Interpolation by integro quintic splines

Recently, Behforooz [1], has introduced a new approach to construct cubic splines by using the integral values, rather than the usual function values at the knots. Also he has established different sets of end conditions for cubic and quintic splines by using the integral values, see Behforooz [2], [3] and [4]. In this paper, we will use the same techniques of [1] to construct integro quintic splines. Although by using the integral values we expected to face a more complicated process for our construction, it turned out that the matrix of the system of linear equations that produces the parameters became a diagonally dominant matrix and the process became very simple. The selection of the required end conditions for our integro quintic splines will be discussed. The numerical examples and computational results illustrate and guarantee a higher accuracy for this approximation.     

Construction of a model towards EOQ and pricing strategy for gradually obsolescent products

Due to the rapid advance of technology, manufacturers’ marketing models and quickly changing consumer tastes, products such as apparel, clothing accessories and consumer electronics are likely to face the problem of short product life cycles. This study deals with the problem of determining order quantity and multi-discount selling prices for these types of gradually obsolescent products, in which two novel proposals are presented as follows: (1) without considering any exogenous factors that could affect demand, we develop a time-dependent demand model that appropriately portrays an integrated demand behavior associated with the characteristic of obsolescence; (2) we then treat the selling price as an exogenous factor influencing demand and, referring to the linear demand D = α − βp, the effect that “the increase of demand due to price change is linearly correlated with the difference between two consecutive selling prices” is incorporated, so as to make the demand model be a function of both time and selling price. Afterwards, optimization models are hereby formulated to predetermine pricing strategy with a limited number of price changes by maximizing retailer’s profit. As a result, numerical examples illustrate that the multi-discount model indeed provides higher total profit than a single discount one. This is presented along with the result analysis conducted to gain some managerial insights.     

Market oscillations induced by the competition between value-based and trend-based investment strategies

We consider financial market using mathematical models which incorporate an excess demand function that depends not only upon the price but on the price derivative. The classical (value-based) motivation for purchasing the equity is augmented with a trend-based strategy of buying due to rising prices. An analysis (based on money flow and the finiteness of assets) of the supply, demand and price as a function of time leads to a system of ordinary differential equations which is mathematically complete. The numerical study of our equations exhibits overshooting, abrupt reversals and oscillations in prices. We examine our models within the context of real markets and economic laboratory experiments by comparing its predictions with a set of Porter and Smith experiments and with all US stock market “crashes” since 1929.     

On fractional integration formulae for Aleph functions

This paper is devoted to the study of a new special function, which is called, according to the symbol used to represent this function, as an Aleph function. This function is an extension of the I-function, which itself is a generalization of the well-known and familiar G- and H-functions in one variable. In this paper, a notation and complete definition of the Aleph function will be presented. Fractional integration of the Aleph functions, in which the argument of the Aleph function contains a factor t^λ(1 − t)^μ, λ, μ > 0, will be investigated. The results derived are of most general character and include many results given earlier by various authors including Kilbas [10], Kilbas and Saigo [11] and Galué [6] and others. The results obtained form the key formulae for the results on various potentially useful special functions of physical and biological sciences and technology available in the literature.     

Kantorovich-type semilocal convergence analysis for inexact Newton methods

We provide a new semilocal convergence analysis for generating an inexact Newton method converging to a solution of a nonlinear equation in a Banach space setting. Our analysis is based on our idea of recurrent functions. Our results are compared favorably to earlier ones by others and us (Argyros (2007, 2009) [5] and [6], Argyros and Hilout (2009) [7], Guo (2007) [15], Shen and Li (2008) [18], Li and Shen (2008) [19], Shen and Li (2009) [20]). Numerical examples are provided to show that our results apply, but not earlier ones [15], [18], [19] and [20].     

A unifying theorem for Newton’s method on spaces with a convergence structure

We present a semilocal convergence theorem for Newton’s method (NM) on spaces with a convergence structure. Using our new idea of recurrent functions, we provide a tighter analysis, with weaker hypotheses than before and with the same computational cost as for Argyros (1996, 1997, 1997, 2007) [1], [2], [3] and [5], Meyer (1984, 1987, 1992) [13], [14] and [15]. Numerical examples are provided for solving equations in cases not covered before.     

Improvement on delay dependent absolute stability of Lurie control systems with multiple time delays

Absolute stability of Lurie control systems with multiple time-delays is studied in this paper. By using extended Lyapunov functionals, we avoid the use of the stability assumption on the main operator and derive improved stability criteria, which are strictly less conservative than the criteria in [2] and [3].     

