can provide a more valuable optimal scheme for real-world applications.
Caixia Guo; Jianmin Guo; Ying Gao; Shugui Kang
2016-01-01
This paper is concerned with the two-point boundary value problems of nonlinear finite discrete fractional differential equations. On one hand, we discuss some new properties of the Green function. On the other hand, by using the main properties of Green function and the Krasnoselskii fixed point theorem on cones, some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established.
Fixed Points of Non-expansive Operators on Weakly Cauchy Normed Spaces
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Sahar M. Ali
2007-01-01
Full Text Available We proved the existence of fixed points of non-expansive operators defined on weakly Cauchy spaces in which parallelogram law holds, the given normed space is not necessarily be uniformly convex Banach space or Hilbert space, we reduced the completeness and the uniform convexity assumptions which imposed on the given normed space.
Elcoro, Luis; Etxebarria, Jesus
2011-01-01
The requirement of rotational invariance for lattice potential energies is investigated. Starting from this condition, it is shown that the Cauchy relations for the elastic constants are fulfilled if the lattice potential is built from pair interactions or when the first-neighbour approximation is adopted. This is seldom recognized in widely used…
Institute of Scientific and Technical Information of China (English)
辛杰
2008-01-01
@@ It is well known that Feynman and Hibbs[1] used path integrals over Brownian paths to derive the standard(nonfractional) Schrodinger equation. Recently, Laskin[5, 6] showed that the path integral over the Lévy-like quantum mechanical paths allows to develop the generalization of the quantum mechanics. Namely, if the path integral over Brownian trajectories leads to the well known Schrodinger equation, then the path integral over Lévy trajectories leads to the fractional Schrodinger equation. Laskin[7] showed the Hermiticity of the fractional Hamilton operator and established the parity conservation law. Xiaoyi Guo and Mingyu Xu[4] studied some physical applications of the fractional Schrodinger equation.
Institute of Scientific and Technical Information of China (English)
TIAN Dongyan; JIN Ming; DUI Guansuo
2006-01-01
A new approach for the derivation of the principal invariants of the stretch tensor with respect to the right Cauchy Green tensor is presented in this paper. According to the definition of the derivation of tensor function, the three first-order derivatives for the principal invariants of the stretch tensor are obtained through derivation directly to the right Cauchy-Green tensor by incremental method. Then the three second-order derivatives are yielded by the derivation to the right Cauchy-Green strain tensor directly. Furthermore, an explicit expression of the tangent modulus of the general Varga material is given as an example.
Energy Technology Data Exchange (ETDEWEB)
Zhou, Ping; Lv, Youbin; Wang, Hong; Chai, Tianyou
2017-09-01
Optimal operation of a practical blast furnace (BF) ironmaking process depends largely on a good measurement of molten iron quality (MIQ) indices. However, measuring the MIQ online is not feasible using the available techniques. In this paper, a novel data-driven robust modeling is proposed for online estimation of MIQ using improved random vector functional-link networks (RVFLNs). Since the output weights of traditional RVFLNs are obtained by the least squares approach, a robustness problem may occur when the training dataset is contaminated with outliers. This affects the modeling accuracy of RVFLNs. To solve this problem, a Cauchy distribution weighted M-estimation based robust RFVLNs is proposed. Since the weights of different outlier data are properly determined by the Cauchy distribution, their corresponding contribution on modeling can be properly distinguished. Thus robust and better modeling results can be achieved. Moreover, given that the BF is a complex nonlinear system with numerous coupling variables, the data-driven canonical correlation analysis is employed to identify the most influential components from multitudinous factors that affect the MIQ indices to reduce the model dimension. Finally, experiments using industrial data and comparative studies have demonstrated that the obtained model produces a better modeling and estimating accuracy and stronger robustness than other modeling methods.
Cn中Cauchy-Stieltjes积分及其乘子的性质%Properties of Cauchy-Stieltjes Integrals and Their Multipliers
Institute of Scientific and Technical Information of China (English)
罗太元; 董新汉
2005-01-01
讨论了n维复空间Cn中Cauchy-Stieltjes积分Fnp及其乘子Mnp的一些性质.通过对Mnp中函数f的径向导数Rf(z)的积分平均估计,证明了Mnp中的函数f是有界的.讨论了同一测度在不同乘子空间的积分之间的联系,从而得到Fnp的一个遗传性质.利用Cn中Dirichlet空间Dnq范数的积分表示证明了Fnp与Dnq的包含关系.
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…
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…
Joseph, Laurice M.; Hunter, Amanda D.
2001-01-01
Explored the differential application of a self-regulatory strategy on mathematics performance among three eighth-graders with learning disabilities and diverse planning abilities. Found that all students improved on fraction probes as a function of using a cue card strategy. The student with high planning ability demonstrated consistent…
Smith, H. V.
2008-01-01
A method is derived for the numerical evaluation of the error term arising in some Gauss-type formulae modified so as to approximate Cauchy Principal Value integrals. The method uses Chebyshev polynomials of the first kind. (Contains 1 table.)
Smith, H. V.
2008-01-01
A method is derived for the numerical evaluation of the error term arising in some Gauss-type formulae modified so as to approximate Cauchy Principal Value integrals. The method uses Chebyshev polynomials of the first kind. (Contains 1 table.)
Yang, Jerry Zhijian
2014-01-01
Irving and Kirkwood formulism (IK formulism) provides a way to compute continuum mechanics quantities at certain location in terms of molecular variables. To make the approach more practical in computer simulation, Hardy proposed to use a spacial kernel function that couples continuum quantities with atomistic information. To reduce irrational fluctuations, Murdoch proposed to use a temporal kernel function to smooth the physical quantities obtained in Hardy's approach. In this paper, we generalize the original IK formulism to systematically incorporate both spacial and temporal average. The Cauchy stress tensor is derived in this generalized IK formulism (g-IK formulism). Analysis is given to illuminate the connection and difference between g-IK formulism and traditional temporal post-process approach. The relationship between Cauchy stress and first Piola-Kirchhoff stress is restudied in the framework of g-IK formulism. Numerical experiments using molecular dynamics are conducted to examine the analysis res...
Energy Technology Data Exchange (ETDEWEB)
Philipp, M; Mueller, U; Jimenez Rioboo, R J; Baller, J; Sanctuary, R; Krueger, J K [Laboratoire de Physique des Materiaux, University of Luxembourg, 162A avenue de la Faiencerie, L-1511 Luxembourg (Luxembourg); Possart, W [Fachbereich Werkstoffwissenschaften, Universitaet des Saarlandes, D-66123 Saarbruecken (Germany)], E-mail: martine.philipp@uni.lu
2009-02-15
The generalized Cauchy relation (gCR) of epoxy/silica nano-composites does not show either the chemically induced sol-gel transition or the chemically induced glass transition in the course of polymerization. Astonishingly, by varying the silica nanoparticles' concentration between 0 and 25 vol% in the composites, the Cauchy parameter A of the gCR remains universal and can be determined from the pure epoxy's elastic moduli. Air-filled porous silica glasses are considered as models for percolated silica particles. A longitudinal modulus versus density representation evidences the aforementioned transition phenomena during polymerization of the epoxy/silica nanocomposites. The existence of optically and mechanically relevant interphases is discussed.
Efficient video sequence matching using the Cauchy function and the modified Hausdorff distance
Kim, Sang Hyun; Park, Rae-Hong
2001-12-01
To manipulate large video databases, effective video indexing and retrieval are required. While most algorithms for video retrieval can be commonly used for frame-wise user query or video content query, video sequence matching has not been investigated much. In this paper, we propose an efficient algorithm to match the video sequences using the Cauchy function of histograms between successive frames and the modified Hausdorff distance. To effectively match the video sequences and to reduce the computational complexity, we use the key frames extracted by the cumulative measure, and compare the set of key frames using the modified Hausdorff distance. Experimental results show that the proposed video sequence matching algorithms using the Cauchy function and the modified Hausdorff distance yield the high accuracy and performances compared with conventional algorithms such as histogram difference and directed divergence methods.
Riemann boundary value problem for hyperanalytic functions
Directory of Open Access Journals (Sweden)
Ricardo Abreu Blaya
2005-01-01
Full Text Available We deal with Riemann boundary value problem for hyperanalytic functions. Furthermore, necessary and sufficient conditions for solvability of the problem are derived. At the end the explicit form of general solution for singular integral equations with a hypercomplex Cauchy kernel in the Douglis sense is established.
Generalized Radar 4-COORDINATES and Equal-Time Cauchy Surfaces for Arbitrary Accelerated Observers
Alba, David; Lusanna, Luca
All existing 4-coordinate systems centered on the world-line of an accelerated observer are only locally defined, as for Fermi coordinates both in special and general relativity. As a consequence, it is not known how non-inertial observers can build equal-time surfaces which (a) correspond to a conventional observer-dependent definition of synchronization of distant clocks, and (b) are good Cauchy surfaces for Maxwell equations. Another type of coordinate singularities generating the same problems are those connected to the relativistic rotating coordinate systems used in the treatment of the rotating disk and the Sagnac effect. We show that the use of Hamiltonian methods based on 3+1 splittings of space-time allows one to define as many observer-dependent globally defined radar 4-coordinate systems as nice foliations of space-time with space-like hyper-surfaces admissible according to Møller (for instance, only the differentially rotating relativistic coordinate system, but not the rigidly rotating ones of non-relativistic physics, are allowed). All these conventional notions of an instantaneous 3-space for an arbitrary observer can be empirically defined by introducing generalizations of the Einstein ½ convention for clock synchronization in inertial frames. Each admissible 3+1 splitting has two naturally associated congruences of time-like observers: as a consequence every 3+1 splitting gives rise to non-rigid non-inertial frames centered on any one of these observers. Only for Eulerian observers are the simultaneity leaves orthogonal to the observer world-line. When there is a Lagrangian description of an isolated relativistic system, its reformulation as a parametrized Minkowski theory allows one to show that all the admissible synchronization conventions are gauge equivalent, as also happens in the canonical metric and tetrad gravity, where, however, the chrono-geometrical structure of space-time is dynamically determined. The framework developed in this
Codomains for the Cauchy-Riemann and Laplace operators in ℝ2
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Lloyd Edgar S. Moyo
2008-01-01
Full Text Available A codomain for a nonzero constant-coefficient linear partial differential operator P(∂ with fundamental solution E is a space of distributions T for which it is possible to define the convolution E*T and thus solving the equation P(∂S=T. We identify codomains for the Cauchy-Riemann operator in ℝ2 and Laplace operator in ℝ2 . The convolution is understood in the sense of the S′-convolution.
Local stability of the Pexiderized Cauchy and Jensen's equations in fuzzy spaces
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Kang Jung Im
2011-01-01
Full Text Available Abstract Lex X be a normed space and Y be a Banach fuzzy space. Let D = {(x, y ∈ X × X : ||x|| + ||y|| ≥ d} where d > 0. We prove that the Pexiderized Jensen functional equation is stable in the fuzzy norm for functions defined on D and taking values in Y. We consider also the Pexiderized Cauchy functional equation. 2000 Mathematics Subject Classification: 39B22; 39B82; 46S10.
On the Existence of a Maximal Cauchy Development for the Einstein Equations - a Dezornification
Sbierski, Jan
2013-01-01
In 1969, Choquet-Bruhat and Geroch established the existence of a unique maximal globally hyperbolic Cauchy development of given initial data for the Einstein equations. Their proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice in the form of Zorn's lemma. In this paper we present a proof that avoids the use of Zorn's lemma. In particular, we provide an explicit construction of this maximal globally hyperbolic development.
