Rate of Convergence for the Empirical Distribution Function and the Empirical Characteristic Function of Mixing Processes

Ｒａｔｅ　ｏｆ　Ｃｏｎｖｅｒｇｅｎｃｅ　ｆｏｒ　ｔｈｅ　Ｅｍｐｉｒｉｃａｌ　Ｄｉｓｔｒｉｂｕｔｉｏｎ　Ｆｕｎｃｔｉｏｎ　ａｎｄ　ｔｈｅ　Ｅｍｐｉｒｉｃａｌ　Ｃｈａｒａｃｔｅｒｉｓｔｉｃ　Ｆｕｎｃｔｉｏｎ　ｏｆ　Ｍｉｘｉｎｇ　ＰｒｏｃｅｓｓｅｓＨｕＳｈ...     

Mathematics Model for Deciding the Possibility of Crack Existence

Ｍａｔｈｅｍａｔｉｃｓ　Ｍｏｄｅｌ　ｆｏｒ　Ｄｅｃｉｄｉｎｇ　ｔｈｅ　Ｐｏｓｓｉｂｉｌｉｔｙ　ｏｆ　Ｃｒａｃｋ　ＥｘｉｓｔｅｎｃｅＭａｔｈｅｍａｔｉｃｓＭｏｄｅｌｆｏｒＤｅｃｉｄｉｎｇｔｈｅＰｏｓｓｉｂｉｌｉｔｙｏｆＣｒａｃｋＥｘｉｓｔｅｎｃｅ￥Ｗｕ...     

Some New Combinatorial Conditions on the Singular Fibre of a Fibration

Ｓｏｍｅ　Ｎｅｗ　Ｃｏｍｂｉｎａｔｏｒｉａｌ　Ｃｏｎｄｉｔｉｏｎｓ　ｏｎ　ｔｈｅ　Ｓｉｎｇｕｌａｒ　Ｆｉｂｒｅ　ｏｆ　ａ　ＦｉｂｒａｔｉｏｎＴａｎｇＭｉｎｇｙｙｕａｎ（唐明元）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａｎｇｈａｉＮｏｒｍ...     

On a Class of Fractal Interpolation Functions

ＯｎａＣｌａｓｏｆＦｒａｃｔａｌＩｎｔｅｒｐｏｌａｔｉｏｎＦｕｎｃｔｉｏｎｓＱｉａｎＸｉａｏｙｕａｎ（Ｉｎｓｔ．ｏｆＭａｔｈ．Ｓｃｉｓ．，ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ，１１６０２４）Ｋｅｙｗｏｒｄｓｉｔｅｒａｔｅｄｆｕｎ...     

Solutions to the Equation of Non－Newtonian Polytropic Filtration Under Optimal Condition On Initial Values

Ｓｏｌｕｔｉｏｎｓ　ｔｏ　ｔｈｅ　Ｅｑｕａｔｉｏｎ　ｏｆ　Ｎｏｎ－Ｎｅｗｔｏｎｉａｎ　Ｐｏｌｙｔｒｏｐｉｃ　Ｆｉｌｔｒａｔｉｏｎ　Ｕｎｄｅｒ　Ｏｐｔｉｍａｌ　Ｃｏｎｄｉｔｉｏｎ　Ｏｎ　Ｉｎｉｔｉａｌ　ＶａｌｕｅｓＹｕａｎＨｏｎｇｊｕｎ（袁洪君）（Ｄｅ...     

FractalpaternsofFractureinSandwichCompositeMaterialsunderBiaxialTension

ＦｒａｃｔａｌｐａｔｅｒｎｓｏｆＦｒａｃｔｕｒｅｉｎＳａｎｄｗｉｃｈＣｏｍｐｏｓｉｔｅＭａｔｅｒｉａｌｓｕｎｄｅｒＢｉａｘｉａｌＴｅｎｓｉｏｎ１ＪｉｎｇＦＡＮＧ，ＸｕｅｆｅｎｇＹＡＯ＆ＪｉａＱＩ（Ｄｅｐｔ．ｏｆＭｅｃｈａｎｉｃｓ＆Ｅｎｇｎｇ．Ｓｃｉｅ...     

