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In 1984, Kyurkchiev et al. constructed an infinite sequence of iterative methods for simultaneous approximation of polynomial zeros (with known multiplicity). The first member of this sequence of iterative methods is the famous root-finding method derived independently by Farmer and Loizou (1977) and Gargantini (1978). For every given positive integer N, the Nth method of this family has the order of convergence 2 N + 1 . In this paper, we prove two new local convergence results for this family of iterative methods. The first one improves the result of Kyurkchiev et al. (1984). We end the paper with a comparison of the computational efficiency, the convergence behavior and the computational order convergence of different methods of the family.

On the basis of the previous work by Tang and Zhang [37], in this paper we present a more effective way to construct high-order symplectic integrators for solving second order Hamiltonian equations. Instead of analyzing order conditions step by step as shown in the previous work, the new technique of this paper is using Legendre expansions to deal with the simplifying assumptions for order conditions. With the new technique, high-order symplectic integrators can be conveniently devised by truncating an orthogonal series.

This paper studies European call option pricing problem under a hard-to-borrow stock model where stock price and buy-in rate are fully coupled. Avellaneda and Lipkin (2009) proposed a simplified solution approach with an independence assumption, and then derived a semi-explicit pricing formula. However, such an approach has limited its application to more general cases. In this paper, we propose a partial differential equation (PDE) approach for pricing European call options, regardless of the independence assumption. A two-dimensional PDE is derived first with a set of appropriate boundary conditions. Then, two numerical schemes are provided with different treatments of the jump term. Through our numerical results, we find that the semi-explicit formula is a good approximate solution when the coupling parameter is small. However, when the stock price and the buy-in rate are significantly coupled, the PDE approach is preferred to solve the option pricing problem under the full hard-to-borrow model.

Delay-dependent stability analysis of linear systems with a time-varying delay is investigated in this paper. Firstly, instead of developing a Lyapunov–Krasovskii functional (LKF) including augmented non-integral quadratic terms as usual, this paper proposes two augmented-integral-function based LKFs, which can reflect closer relationships among system states. Secondly, to bound the derivative of LKF more accurately, a new integral inequality with several free matrices is developed. Compared with the free-matrix-based integral inequality, this inequality can provide additional freedom due to the free matrices introduced. Then, by employing those LKFs and the improved integral inequality, several delay-dependent stability criteria are established for two types of delays. Finally, two numerical examples are given to demonstrate the superiority of the proposed method.

In this paper, a pricing formula for volatility swaps is delivered when the underlying asset follows the stochastic volatility model with jumps and stochastic intensity. By using Feynman-Kac theorem, a partial integral differential equation is obtained to derive the joint moment generating function of the previous model. Moreover, discrete and continuous sampled volatility swap pricing formulas are given by employing transform techniques and the relationship between two pricing formulas is discussed. Finally, some numerical simulations are reported to support the results presented in this paper., 15PAGES

In this paper, we give solution to the quickest drift change detection problem for a Levy process consisting of both a continuous Gaussian part and a jump component. We consider here Bayesian framework with an exponential a priori distribution of the change point using an optimality criterion based on a probability of false alarm and an expected delay of the detection. Our approach is based on the optimal stopping theory and solving some boundary value problem. Paper is supplemented by an extensive numerical analysis related with the construction of the Generalized Shiryaev-Roberts statistics. In particular, we apply this method (after appropriate calibration) to analyse Polish life tables and to model the force of mortality in this population with a drift changing in time.

This paper investigates convergence properties of gradient neural network (GNN) and GNN-based dynamical systems for computing generalized inverses. The main results are exact analytical solutions for the state matrices of the corresponding GNN and GNN-based dynamical systems. The exact solutions are given in each time instant, which enables to express final results as appropriate limiting expressions. This enables more rigorous convergence analysis in terms of exact solutions. Finally, trajectories of state variables in considered dynamical systems can be generated by avoiding time-consuming numerical solving matrix differential equations inside the selected time interval. Using the fact that the stated dynamical systems are in the essence systems of linear equations, explicit solutions can be obtained using known techniques of ordinary differential equations. Main results of the paper are further transformations of obtained exact solutions using main properties of generalized inverses and linear algebra tools. It is important to mention that the convergence results are expressed in terms of expressions involving outer inverses. Several examples are presented and a closed-form expressions for the solutions are given and implemented in package Mathematica. These are compared with the numerical solutions obtained by Matlab Simulink implementation.