A Subspace Embedding Method in L2 Norm via Fast Cauchy Transform
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Xu Xiang
2013-01-01
Full Text Available We propose a subspace embedding method via Fast Cauchy Transform (FCT in L2 norm. It is motivated by and complements the work of the subspace embedding method in Lp norm, for all p∈[1,∞] except p = 2, by K. L. Clarkson (ACM-SIAM, 2013. Unlike the traditionally used orthogonal basis in Johnson-Lindenstrauss (JL embedding, we employ the well-conditioned basis in L2 norm to obtain concentration property of FCT in L2 norm.
Murthy, Ganpathy
2000-01-01
It is well known that the ν = 2/5 state is unpolarized at zero Zeeman energy, while it is fully polarized at large Zeeman energies. A novel state with a charge/spin density wave order for composite fermions is proposed to exist at intermediate values of the Zeeman coupling for ν = 2/5. This state has half the maximum possible polarization, and can be extended to other incompressible fractions. A Hartree-Fock calculation based on the new approach for all fractional quantum Hall states developed by R. Shankar and the author is used to demonstrate the stability of this state to single-particle excitations and to compute gaps. A very recent experiment shows direct evidence for this state.
Murthy
2000-01-10
It is well known that the nu = 2/5 state is unpolarized at zero Zeeman energy, while it is fully polarized at large Zeeman energies. A novel state with a charge/spin density wave order for composite fermions is proposed to exist at intermediate values of the Zeeman coupling for nu = 2/5. This state has half the maximum possible polarization, and can be extended to other incompressible fractions. A Hartree-Fock calculation based on the new approach for all fractional quantum Hall states developed by R. Shankar and the author is used to demonstrate the stability of this state to single-particle excitations and to compute gaps. A very recent experiment shows direct evidence for this state.
Koichu, Boris; Harel, Guershon; Manaster, Alfred
2013-01-01
Twenty-four mathematics teachers were asked to think aloud when posing a word problem whose solution could be found by computing 4/5 divided by 2/3. The data consisted of verbal protocols along with the written notes made by the subjects. The qualitative analysis of the data was focused on identifying the structures of the problems produced and…
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Schümann Kerstin
2016-09-01
Full Text Available Thermoplastic polymers exhibit non-isochoric behaviour during tensile and compression testing as well as particular deformations like local necking (tension or buckling (compression. A method for the determination of Cauchy stresses from tensile and compression tests is presented, that considers the actual deformations of the test specimens. The exact geometry of the specimens in the respective present configuration is determined in photographs, which are taken continuously throughout the test. The engineering stresses at several time points are converted into Cauchy stresses using newly developed formulas in consideration of the actual specimen geometry. For validation finite element analyses of the tensile and compression tests are performed using the identified stress-strain curves. The numerical results show good agreement with the experiments for the tested polymers. Thus a method for conversion of engineering to Cauchy stresses in tensile and compression tests could be established considering the non-isochoric deformation in plasticity. With this method “true” stress-strain curves as input for finite element material models can be identified for arbitrary materials.
Holographic stress-energy tensor near the Cauchy horizon inside a rotating black hole
Ishibashi, Akihiro; Maeda, Kengo; Mefford, Eric
2017-07-01
We investigate a stress-energy tensor for a conformal field theory (CFT) at strong coupling inside a small five-dimensional rotating Myers-Perry black hole with equal angular momenta by using the holographic method. As a gravitational dual, we perturbatively construct a black droplet solution by applying the "derivative expansion" method, generalizing the work of Haddad [Classical Quantum Gravity 29, 245001 (2012), 10.1088/0264-9381/29/24/245001] and analytically compute the holographic stress-energy tensor for our solution. We find that the stress-energy tensor is finite at both the future and past outer (event) horizons and that the energy density is negative just outside the event horizons due to the Hawking effect. Furthermore, we apply the holographic method to the question of quantum instability of the Cauchy horizon since, by construction, our black droplet solution also admits a Cauchy horizon inside. We analytically show that the null-null component of the holographic stress-energy tensor negatively diverges at the Cauchy horizon, suggesting that a singularity appears there, in favor of strong cosmic censorship.
Cauchy's almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow
Frisch, Uriel
2014-01-01
Two prized papers, one by Augustin Cauchy in 1815, presented to the French Academy and the other by Hermann Hankel in 1861, presented to G\\"ottingen University, contain major discoveries on vorticity dynamics whose impact is now quickly increasing. Cauchy found a Lagrangian formulation of 3D ideal incompressible flow in terms of three invariants that generalize to three dimensions the now well-known law of conservation of vorticity along fluid particle trajectories for two-dimensional flow. This has very recently been used to prove analyticity in time of fluid particle trajectories for 3D incompressible Euler flow and can be extended to compressible flow, in particular to cosmological dark matter. Hankel showed that Cauchy's formulation gives a very simple Lagrangian derivation of the Helmholtz vorticity-flux invariants and, in the middle of the proof, derived an intermediate result which is the conservation of the circulation of the velocity around a closed contour moving with the fluid. This circulation the...
The Cauchy Problem for the 3-D Vlasov-Poisson System with Point Charges
Marchioro, Carlo; Miot, Evelyne; Pulvirenti, Mario
2011-07-01
In this paper we establish global existence and uniqueness of the solution to the three-dimensional Vlasov-Poisson system in the presence of point charges with repulsive interaction. The present analysis extends an analogous two-dimensional result (Caprino and Marchioro in Kinet. Relat. Models 3(2):241-254, 2010).
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K. Balachandran
2006-09-01
Full Text Available In this paper we prove the existence of mild and strong solutions of nonlinear time varying delay integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. The results are obtained by using the theory of compact semigroups and Schaefer's fixed point theorem.
On the Cauchy Problem of Evolution p-Laplacian Equation with Nonlinear Gradient Term
Institute of Scientific and Technical Information of China (English)
Mingyu CHEN; Junning ZHAO
2009-01-01
The authors study the existence of solution to p-Laplacian equation with non-linear forcing term under optimal assumptions on the initial data,which are assumed to be measures.The existence of local solution is obtained.
The Cauchy problem for the 3-D Vlasov-Poisson system with point charges
Marchioro, Carlo; Pulvirenti, Mario
2010-01-01
In this paper we establish global existence and uniqueness of the solution to the three-dimensional Vlasov-Poisson system in presence of point charges in case of repulsive interaction. The present analysis extends an analogeous two-dimensional result by Caprino and Marchioro [On the plasma-charge model, to appear in Kinetic and Related Models (2010)].
The Cauchy Problem for the p-Laplacian Equation with a Nonlinear Source
Institute of Scientific and Technical Information of China (English)
LEI Pei-dong
2001-01-01
In this paper we study the existence and uniqueness of positive solutions for the p-Laplacian equation with nonlinear sourceu/ t = div(｜ Du ｜p-2Du) + u-q, p ＞ 2, 0 ＜ q ＜ ∞ in the class of functions with some prescribed growth rate as ｜ x ｜→ ∞. We also give a description of thelarge time behaviour and show that it is determined by the competition between the diffusion and the source.
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Andrei Perjan
2009-07-01
Full Text Available We study the behavior of solutions to perturbed second order abstract evolution equations in Hilbert spaces, when the small parameter, multiplying the second order time derivative, converges to zero.
THE CAUCHY PROBLEM FOR A CLASS OF COUPLED SYSTEMS CONTAINING A CONVOLUTION OPERATOR
Institute of Scientific and Technical Information of China (English)
杨成荣; 伍卓群
2003-01-01
Some results on the invariant regions ,existence and uniqueness of solutions to a class of integrodifferential systems are established. Applying these results to integrodifferential systems with a small parameter e ＞ 0, we obtain,in particular, some estimates of solutions uniform in ε＞ 0.
MAXIMAL SUBSPACES FOR SOLUTIONS OF THE SECOND ORDER ABSTRACT CAUCHY PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
For a continuous, increasing function ω: R+ → R+\\{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R+, and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family.
100 great problems of elementary mathematics their history and solution
Dorrie, Heinrich
2013-01-01
Problems that beset Archimedes, Newton, Euler, Cauchy, Gauss, Monge and other greats, ready to challenge today's would-be problem solvers. Among them: How is a sundial constructed? How can you calculate the logarithm of a given number without the use of logarithm table? No advanced math is required. Includes 100 problems with proofs.
Spectral decomposition of fractional operators and a reflected stable semigroup
Patie, P.; Zhao, Y.
2017-02-01
In this paper, we provide the spectral decomposition in Hilbert space of the C0-semigroup P and its adjoint P ˆ having as generator, respectively, the Caputo and the right-sided Riemann-Liouville fractional derivatives of index 1 operators, which are non-local and non-self-adjoint, appear in many recent studies in applied mathematics and also arise as the infinitesimal generators of some substantial processes such as the reflected spectrally negative α-stable process. Our approach relies on intertwining relations that we establish between these semigroups and the semigroup of a Bessel type process whose generator is a self-adjoint second order differential operator. In particular, from this commutation relation, we characterize the positive real axis as the continuous point spectrum of P and provide a power series representation of the corresponding eigenfunctions. We also identify the positive real axis as the residual spectrum of the adjoint operator P ˆ and elucidate its role in the spectral decomposition of these operators. By resorting to the concept of continuous frames, we proceed by investigating the domain of the spectral operators and derive two representations for the heat kernels of these semigroups. As a by-product, we also obtain regularity properties for these latter and also for the solution of the associated Cauchy problem.
Otra demostración de la desigualdad de Cauchy- Schwarz
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Javier MARTÍN LALANDA
2009-12-01
Full Text Available En todo espacio euclideo E, en el que se haya definido una norma se verifica la siguiente desigualdad, conocida con el nombre Desigualdad de Cauchy-Schwarz. Para el modo usual en que se suele demostrar esta desigualdad, se parte de uno de los axiomas asociados a la definición de producto interno, aquel que dice que el producto interno de todo vector por sí mismo es siempre un número real, mayor o igual que cero
On the Direct Cauchy Theorem in Widom Domains: Positive and Negative Examples
Yuditskii, Peter
2010-01-01
We discuss several questions which remained open in our joint work with M. Sodin "Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character--automorphic functions". In particular, we show that there exists a non-homogeneous set $E$ such that the Direct Cauchy Theorem (DCT) holds in the Widom domain $\\bbC\\setminus E$. On the other hand we demonstrate that the weak homogeneity condition on $E$ (introduced recently by Poltoratski and Remling) does not ensure that DCT holds in the corresponding Widom domain.
Cauchy-Matern Model of Sea Surface Wind Speed at the Lake Worth, Florida
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Ming Li
2012-01-01
Full Text Available We study the Cauchy-Matern (CM process with long-range dependence (LRD. The closed form of its power spectrum density (PSD function is given. We apply it to model the autocovariance function (ACF and the PSD of the sea surface wind speed (wind speed for short observed in the Lake Worth, Florida, over the 1984–2006 period. The present results exhibit that the wind speed at the Lake Worth over 1984–2006 is of LRD. The present results exhibit that the CM process may yet be a novel model to fit the wind speed there.