The Convergance Properties of Quasi Hemite－Fejer Interpolation Polynomial on the Disturbance Chebyshev Knot

Ｔｈｅ　Ｃｏｎｖｅｒｇａｎｃｅ　Ｐｒｏｐｅｒｔｉｅｓ　ｏｆ　Ｑｕａｓｉ　Ｈｅｍｉｔｅ－Ｆｅｊｅｒ　Ｉｎｔｅｒｐｏｌａｔｉｏｎ　Ｐｏｌｙｎｏｍｉａｌ　ｏｎ　ｔｈｅ　Ｄｉｓｔｕｒｂａｎｃｅ　Ｃｈｅｂｙｓｈｅｖ　ＫｎｏｔＴｈｅＣｏｎｖｅｒｇａｎｃｅＰｒｏｐ...     

Fractal Model of the Spheroidal Graphit

ＦｒａｃｔａｌＭｏｄｅｌｏｆｔｈｅＳｐｈｅｒｏｉｄａｌＧｒａｐｈｉｔｅＺ．Ｙ．ＨＥａｎｄＫ．Ｚ．ＨＷＡＮＧＤｅｐｔ．ｏｆＥｎｇｉｎｅｅｒｉｎｇＭｅｃｈａｎｉｃｓ，ＴｓｉｎｇｈｕａＵｎｉｖｅｒｓｉｔｙＢｅｉｊｉｎｇ，Ｐ．Ｒ．Ｃｈｉｎａ１０００８４Ａｂｓ...     

On the Irredundant Forms of Fuzzy Switching Functions

ＯｎｔｈｅＩｒｒｅｄｕｎｄａｎｔＦｏｒｍｓｏｆＦｕｚｚｙＳｗｉｔｃｈｉｎＦｕｎｃｔｉｏｎｓTXＯｎｔｈｅＩｒｒｅｄｕｎｄａｎｔＦｏｒｍｓｏｆＦｕｚｚｙＳｗｉｔｃｈｉｎｇＦｕｎｃｔｉｏｎｓＰｅｉＤａｏｗｕ（ＹａｎｃｈｅｎｇＴｅａｃｈｅｒｓＣｏｌｅｇｅ，...     

On the Set of Element Orders of a Finite Group

ＦｏｒａｎｙｇｒｏｕｐＧ，ｄｅｎｏｔｅｂｙπｅ（Ｇ）ｔｈｅｓｅｔｏｆａｌｅｌｅｍｅｎｔｏｒｄｅｒｓｏｆＧ．Ｇｉｖｅｎａｆｉｎｉｔｅ（ｒｅｓｐ．ｉｎｆｉｎｉｔｅ）ｇｒｏｕｐＧ，ｌｅｔｈ（πｅ（Ｇ））ｂｅｔｈｅｎｕｍｂｅｒｏｆｉｓｏｍｏｒｐｈｉｓｍｃｌａ...     

On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives

In this paper, a class of nonlinear fractional order differential impulsive systems with Hadamard derivative is discussed. First, a reasonable concept on the solutions of fractional impulsive Cauchy problems with Hadamard derivative and the corresponding fractional integral equations are established. Second, two fundamental existence results are presented by using standard fixed point methods. Finally, two examples are given to illustrate our theoretical results.     

Correlation structure of time-changed Pearson diffusions

The stochastic solution to diffusion equations with polynomial coefficients is called a Pearson diffusion. If the time derivative is replaced by a distributed order fractional derivative, the stochastic solution is called a distributed order fractional Pearson diffusion. This paper develops a formula for the covariance function of distributed order fractional Pearson diffusion in the steady state, in terms of generalized Mittag-Leffler functions. The correlation function decays like a power law. That formula shows that distributed order fractional Pearson diffusions exhibits long range dependence.     

A note on fractional derivatives and fractional powers of operators

Definitions of fractional derivatives and fractional powers of positive operators are considered. The connection of fractional derivatives with fractional powers of positive operators is presented. The formula for fractional difference derivative is obtained.     