On the Secant method

We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004)  [2], Bosarge and Falb (1969)  [9], Dennis (1971)  [10], Ezquerro et al. (2010)  [11], Hernández et al. (2005, 2000)   and , Kantorovich and Akilov (1982)  [14], Laasonen (1969)  [15], Ortega and Rheinboldt (1970)  [16], Parida and Gupta (2007)  [17], Potra (1982, 1984–1985, 1985)  ,  and , Proinov (2009, 2010)   and , Schmidt (1978) [23], Wolfe (1978)  [24] and Yamamoto (1987)  [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study.     

Differential operator related to the generalized superradiance integral equation

In this paper we consider a class of problems which are generalized versions of the three-dimensional superradiance integral equation. A commuting differential operator will be found for this generalized problem. For the three-dimensional superradiance problem an alternative set of complete eigenfunctions will also be provided. The kernel for the superradiance problem when restricted to one-dimension is the same as appeared in the works of Slepian, Landau and Pollak (cf. Slepian and Pollak (1961) [1], Landau and Pollak (1961, 1962) [2] and [3], Slepian (1964, 1978) [4] and [5]). The uniqueness of the differential operator commuting with that kernel is indicated.     

A rational approximation based on Bernstein polynomials for high order initial and boundary values problems

We introduce a new method to solve high order linear differential equations with initial and boundary conditions numerically. In this method, the approximate solution is based on rational interpolation and collocation method. Since controlling the occurrence of poles in rational interpolation is difficult, a construction which is found by Floater and Hormann [1] is used with no poles in real numbers. We use the Bernstein series solution instead of the interpolation polynomials in their construction. We find that our approximate solution has better convergence rate than the one found by using collocation method. The error of the approximate solution is given in the case of the exact solution f ∈ C^d+2[a, b].     

A tâtonnement process with fading memory,stabilization and optimal speed of convergence

The purpose of this work is to provide a way to improve stability and convergence rate of a price adjustment mechanism that converges to a Walrasian equilibrium. We focus on a discrete tâtonnement based on a two-agent, two-good exchange economy, and we introduce memory, assuming that the auctioneer adjusts prices not only using the current excess demand, but also making use of the past excess demand functions. In particular, we study the effect of computing a weighted average of the current and the previous excess demands (finite two level memory) and of all the previous excess demands (infinite memory). We show that suitable weights’ distributions have a stabilizing effect, so that the resulting price adjustment process converge toward the competitive equilibrium in a wider range of situations than the process without memory. Finally, we investigate the convergence speed toward the equilibrium of the proposed mechanisms. In particular, we show that using infinite memory with fading weights approaches the competitive equilibrium faster than with a distribution of quasi-uniform weights.     

Hysteresis operators in phase-field models of Penrose-fife type

Phase-field systems as mathematical models for phase transitions have drawn a considerable attention in recent years. However, while they are suitable for capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors have recently proposed a new approach to phase-field models which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. Well-posedness and thermodynamic consistency were proved for a phase-field system with hysteresis which is closely related to the model advanced by Caginalp in a series of papers. In this note the more difficult case of a phase-field system of Penrose-Fife type with hysteresis is investigated. Under slightly more restrictive assumptions than in the Caginalp case it is shown that the system is well-posed and thermodynamically consistent.     

金融衍生产品的力学方法分析（Ⅱ）—期权市场价格基本方程

类似固体力学建立基本方程方法，根据期权特点，采用一些假设，建立期权市场价格基本方程：hv0(t)=m1vo^-1(t)-n1vo(t) F,式中h,m1,n1,F为常数，主要假设有：期权市场价格vo(t)的升降由市场供求决定；影响v0(t)的因素如行使价，期限，波幅等用正或反比关系；买和卖用相反规律。文中给出不同情况下基本方程的解，并和期货市场价基本方程的解vf(t)相比较，以及用隐函数存在定理证明vf与v0(t)存在一一对应关系，为研究期货vf对期权价vo(t)的影响提供理论依据。     

New stability criteria for neural networks with time-varying delays

This paper is concerned with the stability analysis problem of neural networks with time delays. The delay intervals [−d(t), 0] and [−h, 0] are divided into m subintervals with equal length. Some free matrices are introduced to build the relationship among the elements of the resultant matrix inequalities. With the above operations, the new stability criteria are built for the general class of neural networks. The conditions are presented in the form of linear matrix inequalities (LMIs), which can be solved by the numerically efficient Matlab LMI toolbox. Several examples are provided to show that our methods are much less conservative than recently reported ones.     

Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations

This paper examines the numerical solution of the transient nonlinear coupled Burgers’ equations by a Local Radial Basis Functions Collocation Method (LRBFCM) for large values of Reynolds number (Re) up to 10³. The LRBFCM belongs to a class of truly meshless methods which do not need any underlying mesh but works on a set of uniform or random nodes without any a priori node to node connectivity. The time discretization is performed in an explicit way and collocation with the multiquadric radial basis functions (RBFs) are used to interpolate diffusion-convection variable and its spatial derivatives on decomposed domains. Five nodded domains of influence are used in the local support. Adaptive upwind technique [1] and [54] is used for stabilization of the method for large Re in the case of mixed boundary conditions. Accuracy of the method is assessed as a function of time and space discretizations. The method is tested on two benchmark problems having Dirichlet and mixed boundary conditions. The numerical solution obtained from the LRBFCM for different value of Re is compared with analytical solution as well as other numerical methods [15], [47] and [49]. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Reynolds numbers.     

Stability in cellular neural networks with a piecewise constant argument

In this paper, by using the concept of differential equations with piecewise constant arguments of generalized type [1], [2], [3] and [4], a model of cellular neural networks (CNNs) [5] and [6] is developed. The Lyapunov-Razumikhin technique is applied to find sufficient conditions for the uniform asymptotic stability of equilibria. Global exponential stability is investigated by means of Lyapunov functions. An example with numerical simulations is worked out to illustrate the results.     

Minmax regret location–allocation problem on a network under uncertainty

We consider a robust location–allocation problem with uncertainty in demand coefficients. Specifically, for each demand point, only an interval estimate of its demand is known and we consider the problem of determining where to locate a new service when a given fraction of these demand points must be served by the utility. The optimal solution of this problem is determined by the “minimax regret” location, i.e., the point that minimizes the worst-case loss in the objective function that may occur because a decision is made without knowing which state of nature will take place. For the case where the demand points are vertices of a network we show that the robust location–allocation problem can be solved in O(min{p, n − p}n³m) time, where n is the number of demand points, p (p < n) is the fixed number of demand points that must be served by the new service and m is the number of edges of the network.     

二阶中立型微分方程的区间振动准则

利用平均函数技巧 ,对二阶中立型微分方程 [a(t) (x(t) p(t)x(t-τ) )′]′ q(t)f[x(t) ,x(t-σ) ]g[x′(t) ]=0建立了一些区间振动准则 ,这些振动准则不同于已知依赖于整个区间 [t0 ,∞ )的性质的结果 ,而是仅依赖于 [t0 ,∞ )上的子区间列的性质     

Criteria for univalence of the Dziok-Srivastava and the Srivastava-Wright operators in the class A

In the geometric function theory (GFT) much attention is paid to various linear integral operators mapping the class S of the univalent functions and its subclasses into themselves. In [12] and [13] Hohlov obtained sufficient conditions that guarantee such mappings for the operator defined by means of Hadamard product with the Gauss hypergeometric function. In our earlier papers as [20], [19], [17] and [18], etc., we extended his method to the operators of the generalized fractional calculus (GFC, [16]). These operators have product functions of the forms _m+1F_m and _m+1Ψ_m and integral representations by means of the Meijer G- and Fox H-functions. Here we propose sufficient conditions that guarantee mapping of the univalent, respectively of the convex functions, into univalent functions in the case of the celebrated Dziok-Srivastava operator ([8] : J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput.103, No 1 (1999), pp. 1-13) defined as a Hadamard product with an arbitrary generalized hypergeometric function _pF_q. Similar conditions are suggested also for its extension involving the Wright _pΨ_q-function and called the Srivastava-Wright operator (Srivastava, [36]). Since the discussed operators include the above-mentioned GFC operators and many their particular cases (operators of the classical FC), from the results proposed here one can derive univalence criteria for many named operators in the GFT, as the operators of Hohlov, Carlson and Shaffer, Saigo, Libera, Bernardi, Erdélyi-Kober, etc., by giving particular values to the orders p ? q + 1 of the generalized hypergeometric functions and to their parameters.     

Differentiation of generalized inverses for rational and polynomial matrices

Our basic motivation is a direct method for computing the gradient of the pseudo-inverse of well-conditioned system with respect to a scalar, proposed in [13] by Layton. In the present paper we combine the Layton’s method together with the representation of the Moore-Penrose inverse of one-variable polynomial matrix from [24] and developed an algorithm for computing the gradient of the Moore-Penrose inverse for one-variable polynomial matrix. Moreover, using the representation of various types of pseudo-inverses from [26], based on the Grevile’s partitioning method, we derive more general algorithms for computing {1}, {1, 3} and {1, 4} inverses of one-variable rational and polynomial matrices. Introduced algorithms are implemented in the programming language MATHEMATICA. Illustrative examples on analytical matrices are presented.