Symmetries of 2-lattices and second order accuracy of the Cauchy--Born Model
Van Koten, Brian
2012-01-01
We show that the Cauchy--Born model of a single-species 2-lattice is second order if the atomistic and continuum kinematics are connected in a novel way. Our proof uses a generalization to 2-lattices of the point symmetry of Bravais lattices. Moreover, by identifying similar symmetries in multi-species pair interaction models, we construct a new stored energy density, using shift-gradients but not strain gradients, that is also second order accurate. These results can be used to develop highly accurate continuum models and atomistic/continuum coupling methods for materials such as graphene, hcp metals, and shape memory alloys.
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Bernardo Gómez
2009-06-01
Full Text Available Este artículo presenta parte de un estudio que persigue analizar el efecto de las variables de problema y las variables de resolución de problemas en las respuestas de estudiantes de segundo ciclo de educación secundaria obligatoria cuando resuelven problemas multiplicativos de división de fracciones. Se recoge aquí un análisis histórico-epistemológico que ha dado cuenta de los valores de dichas variables y, a partir de resultados preliminares, se sugiere la existencia de interrelaciones entre las variables de problema y las variables de resolución. This article presents part of a study aiming to analyze the effect of problem and problem solving variables in secondary students’ performance when solving multiplicative problems related to division of fractions. Here we present a historic-epistemological analysis that gives account of these variables and some preliminary results that suggest the existence of relations between problem variables and solving variables.
Energy Technology Data Exchange (ETDEWEB)
Bajona-Xandri, C.; Martinez-Legaz, J.E.
1994-12-31
This paper studies the minimax fractional programming problem, assuming quasiconvexity of the objective function, under the lower subdifferentiability viewpoint. Necessary and sufficient optimality conditions and dual properties are found. We present applications of this theory to find the Pareto efficient solutions of a multiobjective fractional problem and to solve several economic models.
The Cauchy-Lagrangian method for numerical analysis of Euler flow
Podvigina, O; Frisch, U
2015-01-01
A novel semi-Lagrangian method is introduced to solve numerically the Euler equation for ideal incompressible flow in arbitrary space dimension. It exploits the time-analyticity of fluid particle trajectories and requires, in principle, only limited spatial smoothness of the initial data. Efficient generation of high-order time-Taylor coefficients is made possible by simple recurrence relations that follow from the Cauchy invariants formulation of the Euler equations (Zheligovsky & Frisch, J. Fluid Mech. 2014, 749, 404-430). Truncated time-Taylor series of very high order allow the use of time steps vastly exceeding the Courant-Friedrichs-Lewy limit, without compromising the accuracy of the solution. Tests performed on the two-dimensional Euler equation indicate that the Cauchy-Lagrangian method is more --- and occasionally much more --- efficient and less prone to instability than Eulerian Runge-Kutta methods and less prone to rapid growth of rounding errors than the high-order Eulerian time-Taylor algor...
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Jian-Guo Zheng
2015-01-01
Full Text Available Artificial bee colony (ABC algorithm is a popular swarm intelligence technique inspired by the intelligent foraging behavior of honey bees. However, ABC is good at exploration but poor at exploitation and its convergence speed is also an issue in some cases. To improve the performance of ABC, a novel ABC combined with grenade explosion method (GEM and Cauchy operator, namely, ABCGC, is proposed. GEM is embedded in the onlooker bees’ phase to enhance the exploitation ability and accelerate convergence of ABCGC; meanwhile, Cauchy operator is introduced into the scout bees’ phase to help ABCGC escape from local optimum and further enhance its exploration ability. Two sets of well-known benchmark functions are used to validate the better performance of ABCGC. The experiments confirm that ABCGC is significantly superior to ABC and other competitors; particularly it converges to the global optimum faster in most cases. These results suggest that ABCGC usually achieves a good balance between exploitation and exploration and can effectively serve as an alternative for global optimization.
Application of Cauchy wavelet transformation to identify time-variant modal parameters of structures
Huang, C. S.; Liu, C. Y.; Su, W. C.
2016-12-01
This work proposes a procedure for accurately identifying instantaneous modal parameters of a linear time-varying system using a time-varying autoregressive with exogenous input (TVARX) model with the continuous Cauchy wavelet transform (CCWT). An appropriate TVARX model is established using the velocity and displacement responses of the system under consideration. The time-varying coefficients of the TVARX are expanded as piecewise polynomial functions. CCWTs with various scale parameters are then applied to the TVARX model to evaluate the instantaneous modal parameters of different modes. The CCWTs of the velocity and displacement responses are analytically obtained from the CCWT of the measured acceleration responses. The effectiveness and accuracy of the proposed procedure are validated by numerical simulations of single and multiple degrees of freedom systems that have periodically varying and sharply varying stiffness and damping coefficients. The effects of noise, the Cauchy wavelet function and the order of the polynomial on the evaluation of the modal parameters are explored in processing the numerically simulated acceleration responses of systems with a single degree of freedom subjected to base excitation. Finally, the proposed procedure is adopted to determine the modal parameters of a five-story symmetric steel frame from its measured acceleration responses in a shaking table test. The measured strains reveal the yielding of columns in the first story. The variations of the identified instantaneous natural frequencies and modal damping ratios with time are consistent with the physical phenomena that are observed from the measured strains and base excitation acceleration.
一个广义的Cauchy型的Taylor公式%A Taylor's Formula of Generalized Cauchy Type
Institute of Scientific and Technical Information of China (English)
苏翃; 赵振华; 董建
2009-01-01
给出了一个高阶导数形式的、广义的Cauchy型的Taylor公式,它将数学分析中一阶微分形式的Cauchy中值定理推广到高阶导数形式,同时它也是Taylor中值定理的推广.%Taylor's formula of having the form of higher order derivative and generalized Cauchy type is obtained, which extends Cauchy's mean-value theorem of differential form of first order in mathematical analysis to the form of higher order derivative, and is the spread of Taylor's mean-value theorem.
Application of a four-step HMX kinetic model to an impact-induced fraction ignition problems
Energy Technology Data Exchange (ETDEWEB)
Perry, William L [Los Alamos National Laboratory; Gunderson, Jake A [Los Alamos National Laboratory; Dickson, Peter M [Los Alamos National Laboratory
2010-01-01
There has been a long history of interest in the decomposition kinetics of HMX and HMX-based formulations due to the widespread use of this explosive in high performance systems. The kinetics allow us to predict, or attempt to predict, the behavior of the explosive when subjected to thermal hazard scenarios that lead to ignition via impact, spark, friction or external heat. The latter, commonly referred to as 'cook off', has been widely studied and contemporary kinetic and transport models accurately predict time and location of ignition for simple geometries. However, there has been relatively little attention given to the problem of localized ignition that results from the first three ignition sources of impact, spark and friction. The use of a zero-order single-rate expression describing the exothermic decomposition of explosives dates to the early work of Frank-Kamanetskii in the late 1930s and continued through the 60's and 70's. This expression provides very general qualitative insight, but cannot provide accurate spatial or timing details of slow cook off ignition. In the 70s, Catalano, et al., noted that single step kinetics would not accurately predict time to ignition in the one-dimensional time to explosion apparatus (ODTX). In the early 80s, Tarver and McGuire published their well-known three step kinetic expression that included an endothermic decomposition step. This scheme significantly improved the accuracy of ignition time prediction for the ODTX. However, the Tarver/McGuire model could not produce the internal temperature profiles observed in the small-scale radial experiments nor could it accurately predict the location of ignition. Those factors are suspected to significantly affect the post-ignition behavior and better models were needed. Brill, et al. noted that the enthalpy change due to the beta-delta crystal phase transition was similar to the assumed endothermic decomposition step in the Tarver/McGuire model. Henson, et
Indian Academy of Sciences (India)
S Baskar; Phoolan Prasad
2006-02-01
For the structure of a sonic boom produced by a simple aerofoil at a large distance from its source we take a physical model which consists of a leading shock (LS), a trailing shock (TS) and a one-parameter family of nonlinear wavefronts in between the two shocks. Then we develop a mathematical model and show that according to this model the LS is governed by a hyperbolic system of equations in conservation form and the system of equations governing the TS has a pair of complex eigenvalues. Similarly, we show that a nonlinear wavefront originating from a point on the front part of the aerofoil is governed by a hyperbolic system of conservation laws and that originating from a point on the rear part is governed by a system of conservation laws, which is elliptic. Consequently, we expect the geometry of the TS to be kink-free and topologically different from the geometry of the LS. In the last section we point out an evidence of kinks on the LS and kink-free TS from the numerical solution of the Euler’s equations by Inoue, Sakai and Nishida [5].
Initial Value Problems for Wave Equations on Manifolds
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Bär, Christian, E-mail: baer@math.uni-potsdam.de [Universität Potsdam, Institut für Mathematik (Germany); Wafo, Roger Tagne, E-mail: rtagnewafo@yahoo.com [University of Douala, Faculty of Science, Department of Mathematics and Computer Science (Cameroon)
2015-12-15
We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hörmander.
A boundary value problem for hypermonogenic functions in Clifford analysis
Institute of Scientific and Technical Information of China (English)
QIAO; Yuying
2005-01-01
This paper deals with a boundary value problem for hypermonogenic functions in Clifford analysis. Firstly we discuss integrals of quasi-Cauchy's type and get the Plemelj formula for hypermonogenic functions in Clifford analysis, and then we address Riemman boundary value problem for hypermonogenic functions.
Giorgio, Ivan
2016-08-01
In order to design the microstructure of metamaterials showing high toughness in extension (property to be shared with muscles), it has been recently proposed (Dell'Isola et al. in Z Angew Math Phys 66(6):3473-3498, 2015) to consider pantographic structures. It is possible to model such structures at a suitably small length scale (resolving in detail the interconnecting pivots/cylinders) using a standard Cauchy first gradient theory. However, the computational costs for such modelling choice are not allowing for the study of more complex mechanical systems including for instance many pantographic substructures. The microscopic model considered here is a quadratic isotropic Saint-Venant first gradient continuum including geometric nonlinearities and characterized by two Lamé parameters. The introduced macroscopic two-dimensional model for pantographic sheets is characterized by a deformation energy quadratic both in the first and second gradient of placement. However, as underlined in Dell'Isola et al. (Proc R Soc Lond A 472(2185):20150790, 2016), it is needed that the second gradient stiffness depends on the first gradient of placement if large deformations and large displacements configurations must be described. The numerical identification procedure presented in this paper consists in fitting the macro-constitutive parameters using several numerical simulations performed with the micro-model. The parameters obtained by the best fit identification in few deformation problems fit very well also in many others, showing that the reduced proposed model is suitable to get an effective model at relevantly lower computational effort. The presented numerical evidences suggest that a rigorous mathematical homogenization result most likely holds.
Riemann Boundary Value Problems for Koch Curve
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Zhengshun Ruanand
2012-11-01
Full Text Available In this study, when L is substituted for Koch curve, Riemann boundary value problems was defined, but generally speaking, Cauchy-type integral is meaningless on Koch curve. When some analytic conditions are attached to functions G (z and g (z, through the limit function of a sequence of Cauchytype integrals, the homogeneous and non-homogeneous Riemann boundary problems on Koch curve are introduced, some similar results was attained like the classical boundary value problems for analytic functions.
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Yekini Shehu
2010-01-01
real Banach space which is also uniformly smooth using the properties of generalized f-projection operator. Using this result, we discuss strong convergence theorem concerning general H-monotone mappings and system of generalized mixed equilibrium problems in Banach spaces. Our results extend many known recent results in the literature.