Coupled oscillators with power-law interaction and their fractional dynamics analogues

A one-dimensional chain of coupled oscillators with the long-range power-law interactions is considered. Equations of motion in the infrared limit are mapped onto the continuum equation with the Riesz fractional derivative of order α, when 0 < α < 2. The evolution of soliton-like and breather-like structures is obtained numerically and compared for two types of simulations: using the chain of oscillators and using the continuous medium equation with the fractional derivative.     

Approximate analytical solutions of fractional gas dynamic equations

In this article, differential transform method (DTM) has been successfully applied to obtain the approximate analytical solutions of the nonlinear homogeneous and non-homogeneous gas dynamic equations, shock wave equation and shallow water equations with fractional order time derivatives. The true beauty of the article is manifested in its emphatic application of Caputo fractional order time derivative on the classical equations with the achievement of the highly accurate solutions by the known series solutions and even for more complicated nonlinear fractional partial differential equations (PDEs). The method is really capable of reducing the size of the computational work besides being effective and convenient for solving fractional nonlinear equations. Numerical results for different particular cases of the equations are depicted through graphs.     

Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations

Stock exchange dynamics of fractional order are usually modeled as a non-random exponential growth process driven by a fractional Brownian motion. Here we propose to use rather a non-random fractional growth driven by a (standard) Brownian motion. The key is the Taylor’s series of fractional order where E_α(.) denotes the Mittag-Leffler function, and is the so-called modified Riemann-Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration. Various models of fractional dynamics for stock exchange are proposed, and their solutions are obtained. Mainly, the Itô’s lemma of fractional order is illustrated in the special case of a fractional growth with white noise. Prospects for the Merton’s optimal portfolio are outlined, the path probability density of fractional stock exchange dynamics is obtained, and two fractional Black-Scholes equations are derived. This approach avoids using fractional Brownian motion and thus is of some help to circumvent the mathematical difficulties so involved.     

A sufficient condition of viability for fractional differential equations with the Caputo derivative

In this paper viability results for nonlinear fractional differential equations with the Caputo derivative are proved. We give the sufficient condition that guarantees fractional viability of a locally closed set with respect to nonlinear function. As an example we discuss positivity of solutions, particularly in linear case.     

Nonlinear fractional cone systems with the Caputo derivative

Nonlinear fractional cone systems involving the Caputo fractional derivative are considered. We establish sufficient conditions for the existence of at least one cone solution to such systems. Sufficient conditions for the unique existence of the cone solution to a nonlinear fractional cone system are given.     

A general finite element formulation for fractional variational problems

This paper presents a general finite element formulation for a class of Fractional Variational Problems (FVPs). The fractional derivative is defined in the Riemann-Liouville sense. For FVPs the Euler-Lagrange and the transversality conditions are developed. In the Fractional Finite Element Formulation (FFEF) presented here, the domain of the equations is divided into several elements, and the functional is approximated in terms of nodal variables. Minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme. Three examples are considered to show the performance of the algorithm. Results show that as the number of discretization is increased, the numerical solutions approach the analytical solutions, and as the order of the derivative approaches an integer value, the solution for the integer order system is recovered. For unspecified boundary conditions, the numerical solutions satisfy the transversality conditions. This indicates that for the class of problems considered, the numerical solutions can be obtained directly from the functional, and there is no need to solve the fractional Euler-Lagrange equations. Thus, the formulation extends the traditional finite element approach to FVPs.     

Formulation of Euler-Lagrange equations for fractional variational problems

This paper presents extensions to traditional calculus of variations for systems containing fractional derivatives. The fractional derivative is described in the Riemann-Liouville sense. Specifically, we consider two problems, the simplest fractional variational problem and the fractional variational problem of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives and unknown functions. For the second problem, we also present a Lagrange type multiplier rule. For both problems, we develop the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Two problems are considered to demonstrate the application of the formulation. The formulation presented and the resulting equations are very similar to those that appear in the field of classical calculus of variations.