In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applying the classical technique of the heat balance integral method and the refined integral method, assuming a quadratic temperature profile in space. We develop variations of these methods which turn out to be optimal in some cases. Throughout this paper, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to estimate the errors committed by each approach for the corresponding free boundary and temperature profiles.

In this paper we present a new class of direct numerical integrators of hybrid type for special third order ordinary differential equations (ODEs), y ′ ′ ′ = f ( x , y ) ; namely, hybrid methods for solving third order ODEs directly (HMTD). Using the theory of B-series, order of convergence of the HMTD methods is investigated. The main result of the paper is a theorem that generates algebraic order conditions of the methods that are analogous to those of two-step hybrid method. A three-stage explicit HMTD is constructed. Results from numerical experiment suggest the superiority of the new method over several existing methods considered in the paper.

Different Engineering applications require dealing with improper integral on unbounded domains (improper integrals of the first kind). The classical way for solving these integrals is by means of elementary Calculus (antiderivatives and limit computations) or using numerical approaches. In both situations different problems can arise. For example: the non-existence of antiderivative, the corresponding limit does not exist or integrals depending on parameters which complicate the use of numerical approaches. In order to solve this situation, Advanced Calculus techniques, such as Laplace or Fourier Transforms and the Residue Theorem can be applied. A brief review of the corresponding theoretical frame is included in this paper. Computer Algebra Systems ( Cas ) are pieces of software that allow symbolic computations. Nowadays, there are many Cas in the market with an increasing level of sophistication which make them a very powerful tool in Engineering. In this paper, a brief review on the history of Cas evolution is introduced. In this work, some Advanced Calculus techniques for computing improper integrals of the first kind which cannot be solved with standard procedures are described. These techniques could be easily integrated in almost every Cas . However, we have detected some lacks of these techniques in many widely used Cas . In this paper, some tests involving improper integrals have been developed. These tests have been used to check the capabilities of some Cas and the results have provided a classification of the Cas with respect to this field. One of the main contributions in this work is the generation of different rules to compute improper integrals of the first kind. The rules have been classified in specific rules (those coming from the tests) and general rules (those coming from theoretical frames and generalizations of the specific rules). These rules are easy to include in a Cas increasing the facilities of the Cas in the field of improper integral computation.

[EN] This paper deals with weighted G-Drazin inverses, which is a new class of matrices introduced to extend (to the rectangular case) G-Drazin inverses recently considered by Wang and Liu for square matrices. First, we define and characterize weighted G-Drazin inverses. Next, we consider a new pre-order defined on complex rectangular matrices based on weighted G-Drazin inverses. Finally, we characterize this pre-order and relate it to the minus partial order and to the weighted Drazin pre-order. (C) 2017 Elsevier Inc. All rights reserved., This paper was partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria, grant resol. no. 155/14. The first and third authors were partially supported by Ministerio de Economia y Competitividad of Spain (grant no. DGI MTM2013-43678-P) and the third author was also partially supported by Ministerio de Economia y Competitividad of Spain (Red de Excelencia MTM2015-68805-REDT).

Survey of fixed point iteration processes.Extension of the pseudo-Newton method idea to other root finding methods.Modifications of the MMP-methods and their convergence visualizations.Non-trivial and intriguing polynomiographs have been obtained. In this paper, an iteration process, referred to in short as MMP, will be considered. This iteration is related to finding the maximum modulus of a complex polynomial over a unit disc on the complex plane creating intriguing images. Kalantari calls these images polynomiographs independently from whether they are generated by the root finding or maximum modulus finding process applied to any polynomial. We show that the images can be easily modified using different MMP methods (pseudo-Newton, MMP-Householder, methods from the MMP-Basic, MMP-Parametric Basic or MMP-EulerSchrder Families of Iterations) with various kinds of non-standard iterations. Such images are interesting from three points of views: scientific, educational and artistic. We present the results of experiments showing automatically generated non-trivial images obtained for different modifications of root finding MMP-methods. The colouring by iteration reveals the dynamic behaviour of the used root finding process and its speed of convergence. The results of the present paper extend Kalantaris recent results in finding the maximum modulus of a complex polynomial based on Newtons process with the Picard iteration to other MMP-processes with various non-standard iterations.

In this paper the Hamiltonian approximate analytical approach is extended for solving vibrations of conservative oscillators with the sum of integer and/or non-integer order strong nonlinearities. Solution of the nonlinear differential equation is assumed in the form of a trigonometric function with unknown frequency. The frequency equation is obtained based on the hypothesis that the derivative of the Hamiltonian in amplitude of vibration is zero. The accuracy of the approximate solution is treated with two different approaches: comparing the analytical value for period of vibration with the period obtained numerically and developing an error estimation method based on the ratio between the averaged residual function and the total constant energy of system. The procedures given in the paper are applied for two types of examples: an oscillator with a strong nonlinear term and an oscillator where the nonlinearity is of polynomial type.