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The properties of eigenvalues and eigenfunctions of the infinite dimensional Hamiltonian operators are studied, and the suffcient conditions of the completeness in the sense of Cauchy principal value of the eigenfunction systems of the infinite dimensional Hamiltonian operators are given. In the end, concrete examples are constructed to justify the effectiveness of the criterion.
Institute of Scientific and Technical Information of China (English)
Alatancang; WU DeYu
2009-01-01
The properties of eigenvalues and eigenfunctions of the infinite dimensional Hamiltonian operators are studied,and the sufficient conditions of the completeness in the sense of Cauchy principal value of the eigenfunction systems of the infinite dimensional Hamiltonian operators are given.In the end,concrete examples are constructed to justify the effectiveness of the criterion.
Dechoum, K; Khoury, A Z; 10.1103/PhysRevA.83.063843
2011-01-01
We analyze the three-mode correlation properties of the electromagnetic field in a optical parametric oscillator below threshold. We employ a perturbative expansion of the It\\^o equations derived from the positive-P representation of the density matrix. Using the generalized Cauchy-Schwarz inequality, we investigate the genuine quantum nature of the triple correlations between the interacting fields, since in this case continuous variable entanglement is not detected by the van Loock-Furusawa criterion [Phys. Rev. A {\\bf 67}, 052315 (2003)]. Although not being a necessary condition, these triple correlations are a sufficient evidence of tripartite entanglement. Of course, our characterization of the quantum correlations is applicable to non-Gaussian states, which we show to be the case of the optical parametric oscillator below threshold, provided nonlinear quantum fluctuations are properly taken into account.
Institute of Scientific and Technical Information of China (English)
时统业; 周本虎
2006-01-01
设f(x),g(x)均在[a,b]上可积,则Cauchy-Schwarz不等式可加强为:[∫baf(x)g(x)dx]2 ≤∫baf2(x)dx∫ba g2(x)dx + 2/b-a∫ba f(x)g(x)dx∫ba f(x)d(x)∫ba g(x)dx - 1/b-a∫ba f2(x)dx ·[∫ba g(x)dx]2-1/b-a∫ba g2(x)dx[∫ba f(x)dx]2.由此推广了文[1]结果
Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator
Kravchenko, Vladislav V
2009-01-01
In [8] it was shown that the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schr\\"odinger equation. In this paper we demonstrate a fundamental property of this operator: it transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a "complicated" Vekua equation from positive formal powers and a generating sequence of "simpler" Vekua equation. Similar results is obtained regarding construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theory are considered and examples are given to illustrate the procedure.
Different glassy states, as indicated by a violation of the generalized Cauchy relation
Energy Technology Data Exchange (ETDEWEB)
Krueger, J K [Laboratoire Europeen de Recherche Universitaire Saarland-Lorraine (LERUSL), Universitaet des Saarlandes, Fakultaet fuer Physik und Elektrotechnik 7.2, Gebaeude 38, D-66041 Saarbruecken (Germany); Britz, T [Laboratoire Europeen de Recherche Universitaire Saarland-Lorraine (LERUSL), Universitaet des Saarlandes, Fakultaet fuer Physik und Elektrotechnik 7.2, Gebaeude 38, D-66041 Saarbruecken (Germany); Coutre, A le [Laboratoire Europeen de Recherche Universitaire Saarland-Lorraine (LERUSL), Universitaet des Saarlandes, Fakultaet fuer Physik und Elektrotechnik 7.2, Gebaeude 38, D-66041 Saarbruecken (Germany); Baller, J [Laboratoire Europeen de Recherche Universitaire Saarland-Lorraine (LERUSL), Universitaet des Saarlandes, Fakultaet fuer Physik und Elektrotechnik 7.2, Gebaeude 38, D-66041 Saarbruecken (Germany); Possart, W [Universitaet des Saarlandes, Fakultaet fuer Chemie, Pharmazie und Werkstoffwissenschaften 8.15, Gebaeude 22, D-66041 Saarbruecken (Germany); Alnot, P [Laboratoire Europeen de Recherche Universitaire Saarland-Lorraine (LERUSL), Universitaet des Saarlandes, Fakultaet fuer Physik und Elektrotechnik 7.2, Gebaeude 38, D-66041 Saarbruecken (Germany); Sanctuary, R [Centre Universitaire de Luxembourg, Departement des Sciences, Laboratoire 1.19, 162a Avenue de la Faiencerie, L-1511, Luxembourg (Luxembourg)
2003-07-01
Using Brillouin spectroscopy as a probe for high-frequency clamped acoustic properties, a shear modulus c{sub 44}{sup {infinity}} can be measured in addition to the longitudinal modulus c{sub 11}{sup {infinity}} already well above the thermal glass transition. On slow cooling of the liquid through the thermal glass transition temperature T{sub g}, both moduli show a kink-like behaviour and the function c{sub 11}{sup {infinity}} = c{sub 11}{sup {infinity}}(c{sub 44}{sup {infinity}}) follows a generalized Cauchy relation (gCR) defined by the linear relation c{sub 11}{sup {infinity}} = 3c{sub 44}{sup {infinity}} + constant, which completely hides the glass transition. In this work we show experimentally that on fast cooling this linear transformation becomes violated within the glassy state, but that thermal ageing drives the elastic coefficients towards the gCR, i.e. towards a unique glassy state.
Newton`s iteration for inversion of Cauchy-like and other structured matrices
Energy Technology Data Exchange (ETDEWEB)
Pan, V.Y. [Lehman College, Bronx, NY (United States); Zheng, Ailong; Huang, Xiaohan; Dias, O. [CUNY, New York, NY (United States)
1996-12-31
We specify some initial assumptions that guarantee rapid refinement of a rough initial approximation to the inverse of a Cauchy-like matrix, by mean of our new modification of Newton`s iteration, where the input, output, and all the auxiliary matrices are represented with their short generators defined by the associated scaling operators. The computations are performed fast since they are confined to operations with short generators of the given and computed matrices. Because of the known correlations among various structured matrices, the algorithm is immediately extended to rapid refinement of rough initial approximations to the inverses of Vandermonde-like, Chebyshev-Vandermonde-like and Toeplitz-like matrices, where again, the computations are confined to operations with short generators of the involved matrices.
An Analysis of Surface Relaxation in the Surface Cauchy--Born Model
Jayawardana, K; Ortner, C; Park, H S
2011-01-01
The Surface Cauchy-Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is O(1) in the mesh size; however, we are able to identify an alternative "approximation parameter" - the stiffness of the interaction potential - with respect to which the error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary.
Modelling mortality of a stored grain insect pest with fumigation: probit, logistic or Cauchy model?
Shi, Mingren; Renton, Michael
2013-06-01
Computer simulation models can provide a relatively fast, safe and inexpensive means to judge and weigh the merits of various pest control management options. However, the usefulness of such simulation models relies on the accurate estimation of important model parameters, such as the pest mortality under different treatments and conditions. Recently, an individual-based simulation model of population dynamics and resistance evolution has been developed for the stored grain insect pest Rhyzopertha dominica, based on experimental results showing that alleles at two different loci are involved in resistance to the grain fumigant phosphine. In this paper, we describe how we used three generalized linear models, probit, logistic and Cauchy models, each employing two- and four-parameter sub-models, to fit experimental data sets for five genotypes for which detailed mortality data was already available. Instead of the usual statistical iterative maximum likelihood estimation, a direct algebraic approach, generalized inverse matrix technique, was used to estimate the mortality model parameters. As this technique needs to perturb the observed mortality proportions if the proportions include 0 or 1, a golden section search approach was used to find the optimal perturbation in terms of minimum least squares (L2) error. The results show that the estimates using the probit model were the most accurate in terms of L2 errors between observed and predicted mortality values. These errors with the probit model ranged from 0.049% to 5.3%, from 0.381% to 8.1% with the logistic model and from 8.3% to 48.2% with the Cauchy model. Meanwhile, the generalized inverse matrix technique achieved similar results to the maximum likelihood estimation ones, but is less time consuming and computationally demanding. We also describe how we constructed a two-parameter model to estimate the mortalities for each of the remaining four genotypes based on realistic genetic assumptions.
Goursat problem for the Yang-Mills-Vlasov system in temporal gauge
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Marcel Dossa
2011-12-01
Full Text Available This article studies the characteristic Cauchy problem for the Yang-Mills-Vlasov (YMV system in temporal gauge, where the initial data are specified on two intersecting smooth characteristic hypersurfaces of Minkowski spacetime $(mathbb{R}^{4},eta $. Under a $mathcal{C}^{infty }$ hypothesis on the data, we solve the initial constraint problem and the evolution problem. Local in time existence and uniqueness results are established thanks to a suitable combination of the method of characteristics, Leray's Theory of hyperbolic systems and techniques developed by Choquet-Bruhat for ordinary spatial Cauchy problems related to (YMV systems.
Instantaneous blow-up of semilinear non-autonomous equations with fractional diffusion
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Jose Villa-Morales
2017-05-01
Full Text Available We consider the Cauchy initial value problem $$\\displaylines{ \\frac{\\partial }{\\partial t}u(t,x =k(t\\Delta _{\\alpha}u(t,x+h(tf(u(t,x, \\cr u(0,x = u_0(x, }$$ where $\\Delta _{\\alpha }$ is the fractional Laplacian for $0<\\alpha \\leq 2$. We prove that if the initial condition $u_0$ is non-negative, bounded and measurable then the problem has a global integral solution when the source term f is non-negative, locally Lipschitz and satisfies the generalized Osgood's condition $$ \\int_{\\|u_0\\|_{\\infty }}^{\\infty }\\frac{ds}{f(s}\\geq \\int_0^{\\infty}h(sds. $$ Also, we prove that if the initial data is unbounded then the generalized Osgood's condition does not guarantee the existence of a global solution. It is important to point out that the proof of the existence hinges on the role of the function h. Analogously, the function k plays a central role in the proof of the instantaneous blow-up.
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...
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…
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…
Shuji, OBATA; Shigeru, OHKURO; Physics Laboratory, Faculty of Science aud Engineering, Tohyo Denki University; Laboratory of Information aud System Engineering, Hachinohe Institute of Technology
1999-01-01
We have been studying chaotic behavior and chaos-like behavior in continued fractions. In this paper, such chaos-like behavior is investigated in detail. This behavior originates in the complex numbers that determine the Cauchy distributions, where cyclic terms discretely appear at isolated parameter values. The distributions are formed along with alternate tangent functions that are dominated by the cyclic terms characterized by double-Markov processes. Finally, the probability densities of ...
THE NONLOCAL INITIAL PROBLEMS OF A SEMILINEAR EVOLUTION EQUATION
Institute of Scientific and Technical Information of China (English)
王远弟; 冉启康
2004-01-01
The purpose of this paper is to investigate the existence of solutions to a nonlocal Cauchy problem for an evolution equation. The methods used here include the abstract semigroup methods in proper spaces and Schauder's theorem.And the abstract results are applied to a system of nonlinear partial differential equations with nonlinear boundary conditions.
D'Inverno, R A; Sarkies, E A
2000-01-01
This paper is part of a long term program to Cauchy-characteristic matching(CCM) codes as investigative tools in numerical relativity. The approach hastwo distinct features: (i) it dispenses with an outer boundary condition andreplaces this with matching conditions at an interface between the Cauchy andcharacteristic regions, and (ii) by employing a compactified coordinate, itproves possible to generate global solutions. In this paper CCM is applied toan exact two-parameter family of cylindrically symmetric vacuum solutionspossessing both gravitational degrees of freedom due to Piran, Safier and Katz.This requires a modification of the previously constructed CCM cylindrical codebecause, even after using Geroch decomposition to factor out the $z$-direction,the family is not asymptotically flat. The key equations in the characteristicregime turn out to be regular singular in nature.
N{sup ±}-integrals and boundary values of Cauchy-type integrals of finite measures
Energy Technology Data Exchange (ETDEWEB)
Aliev, R. A., E-mail: aliyevrashid@hotmail.ru, E-mail: alievrashid@box.az [Baku State University (Azerbaijan)
2014-07-31
Let Γ be a simple closed Lyapunov contour with finite complex measure ν, and let G{sup +} be the bounded and G{sup −} the unbounded domains with boundary Γ. Using new notions (so-called N-integration and N{sup +}- and N{sup −}-integrals), we prove that the Cauchy-type integrals F{sup +}(z), z∈G{sup +}, and F{sup −}(z), z∈G{sup −}, of ν are Cauchy N{sup +}- and N{sup −}-integrals, respectively. In the proof of the corresponding results, the additivity property and the validity of the change-of-variable formula for the N{sup +}- and N{sup −}-integrals play an essential role. Bibliography: 21 titles. (paper)
Karras, G; Billard, F; Lavorel, B; Siour, G; Hartmann, J -M; Faucher, O; Gershnabel, Erez; Prior, Yehiam; Averbukh, Ilya Sh
2016-01-01
We report the observation of fractional echoes in a double-pulse excited nonlinear system. Unlike standard echoes which appear periodically at delays which are integer multiple of the delay between the two exciting pulses, the fractional echoes appear at rational fractions of this delay. We discuss the mechanism leading to this phenomenon, and provide the first experimental demonstration of fractional echoes by measuring third harmonic generation in a thermal gas of CO2 molecules excited by a pair of femtosecond laser pulses.
Institute of Scientific and Technical Information of China (English)
张毅
2013-01-01
The Noether symmetry and the conserved quantity for a fractional action-like variational problem in phase space are studied based on the method of fractional dynamics modeling presented by ElNabulsi,namely fractional action-like variational approach.First,the fractional action-like variational problem in phase space is established,and the fractional action-like Hamilton canonical equations are obtained.Secondly,the definitions and criteria of the fractional action-like Noether (quasi-) symmetrical transformations are presented in terms of the invariance of the fractional action-like integral of Hamilton under the infinitesimal transformation of group.Finally,the Noether theorems for the fractional actionlike Hamiltonian system are given,the relationship between the Noether symmetry and the conserved quantity of the system is established.An example is given to illustrate the application of the results.%基于El-Nabulsi提出的分数阶动力学建模方法,即类分数阶变分方法,研究相空间中类分数阶变分问题与Noether对称性和守恒量.建立了相空间中类分数阶变分问题,得到了类分数阶Hamilton正则方程；基于类分数阶Hamilton作用量在无限小群变换下的不变性,提出了相空间中类分数阶Noether(准)对称变换的定义和判据；给出了类分数阶Hamilton系统的Noether定理,建立了类分数阶Noether对称性与守恒量之间的内在关系,并举例说明结果的应用.
Institute of Scientific and Technical Information of China (English)
孙建设; 叶留青
2006-01-01
In this article,the authors discuss the optimal conditions of the linear fractional programming problem and prove that a locally optional solution is a globally optional solution and the locally optimal solution can be attained at a basic feasible solution with constraint condition.
Institute of Scientific and Technical Information of China (English)
杨朝强
2013-01-01
利用混合分数布朗运动的Itó公式和复合泊松过程驱动的随机微分方程,建立了一类混合跳-扩散分数布朗运动环境下的价格模型,在Merton假设条件下对其随机微分方程的Cauchy初值问题采用迭代法作了估计,得到了混合跳-扩散模型下的欧式看跌期权定价的Merton公式,从而给出了混合跳-扩散分数布朗运动欧式浮动履约价的看涨回望期权和看跌回望期权定价公式.%The mixed jump-diffusion fractional Brownian motion model under the Itó formula and fractional diffusion process with non-homogeneous Poisson process was proposed.By using the iterative method,the Cauchy initial problem of stochastic differential equations were estimated under the conditions of Merton assumptions.Then the pricing Merton-formula of European option that meets the pricing model for the European floating strike price of the lookback option was obtained.Finally the pricing formulas of floating strike lookback call option and lookback put option were proofed.
Generalized Moisil-Théodoresco Systems and Cauchy Integral Decompositions
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Ricardo Abreu Blaya
2008-01-01
Full Text Available Let ℝ0,m+1(s be the space of s-vectors (0≤s≤m+1 in the Clifford algebra ℝ0,m+1 constructed over the quadratic vector space ℝ0,m+1, let r,p,q∈ℕ with 0≤r≤m+1, 0≤p≤q, and r+2q≤m+1, and let ℝ0,m+1(r,p,q=∑j=pq⨁ ℝ0,m+1(r+2j. Then, an ℝ0,m+1(r,p,q-valued smooth function W defined in an open subset Ω⊂ℝm+1 is said to satisfy the generalized Moisil-Théodoresco system of type (r,p,q if ∂xW=0 in Ω, where ∂x is the Dirac operator in ℝm+1. A structure theorem is proved for such functions, based on the construction of conjugate harmonic pairs. Furthermore, if Ω is bounded with boundary Γ, where Γ is an Ahlfors-David regular surface, and if W is a ℝ0,m+1(r,p,q-valued Hölder continuous function on Γ, then necessary and sufficient conditions are given under which W admits on Γ a Cauchy integral decomposition W=W++W−.
Poisson kernel and Cauchy formula of a non-symmetric transitive domain
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In 1965, Lu Yu-Qian discovered that the Poisson kernel of the homogenous domain S m,p,q={Z∈Cm×m, Z1∈Cm×p,Z2 ∈Cq×m|2i1( Z-Z+)-Z1Z1′-Z2′Z2>0} does not satisfy the Laplace-Beltrami equation associated with the Bergman metric when S m,p,q is not symmetric. However the map T0:Z→Z, Z1→Z1 , Z2→Z2 transforms S m,p,q into a domain S I (m, m + p + q) which can be mapped by the Cayley transformation into the classical domains R I (m, m + p + q). The pull back of the Bergman metric of R I (m, m + p + q) to S m,p,q is a Riemann metric ds 2 which is not a Khler metric and even not a Hermitian metric in general. It is proved that the Laplace-Beltrami operator associated with the metric ds 2 when it acts on the Poisson kernel of S m,p,q equals 0. Consequently, the Cauchy formula of S m,p,q can be obtained from the Poisson formula.
Generalized Cauchy model of sea level fluctuations with long-range dependence
Li, Ming; Li, Jia-Yue
2017-10-01
This article suggests the contributions with two highlights. One is to propose a novel model of sea level fluctuations (sea level for short), which is called the generalized Cauchy (GC) process. It provides a new outlook for the description of local and global behaviors of sea level from a view of fractal in that the fractal dimension D that measures the local behavior of sea level and the Hurst parameter H which characterizes the global behavior of sea level are independent of each other. The other is to show that sea level appears multi-fractal in both spatial and time. Such a meaning of multi-fractal is new in the sense that a pair of fractal parameters (D, H) of sea level is varying with measurement sites and time. This research exhibits that the ranges of D and H of sea level, in general, are 1 ≤ D 0 . 96 for all data records at all measurement sites, implying that strong LRD may be a general phenomenon of sea level. On the other side, regarding with the local behavior, we will reveal that there appears D = 1 or D ≈ 1 for data records at a few stations and at some time, but D > 0 . 96 at most stations and at most time, meaning that sea level may appear highly local irregularity more frequently than weak local one.
Multipoint Singular Boundary-Value Problem for Systems of Nonlinear Differential Equations
Directory of Open Access Journals (Sweden)
Zdeněk Šmarda
2009-01-01
Full Text Available A singular Cauchy-Nicoletti problem for a system of nonlinear ordinary differential equations is considered. With the aid of combination of Ważewski's topological method and Schauder's principle, the theorem concerning the existence of a solution of this problem (having the graph in a prescribed domain is proved.
Institute of Scientific and Technical Information of China (English)
高浩; 吴冬梅; 丁洁
2012-01-01
Comparing with the other optimization algorithms, particle swarm optimization (PSO)has the merits of fast convergence rata and easy application, but the results of research have shown that the PSO has the shortcoming that it is easily be trapped into local optimum. A quantum-behaved particle swarm optimization with Cauchy mutation (CQPSO) is presented. The results of experiment show that the new algorithm not only accelerate the global search ability but also accelerate the convergence rate of the PSO, so it can be applied in engineering optimization problems.%和其他优化算法相比，粒子群算法有着简单易实现以及寻优结果快的优点，但研究结果表明标准粒子群算法在优化过程中存在着易于陷入最小的缺陷。文章提出了一种基于Cauchy策略的量子-粒子群算法。标准测试函数的仿真结果表明，新的算法不仅能够提高算法的全局搜索能力，而且能够加快算法的寻优速度，能够应用在实际工程中的函数优化问题。
Energy Technology Data Exchange (ETDEWEB)
Eliazar, Iddo I., E-mail: eliazar@post.tau.ac.il [Holon Institute of Technology, P.O. Box 305, Holon 58102 (Israel); Shlesinger, Michael F., E-mail: mike.shlesinger@navy.mil [Office of Naval Research, Code 30, 875 N. Randolph St., Arlington, VA 22203 (United States)
2013-06-10
Brownian motion is the archetypal model for random transport processes in science and engineering. Brownian motion displays neither wild fluctuations (the “Noah effect”), nor long-range correlations (the “Joseph effect”). The quintessential model for processes displaying the Noah effect is Lévy motion, the quintessential model for processes displaying the Joseph effect is fractional Brownian motion, and the prototypical model for processes displaying both the Noah and Joseph effects is fractional Lévy motion. In this paper we review these four random-motion models–henceforth termed “fractional motions” –via a unified physical setting that is based on Langevin’s equation, the Einstein–Smoluchowski paradigm, and stochastic scaling limits. The unified setting explains the universal macroscopic emergence of fractional motions, and predicts–according to microscopic-level details–which of the four fractional motions will emerge on the macroscopic level. The statistical properties of fractional motions are classified and parametrized by two exponents—a “Noah exponent” governing their fluctuations, and a “Joseph exponent” governing their dispersions and correlations. This self-contained review provides a concise and cohesive introduction to fractional motions.
Institute of Scientific and Technical Information of China (English)
白定勇
2014-01-01
考虑Riemann-Liouville分数导数意义下的分数变分问题。首先，对于这类分数变分计算，证明了与古典Du Bois-Reymond 引理相对应的结果。然后，应用该结果建立了分数变分泛函的Euler必要条件。最后，讨论了全局极值问题，得到了一些全局极值存在的充分必要条件。%The paper concerns with fractional variational problems in terms of the Riemann-Liouville fractional derivative.First, for such kinds of fractional variational calculus , we prove a counterpart of the Du Bois-Rey-mond lemma in the classical calculus of variations .Then, this result is applied to establish the Euler necessary conditions on fractional variational functionals .Finally, we discuss the global minimum problems and obtain some sufficient and necessary conditions on the existence of global minimum .
Fractional calculus in bioengineering.
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
Energy Technology Data Exchange (ETDEWEB)
Hadermann, A. F.
1985-04-09
Soluble polymers are fractionated according to molecular weight by cryogenically comminuting the polymer and introducing the polymer particles, while still in the active state induced by cryogenic grinding, into a liquid having a solvent power selected to produce a coacervate fraction containing high molecular weight polymer species and a dilute polymer solution containing lower molecular weight polymer species. The coacervate may be physically separated from the solution and finds use in the production of antimisting jet fuels and the like.
基于高阶Cauchy-Born准则的单壁碳纳米管本构模型%A higher order Cauchy-Born rule based constitutive model of SWNT
Institute of Scientific and Technical Information of China (English)
郭旭; 王晋宝; 张洪武
2005-01-01
提出了一种基于高阶Cauchy-Born准则建立单壁碳纳米管本构模型的方法.通过引入高阶变形梯度,合理地修正了传统Cauchy-Born准则在描述纳米管变形几何关系时所存在的缺陷.利用原子间相互作用势以及能量等效原理,得到了基于广义连续介质模型的单壁碳纳米管的本构关系.由此得到的本构参数不仅与变形梯度张量F,而且与其梯度 F相关,因此是一种广义连续介质模型.利用这样的本构模型,本文还对单壁碳纳米管的杨氏模量进行了预测,并与采用其他方法得到的结果进行了对比,从而证实了所提出方法的有效性.
LogCauchy, log-sech and lognormal distributions of species abundances in forest communities
Yin, Z.-Y.; Peng, S.-L.; Ren, H.; Guo, Q.; Chen, Z.-H.
2005-01-01
Species-abundance (SA) pattern is one of the most fundamental aspects of biological community structure, providing important information regarding species richness, species-area relation and succession. To better describe the SA distribution (SAD) in a community, based on the widely used lognormal (LN) distribution model with exp(-x2) roll-off on Preston's octave scale, this study proposed two additional models, logCauchy (LC) and log-sech (LS), respectively with roll-offs of simple x-2 and e-x. The estimation of the theoretical total number of species in the whole community, S*, including very rare species not yet collected in sample, was derived from the left-truncation of each distribution. We fitted these three models by Levenberg-Marquardt nonlinear regression and measured the model fit to the data using coefficient of determination of regression, parameters' t-test and distribution's Kolmogorov-Smirnov (KS) test. Examining the SA data from six forest communities (five in lower subtropics and one in tropics), we found that: (1) on a log scale, all three models that are bell-shaped and left-truncated statistically adequately fitted the observed SADs, and the LC and LS did better than the LN; (2) from each model and for each community the S* values estimated by the integral and summation methods were almost equal, allowing us to estimate S* using a simple integral formula and to estimate its asymptotic confidence internals by regression of a transformed model containing it; (3) following the order of LC, LS, and LN, the fitted distributions became lower in the peak, less concave in the side, and shorter in the tail, and overall the LC tended to overestimate, the LN tended to underestimate, while the LS was intermediate but slightly tended to underestimate, the observed SADs (particularly the number of common species in the right tail); (4) the six communities had some similar structural properties such as following similar distribution models, having a common
Cavalcanti, Marcelo M.; Corrêa, Wellington J.; Domingos Cavalcanti, Valéria N.; Tebou, Louis
2017-02-01
In this paper, we study the existence at the H1-level as well as the stability for the damped defocusing Schrödinger equation in Rd. The considered damping coefficient is time-dependent and may vanish at infinity. To prove the existence, we employ the method devised by Özsarı, Kalantarov and Lasiecka [27], which is based on monotone operators theory. In particular, when d = 1 or d = 2, we obtain the uniqueness. Decay estimates for the L2-level and (H1 ∩L p + 2)-level energies are established with the help of direct multipliers method, coupled with refined energy estimates and a lower semi-continuity argument.
Rahali, Radouane
2013-03-01
In this paper, we investigate the decay property of a Timoshenko system in thermoelasticity of type III in the whole space where the heat conduction is given by the Green and Naghdi theory. Surprisingly, we show that the coupling of the Timoshenko system with the heat conduction of Green and Naghdi\\'s theory slows down the decay of the solution. In fact we show that the L-2-norm of the solution decays like (1 + t)(-1/8), while in the case of the coupling of the Timoshenko system with the Fourier or Cattaneo heat conduction, the decay rate is of the form (1 + t)(-1/4) [25]. We point out that the decay rate of (1 + t)(-1/8) has been obtained provided that the initial data are in L-1 (R) boolean AND H-s (R); (s >= 2). If the wave speeds of the fi rst two equations are di ff erent, then the decay rate of the solution is of regularity-loss type, that is in this case the previous decay rate can be obtained only under an additional regularity assumption on the initial data. In addition, by restricting the initial data to be in H-s (R) boolean AND L-1,L-gamma (R) with gamma is an element of [0; 1], we can derive faster decay estimates with the decay rate improvement by a factor of t(-gamma/4).
Understanding Multiplication of Fractions.
Sweetland, Robert D.
1984-01-01
Discussed the use of Cuisenaire rods in teaching the multiplication of fractions. Considers whole number times proper fraction, proper fraction multiplied by proper fraction, mixed number times proper fraction, and mixed fraction multiplied by mixed fractions. (JN)
THE INTERACTION PROBLEM BETWEEN THE ELASTIC LINE INCLUSIONS
Institute of Scientific and Technical Information of China (English)
陶昉敏; 张明焕; 汤任基
2002-01-01
Using the engineering model of elastic line inclusion and the basic solutions of a single inclusion, the interaction problem between line inclusions in an elastic solid was investigated. A set of standard Cauchy-type singular equations of the problem was presented. The stress intensity factors at points of inclusions and the interface stresses of two sides of the inclusion were calculated. Several numerical examples were given. The results could be regarded as a reference to engineering.
Institute of Scientific and Technical Information of China (English)
A.Kanoune; N.Mehidi
2008-01-01
We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Q of RN with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy-ventcel type.Under suitable and natural assumptions on the nonlinearity,we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity.Subcriticality means,roughly speaking,that the nonlinearity grows at infinity at most as a power P＜5.The results obtained in R3 and RN by B.Dehman,G.Lebeau and E.Zuazua on the inequalities of the classical energv(which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball)and on Strichartz's estimates,allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of RN with a subcritical nonlinearity on the domain and its boundary,and conditions on the boundary of Cauchy-Vlentcel type.
Elementary proof of Cauchy integral theorem%柯西积分定理的初等证明
Institute of Scientific and Technical Information of China (English)
王信松; 张节松
2011-01-01
利用数学分析的知识构造一个简单的恒同逼近函数,由此用分析中的逼近思想,成功地用满足柯西-黎曼条件的连续可微的函数逼近一般的可微函数,给出了柯西积分定理的一个初等证明,克服了复变函数论中这一关键性定理证明繁琐或者超纲的困难.%In this paper, a simple identity approximation function was constructed by using the knowledge of mathematic analysis, thereby, applying the approximation idea in analysis, we successfully obtained that ordinary differential functions were well approximated by continuously differentiable functions satisfying the Cauchy-Riemann equations, and an elementary proof of Cauchy integral theorem was presented, which overcame some troublesome or over-syllabus difficulties in the proof of the theorem in the text of complex variable functions, and was also beneficial to the teaching.
Hart, Kathleen; Kerslake, Daphne
The Concepts in Secondary Mathematics and Science (CSMS) and Strategies and Errors in Secondary Mathematics (SESM) research projects based at Chelsa College, England, have shown the marked reluctance of secondary school students to use fractions when solving mathematical problems, even though they have been taught the topic for a number of years.…
Stephens, Ana C.; Bottge, Brian A.; Rueda, Enrique
2009-01-01
This article describes a technology-based and hands-on instructional intervention designed to advance middle school students' understandings of fractions. This problem-solving experience is based on the principles of Enhanced Anchored Instruction (EAI) and proved instructionally worthwhile and motivating to teachers and students in both inclusive …
Institute of Scientific and Technical Information of China (English)
李耀红; 张海燕
2014-01-01
研究了一类具有Riemann-Liouville分数阶积分条件的新分数阶微分方程边值问题，其非线性项包含Caputo型分数阶导数。将该问题转化为等价的积分方程，应用Leray-Schauder不动点定理结合一个范数形式的新不等式，获得了解的存在性充分条件，推广和改进了已有的结果，并给出了应用实例。%A class of boundary value problem of fractional differential equation with Riemann-Liouville fractional integral conditions is investigated, which involves the Caputo fractional derivative in nonlinear terms and can be reduced to the equivalent integral equation. By using Leray-Schauder fixed point theory combined with a new inequality of norm form, some sufficient conditions on the exitence of solution for boundary value problem are established. Some known results are extended and improved. An example is given to illustrate the application of the result.
An improved fractional divider for fractional-N frequency synthesizers
Zhang, Wei; Liu, Yang; Zhou, Yongqi
2009-07-01
This paper presents an improved fractional divider used in 1.8~2GHz fractional-N frequency synthesizers. A new clock setting for delta-sigma modulator (DSM) is proposed to prevent the potential problems of traditional fractional dividers: the DSM output would be wrongly loaded and the action of DSM circuit may affect the phase-detection of phase-frequency-detector (PFD). Simulation result shows the effectiveness of this improvement.
Using Number Sense to Compare Fractions
Bray, Wendy S.; Abreu-Sanchez, Laura
2010-01-01
One mathematical focus for third graders is to develop deep understanding of fractions and fraction equivalence, including comparing fractions through use of models and reasoning strategies. Before reading further, consider how you might solve the following problem: Which fraction is greater, 14/24 or 17/36? The initial impulse of many adults is…
The Richness of Children's Fraction Strategies
Kent, Laura B.; Empson, Susan B.; Nielsen, Lynne
2015-01-01
In this article, the authors discuss a special type of multiplication-and-division-of-fractions problem that elementary school teachers can use to promote children's understanding of fractional quantities and their relationships. These problems are accessible to students working at different levels of fraction understanding, and they can be solved…
Institute of Scientific and Technical Information of China (English)
郭秀清; 王旭焕
2013-01-01
讨论了分数阶Langevin方程的非局部狄利克雷边值问题,利用Leray-Schauder's和压缩映像原理,分别得到了方程的解的存在及唯一性.%In this paper,a new type of Langevin equation with fractional orders with Nonlocal Dirichlet Boundary Value Problems is considered.By using Leray-Schauder's fixed point theorem and Banach's contraction mapping principle,we obtain the existence and uniqueness results of the solution.
Uniqueness of Bounded Variation Solutions of Cauchy Systems Involving Parameter%含参量Cauchy系统有界变差解的唯一性
Institute of Scientific and Technical Information of China (English)
马学敏; 李宝麟; 林长伟
2013-01-01
By using Henstock-Kurzweil integral, uniqueness of bounded variation solutions of Cauchy systems involving parameter is discussed in the condition of generalization of ω.%利用Henstock-Kurzweil积分，在ω较弱的条件下，讨论了含参量Cauchy系统有界变差解的唯一性。
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.
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.
Financial Planning with Fractional Goals
M.H. Goedhart; J. 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 mult
Financial Planning with Fractional Goals
Goedhart, M.H.; 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.
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…
Gerr, S
1982-01-01
An attempt is made to draw an analogy between contour drawing and a particular mathematical theorem. The analogy is seen to depend on the fact that both methods use definite values along a contour to imply a totality of values within the contour; thus, the use of a part to suggest the whole, by way of a hypothetical 'gestalt-like integration' in the case of the art contour, and the usual process of mathematical integration in the case of Cauchy's formula. Examples illustrating the analogy are drawn from a wide range of artistic work: a modern American drawing, a Cro-Magnon cave painting, and two Chinese works. The traditional Chinese philosophy of painting is invoked in support of the analogy because of its explicit emphasis on the primacy of outline drawing in Chinese painting. Some speculations are offered on further development and application of the analogy.
Some improperly posed problems of mathematical physics
Lavrentiev, M M
1967-01-01
This monograph deals with the problems of mathematical physics which are improperly posed in the sense of Hadamard. The first part covers various approaches to the formulation of improperly posed problems. These approaches are illustrated by the example of the classical improperly posed Cauchy problem for the Laplace equation. The second part deals with a number of problems of analytic continuations of analytic and harmonic functions. The third part is concerned with the investigation of the so-called inverse problems for differential equations in which it is required to determine a dif ferential equation from a certain family of its solutions. Novosibirsk June, 1967 M. M. LAVRENTIEV Table of Contents Chapter I Formu1ation of some Improperly Posed Problems of Mathematic:al Physics § 1 Improperly Posed Problems in Metric Spaces. . . . . . . . . § 2 A Probability Approach to Improperly Posed Problems. . . 8 Chapter II Analytic Continuation § 1 Analytic Continuation of a Function of One Complex Variable fro...
Tatiana Odzijewicz; Malinowska, Agnieszka B.; Torres, Delfim F. M.
2012-01-01
We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed. Copyright 2012 Tatiana Odzijewicz et al.
A multi-step method for partial eigenvalue assignment problem of high order control systems
Liu, Hao; Xu, Jiajia
2017-09-01
In this paper, we consider the partial eigenvalue assignment problem of high order control systems. Based on the orthogonality relations, we propose a new method for solving this problem by which the undesired eigenvalues are moved to desired values and keep the remaining eigenvalues unchanged. Using the inverse of Cauchy matrix, we give the solvable condition and the explicit solutions to this problem. Numerical examples show that our method is effective.
On $L^1$ extremal problem for entire functions
Yuditskii, Peter
2012-01-01
We generalized the Korkin-Zolotarev theorem to the case of entire functions having the smallest $L^1$ norm on a system of intervals $E$. If $\\bbC\\setminus E$ is a domain of Widom type with the Direct Cauchy Theorem we give an explicit formula for the minimal deviation. Important relations between the problem and the theory of canonical systems with reflectionless resolvent functions are shown.
Fractional conservation laws in optimal control theory
Frederico, Gastao S F
2007-01-01
Using the recent formulation of Noether's theorem for the problems of the calculus of variations with fractional derivatives, the Lagrange multiplier technique, and the fractional Euler-Lagrange equations, we prove a Noether-like theorem to the more general context of the fractional optimal control. As a corollary, it follows that in the fractional case the autonomous Hamiltonian does not define anymore a conservation law. Instead, it is proved that the fractional conservation law adds to the Hamiltonian a new term which depends on the fractional-order of differentiation, the generalized momentum, and the fractional derivative of the state variable.
Institute of Scientific and Technical Information of China (English)
丁金凤; 张毅
2014-01-01
基于El-Nabulsi动力学模型，提出并研究了Birkhoff系统基于按指数律拓展的分数阶积分的变分问题的Noether对称性与守恒量。基于按指数律拓展的分数阶积分的El-Nabulsi-Pfaff-Birkhoff变分问题，建立起与之对应的El-Nabulsi-Birkhoff方程；基于El-Nabulsi-Pfaff作用量在无限小变换下的不变性，给出系统的Noether对称变换和Noether准对称变换的定义和判据。该研究建立Birkhoff系统基于按指数律拓展的分数阶积分的变分问题的Noether定理，揭示了该模型下系统的Noether对称性和守恒量之间的关系。文末举例说明结果的应用。%Based on El-Nabulsi dynamical model,the Noether symmetries and the conserved quantities for the variational problem of Birkhoffian system from extended exponentially fractional integral are pres-ented and studied.Firstly,the El-Nabulsi-Pfaff-Birkhoff variational problem from extended exponentially fractional integral is presented,then the corresponding El-Nabulsi-Birkhoff equations are derived.Sec-ondly,the definitions and the criteria of the Noether symmetric transformations and the Noether quasi-symmetric transformations of the system are given,which are based on the invariance of El-Nabulsi-Pfaff action under the infinitesimal transformations of group.Finally,the Noether theorem for the variational problem of Birkhoffian system from extended exponentially fractional integral is established,which reveals the inner relationship between a Noether symmetry and a conserved quantity.An example is given to il-lustrate the application of the results.
A Note on Fractional Differential Equations with Fractional Separated Boundary Conditions
Directory of Open Access Journals (Sweden)
Bashir Ahmad
2012-01-01
Full Text Available We consider a new class of boundary value problems of nonlinear fractional differential equations with fractional separated boundary conditions. A connection between classical separated and fractional separated boundary conditions is developed. Some new existence and uniqueness results are obtained for this class of problems by using standard fixed point theorems. Some illustrative examples are also discussed.
Fractional complex transforms for fractional differential equations
National Research Council Canada - National Science Library
Ibrahim, Rabha W
2012-01-01
The fractional complex transform is employed to convert fractional differential equations analytically in the sense of the Srivastava-Owa fractional operator and its generalization in the unit disk...
Fractional complex transform for fractional differential equations
National Research Council Canada - National Science Library
Lİ, Zheng Biao; HE, Ji Huan
2010-01-01
Fractional complex transform is proposed to convert fractional differential equations into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily...
Feller Fractional Advection-Dispersion and Levy Distribution%Feller算子下的分数阶对流-弥散过程与Levy分布
Institute of Scientific and Technical Information of China (English)
朱波; 韩宝燕
2011-01-01
The paper discusses the fractional Levy-Feller diffusion equation,draws the Green function with Cauchy problem by means of Fourier transform. Here Green function is represented by Levy stable probability densities function with index and skewness θ. It turns out that the presence of asymmetry (θ →0) plays a fundamental role: it produces shift of the maximum concentration location and long tail. When α→ 2,θ → 0,the analytical solution is same to the solution of the classical advection-dispersion equation.%建立了Levy-Feller分数阶扩散方程,利用Fourier变换及其逆变换,给出其Cauchy问题的带有分数阶导数阶数α(1＜α≤2)和扭曲参数θ(｜θ｜≤α-2)的Levy平稳概率密度函数表示的Green函数解.结果表明,在非均匀(θ≠0)扩散过程中,主要由扭曲参数导致了最大浓度位置的偏移和拖尾现象;当α→2,即θ→0时,问题的解与相应整数阶对流-弥散方程的解一致.
Sucesión convergente y sucesión de Cauchy : equivalencia matemática y equivalencia fenomenológica
Claros Mellado, Francisco Javier; Sánchez Compaña, María Teresa; Coriat Benarroch, Moisés
2013-01-01
Esta investigación está enmarcada por la Fenomenología, el Pensamiento Matemático Avanzado y los Sistemas de Representación; en ella estudiamos una equivalencia fenomenológica, junto a la conocida equivalencia matemática, entre sucesión convergente y sucesión de Cauchy. Enunciamos dos fenómenos organizados por definiciones de límite finito de una sucesión y de sucesión de Cauchy; con el apoyo de libros de texto de secundaria, reconocemos en los libros de texto españoles dos fenómenos organiza...
Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes
Orsingher, Enzo; Polito, Federico
2012-08-01
In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes N α ( t), N β ( t), t>0, we have that N_{α}(N_{β}(t)) stackrel{d}{=} sum_{j=1}^{N_{β}(t)} Xj, where the X j s are Poisson random variables. We present a series of similar cases, where the outer process is Poisson with different inner processes. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns compositions of the form N_{α}(tauk^{ν}), ν∈(0,1], where tauk^{ν} is the inverse of the fractional Poisson process, and we show how these compositions can be represented as random sums. Furthermore we study compositions of the form Θ( N( t)), t>0, which can be represented as random products. The last section is devoted to studying continued fractions of Cauchy random variables with a Poisson number of levels. We evaluate the exact distribution and derive the scale parameter in terms of ratios of Fibonacci numbers.
Nal, P L
2002-01-01
We consider the asymptotic stability and the boundedness with probability one of solutions of linear lto stochastic differential equations not reduced to the Cauchy form and give some numerical examples to show that our sufficient conditions for the asymptotic stability with probability one of solutions are more general and more effective than those of Korenevskij and Mitropoloshij. Moreover, our results can also be applied to the case when the unperturbed linear deterministic system is not assumed to be stable.
Directory of Open Access Journals (Sweden)
M. G. Crandall
1999-07-01
Full Text Available We study existence of continuous weak (viscosity solutions of Dirichlet and Cauchy-Dirichlet problems for fully nonlinear uniformly elliptic and parabolic equations. Two types of results are obtained in contexts where uniqueness of solutions fails or is unknown. For equations with merely measurable coefficients we prove solvability of the problem, while in the continuous case we construct maximal and minimal solutions. Necessary barriers on external cones are also constructed.
Zhang, L. W.; Ademiloye, A. S.; Liew, K. M.
In normal physiological and healthy conditions, red blood cells (RBCs) deform readily as they pass through the microcapillaries and the spleen, however, upon invasion by the malaria parasite, the host RBC membrane begins to lose their deformability. In spite of the progress in understanding malaria pathogenesis, the primary mechanism responsible for the loss of deformability remains unclear. In this paper, we examine the effects of Plasmodium falciparum infection and maturation on the deformability of parasitized or infected red blood cells (iRBCs) by means of a three-dimensional (3D) multiscale red blood cell (RBC) framework. This multiscale framework is developed based on the Cauchy-Born rule and the meshfree IMLS-Ritz method. The atomistic scale strain energy density function of the RBC membrane was computed using a selected representative cell based on the membrane spectrin network. The results obtained from our numerical simulations affirm that the presence of malaria infection significantly increases the rigidity of RBC membrane. It was observed that in the trophozoite and schizont infection stages, biconcave cell geometry leads to better prediction than nearly spherical geometry in comparison with experimental studies. Furthermore, we confirm that increase in temperature also results to increased stiffening of the cell membrane. Lastly, the observed decrease in the deformability of iRBC membrane may be primarily due to the structural remodeling and changes in the microstructure of the membrane rather than the change in cell shape.
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.
On explicit and numerical solvability of parabolic initial-boundary value problems
Directory of Open Access Journals (Sweden)
Lepsky Olga
2006-01-01
Full Text Available A homogeneous boundary condition is constructed for the parabolic equation in an arbitrary cylindrical domain ( being a bounded domain, and being the identity operator and the Laplacian which generates an initial-boundary value problem with an explicit formula of the solution . In the paper, the result is obtained not just for the operator , but also for an arbitrary parabolic differential operator , where is an elliptic operator in of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation in is reduced to an integral equation in a thin lateral boundary layer. An approximate solution to the integral equation generates a rather simple numerical algorithm called boundary layer element method which solves the 3D Cauchy-Dirichlet problem (with three spatial variables.
Finding Solutions to the Picard Boundary Value Problem via Homotopy Method%用同伦方法求Picard边值问题的解
Institute of Scientific and Technical Information of China (English)
李兰; 徐旭
2008-01-01
This paper deals with the problem of finding solutions to the Picard boundary problem. In our approach, by means of the homotopy method, the equation considered is linked to a simpler equation by introducing a parameter. We first find the solutions of the simpler equation, and give a priori estimates of" the equa tion we considered, and then one can obtain the solutions of Picard boundary problem by following the path of solutions of Cauchy problem.
Quantization of area for event and Cauchy horizons of the Kerr-Newman black hole
Visser, Matt
2012-01-01
Based on various string theoretic constructions, there have been repeated suggestions that the areas of black hole event horizons should be quantized in a quite specific manner, involving linear combinations of square roots of natural numbers. It is important to realise the significant physical limitations of such proposals when one attempts to extend them outside their original framework. Specifically, in their most natural and direct physical interpretations, these specific proposals for horizon areas fail for the ordinary Kerr-Newman black holes in (3+1) dimensions, essentially because the fine structure constant is not an integer. A more baroque interpretation involves asserting the fine structure constant is the square root of a rational number; but such a proposal has its own problems. Insofar as one takes (3+1) general relativity (plus the usual quantization of angular momentum and electric charge) as being paramount, the known explicitly calculable spectra of horizon areas for the physically compellin...
Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues
Davit, Y.
2013-04-30
The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to continuum methods based on partial differential equations, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for noncrystalline materials, it may still be used as a first-order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to two-dimensional cellular-scale models by assessing the mechanical behavior of model biological tissues, including crystalline (honeycomb) and noncrystalline reference states. The numerical procedure involves applying an affine deformation to the boundary cells and computing the quasistatic position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For center-based cell models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based cell models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete and continuous modeling, adaptation of atom-to-continuum techniques to biological
Fractional Laplace Transforms - A Perspective
Treumann, R A
2014-01-01
A 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.
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.
On the Definitions of Nabla Fractional Operators
Directory of Open Access Journals (Sweden)
Thabet Abdeljawad
2012-01-01
properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced.
Tenreiro Machado, J. A.
2015-08-01
This paper addresses the matrix representation of dynamical systems in the perspective of fractional calculus. Fractional elements and fractional systems are interpreted under the light of the classical Cole-Cole, Davidson-Cole, and Havriliak-Negami heuristic models. Numerical simulations for an electrical circuit enlighten the results for matrix based models and high fractional orders. The conclusions clarify the distinction between fractional elements and fractional systems.
Energy Technology Data Exchange (ETDEWEB)
Mehta, Sheetal, E-mail: smehta-29@yahoo.com; Das, Kallol, E-mail: smehta-29@yahoo.com; Keller, Jag Mohan, E-mail: smehta-29@yahoo.com [Department of Physics, St. Aloysius College (Autonomous), Jabalpur-482001, Madhya Pradesh, India and Department of Physics and Electronics, Rani Durgawati University, Jabalpur-482001, Madhya Pradesh (India)
2014-04-24
Poly (methyl methacrylate) / Polystyrene and iodine / selenium hybrid matrixes have been prepared and characterized. Refractive index measurements were done at 390, 535, 590, 635 nm wavelengths. Abbe's number and Cauchy's constants of the iodine / selenium doped poly (methylmethacrylate) and polystyrene samples are being reported. The results also showed that the refractive index of the composite varies non-monotonically with the doping concentration at low iodine concentration or in the region of nanoparticles formation and is also dependent on thermal annealing.
关于柯西sn-对称空间的一个注记%A Note on Cauchy sn-Symmetric Spaces
Institute of Scientific and Technical Information of China (English)
陈内萍
2013-01-01
It is proved that a space is a Cauchy sn-symmetric space if and only if it has a sn-devel-opment consisting of cs-covers ( or sn-covers) if and only if it is a sequence-covering π-image of a metric space .%证明了一个空间是柯西sn-对称空间当且仅当它有一个由cs-复盖或sn-复盖组成sn-展开当且仅当它是度量空间的序列复盖π映射。
Stability of the Cauchy-Jensen Functional Equation in CÃ¢ÂˆÂ—-Algebras: A Fixed Point Approach
Directory of Open Access Journals (Sweden)
Jong Su An
2008-06-01
Full Text Available we prove the Hyers-Ulam-Rassias stability of CÃ¢ÂˆÂ—-algebra homomorphisms and of generalized derivations on CÃ¢ÂˆÂ—-algebras for the following Cauchy-Jensen functional equation 2f((x+y/2+z=f(x+f(y+2f(z, which was introduced and investigated by Baak (2006. The concept of Hyers-Ulam-Rassias stability originated from the stability theorem of Th. M. Rassias that appeared in (1978.
关于Cauchy型积分与Fourier积分的研究%On the Investigation with Cauchy Type Integral and Fourier Integral
Institute of Scientific and Technical Information of China (English)
李平润
2014-01-01
利用复分析中推广的Cauchy留数定理与奇异积分方程中的Plemelj 公式，首次给出了 Cauchy型积分与 Fourier积分之间的关系，并得出了单侧的 Fourier积分的性质，然后给予证明。%In this paper,by using the Residue theorem and Plemelj formula,the relation of Cauchy type integral and Fourier integral is given,and the properties of the one-sided Fourier integral are proved.
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.
Directory of Open Access Journals (Sweden)
Sajewski Łukasz
2017-03-01
Full Text Available Reachability and minimum energy control of descriptor fractional discrete-time linear systems with different fractional orders are addressed. Using the Weierstrass–Kronecker decomposition theorem of the regular pencil, a solution to the state equation of descriptor fractional discrete-time linear systems with different fractional orders is given. The reachability condition of this class of systems is presented and used for solving the minimum energy control problem. The discussion is illustrated with numerical examples.
Approximations of the operator exponential in a periodic diffusion problem with drift
Energy Technology Data Exchange (ETDEWEB)
Pastukhova, Svetlana E [Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University), Moscow (Russian Federation)
2013-02-28
A Cauchy problem for a parabolic diffusion equation with 1-periodic coefficients containing first order terms is studied. For the corresponding semigroup we construct approximations in the L{sup 2}-operator norm on sections t=const of order O(t{sup -m/2}) as t{yields}{infinity} for m=1 or m=2. The spectral method based on the Bloch representation of an operator with periodic coefficients is used. Bibliography: 25 titles.
Directory of Open Access Journals (Sweden)
Domoshnitsky Alexander
2009-01-01
Full Text Available We obtain the maximum principles for the first-order neutral functional differential equation where , and are linear continuous operators, and are positive operators, is the space of continuous functions, and is the space of essentially bounded functions defined on . New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.
Initial data problems for the two-component Camassa-Holm system
Directory of Open Access Journals (Sweden)
Xiaohuan Wang
2014-06-01
Full Text Available This article concerns the study of some properties of the two-component Camassa-Holm system. By constructing two sequences of solutions of the two-component Camassa-Holm system, we prove that the solution map of the Cauchy problem of the two-component Camassa-Holm system is not uniformly continuous in $H^s(\\mathbb{R}$, $s>5/2$.
Variable Order Fractional Variational Calculus for Double Integrals
Odzijewicz, Tatiana; Malinowska, Agnieszka B.; Torres, Delfim F. M.
2012-01-01
We introduce three types of partial fractional operators of variable order. An integration by parts formula for partial fractional integrals of variable order and an extension of Green's theorem are proved. These results allow us to obtain a fractional Euler-Lagrange necessary optimality condition for variable order two-dimensional fractional variational problems.
Variable Order Fractional Variational Calculus for Double Integrals
Odzijewicz, Tatiana; Malinowska, Agnieszka B.; Torres, Delfim F. M.
2012-01-01
We introduce three types of partial fractional operators of variable order. An integration by parts formula for partial fractional integrals of variable order and an extension of Green's theorem are proved. These results allow us to obtain a fractional Euler-Lagrange necessary optimality condition for variable order two-dimensional fractional variational problems.
A Fractional Micro-Macro Model for Crowds of Pedestrians based on Fractional Mean Field Games
Cao, Ke-cai; Stuart, Dan
2016-01-01
Modeling of crowds 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 micro-macro 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.
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.
Directory of Open Access Journals (Sweden)
Jing Zhao
2013-01-01
Full Text Available We study a boundary value problem for fractional equations involving two fractional orders. By means of a fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. In addition, we describe the dynamic behaviors of the fractional Langevin equation by using the G2 algorithm.
Directory of Open Access Journals (Sweden)
Bashir Ahmad
2013-02-01
Full Text Available In this article, we discuss the existence of solutions for a boundary-value problem of integro-differential equations of fractional order with nonlocal fractional boundary conditions by means of some standard tools of fixed point theory. Our problem describes a more general form of fractional stochastic dynamic model for financial asset. An illustrative example is also presented.
Fractional linear systems and electrical circuits
Kaczorek, Tadeusz
2015-01-01
This monograph covers some selected problems of positive and fractional electrical circuits composed of resistors, coils, capacitors and voltage (current) sources. The book consists of 8 chapters, 4 appendices and a list of references. Chapter 1 is devoted to fractional standard and positive continuous-time and discrete-time linear systems without and with delays. In chapter 2 the standard and positive fractional electrical circuits are considered and the fractional electrical circuits in transient states are analyzed. Descriptor linear electrical circuits and their properties are investigated in chapter 3, while chapter 4 is devoted to the stability of fractional standard and positive linear electrical circuits. The reachability, observability and reconstructability of fractional positive electrical circuits and their decoupling zeros are analyzed in chapter 5. The fractional linear electrical circuits with feedbacks are considered in chapter 6. In chapter 7 solutions of minimum energy control for standa...
Gerhardt, Claus
2016-01-01
In a recent paper we quantized the interaction of gravity with a Yang-Mills and Higgs field and obtained as a result a gravitational wave equation in a globally hyperbolic spacetime. Assuming that the Cauchy hypersurfaces are compact we proved a spectral resolution for the wave equation by applying the method of separation of variables. In this paper we extend the results to the case when the Cauchy hypersurfaces are non-compact by considering a Gelfand triplet and applying the nuclear spectral theorem.
Computational experiment on the numerical solution of some inverse problems of mathematical physics
Vasil'ev, V. I.; Kardashevsky, A. M.; Sivtsev, PV
2016-11-01
In this article the computational experiment on the numerical solution of the most popular linear inverse problems for equations of mathematical physics are presented. The discretization of retrospective inverse problem for parabolic equation is performed using difference scheme with non-positive weight multiplier. Similar difference scheme is also used for the numerical solution of Cauchy problem for two-dimensional Laplace equation. The results of computational experiment, performed on model problems with exact solution, including ones with randomly perturbed input data are presented and discussed.
Approximating fractional derivatives through the generalized mean
Tenreiro Machado, J. A.; Galhano, Alexandra M.; Oliveira, Anabela M.; Tar, József K.
2009-11-01
This paper addresses the calculation of fractional order expressions through rational fractions. The article starts by analyzing the techniques adopted in the continuous to discrete time conversion. The problem is re-evaluated in an optimization perspective by tacking advantage of the degree of freedom provided by the generalized mean formula. The results demonstrate the superior performance of the new algorithm.