The joint distribution of a geometric Brownian motion and its time-integral was derived in a seminal paper by Yor (1992) using Lamperti’s transformation, leading to explicit solutions in terms of modified Bessel functions. In this paper, we revisit this classic result using the simple Laplace transform approach in connection to the Heun differential equation. We extend the methodology to the geometric Brownian motion with affine drift and show that the joint distribution of this process and its time-integral can be determined by a doubly-confluent Heun equation. Furthermore, the joint Laplace transform of the process and its time-integral is derived from the asymptotics of the solutions. In addition, we provide an application by using the results for the asymptotics of the double-confluent Heun equation in pricing Asian options. Numerical results show the accuracy and efficiency of this new method.

Inversion sequences are finite sequences of non-negative integers, where the value of each entry is bounded from above by its position. Patterns in inversion sequences have been studied by Corteel-Martinez-Savage-Weselcouch and Mansour-Shattuck in the classical case, where patterns can occur in any positions, and by Auli-Elizalde in the consecutive case, where only adjacent entries can form an occurrence of a pattern. These papers classify classical and consecutive patterns of length 3 into Wilf equivalence classes according to the number of inversion sequences avoiding them. In this paper we consider vincular patterns in inversion sequences, which, in analogy to Babson-Steingr\'{\i}msson patterns in permutations, require only certain entries of an occurrence to be adjacent, and thus generalize both classical and consecutive patterns. Solving a conjecture of Lin and Yan, we provide a complete classification of vincular patterns of length 3 in inversion sequences into Wilf equivalence classes, and into more restrictive classes that consider the number of occurrences of the pattern and the positions of such occurrences. We find the first known instance of patterns in inversion sequences where these two more restrictive classes do not coincide., Comment: 18 pages, 9 figures

Due to instability being induced easily by parameter disturbances of network systems, this paper investigates the multistability of memristive Cohen-Grossberg neural networks (MCGNNs) under stochastic parameter perturbations. It is demonstrated that stable equilibrium points of MCGNNs can be flexibly located in the odd-sequence or even-sequence regions. Some sufficient conditions are derived to ensure the exponential multistability of MCGNNs under parameter perturbations. It is found that there exist at least ( w + 2 ) l (or ( w + 1 ) l ) exponentially stable equilibrium points in the odd-sequence (or the even-sequence) regions. In the paper, two numerical examples are given to verify the correctness and effectiveness of the obtained results.

The main aim of the paper is to prove that the implicit numerical approximation can converge to the true solution to highly nonlinear hybrid stochastic pantograph differential equation. After providing the boundedness of the exact solution, the paper proves that the backward Euler-Maruyama numerical method can preserve boundedness of moments, and the numerical approximation converges strongly to the true solution. Finally, the exponential stability criterion on the backward Euler-Maruyama scheme is given, and a high order example is provided to illustrate the main result.

In this paper, we study a class of approximation problems appearing in data approximation and signal processing. The approximations are constructed as combinations of polynomial splines (piecewise polynomials) whose parameters are subject to optimisation and so called prototype functions whose choice is based on the application rather than optimisation. We investigate two types of models, namely Model 1 and Model 2 in order to analyse the singularity of their system matrices. The main difference between these two models is that in Model 2, the signal is shifted vertically (signal biasing) by a polynomial spline function. The corresponding optimisation problems can be formulated as Linear Least Squares Problems (LLSPs). If the system matrix is non-singular, then, the corresponding problem can be solved inexpensively and efficiently, while for singular cases, slower (but more robust) methods have to be used. To choose a better suited method for solving the corresponding LLSPs we have developed a singularity verification rule. In this paper, we develop necessary and sufficient conditions for non-singularity of Model 1 and sufficient conditions for non-singularity of Model 2. These conditions can be verified much faster than the direct singularity verification of the system matrices. Therefore, the algorithm efficiency can be improved by choosing a suitable method for solving the corresponding LLSPs.

This paper considers the three-dimensional path planning problem for horizontal oil wells. The decision variables in this problem are the curvature, tool-face angle and switching points for each turn segment in the path, and the optimization objective is to minimize the path length and target error. The optimal curvatures, tool-face angles and switching points can be readily determined using existing gradient-based dynamic optimization techniques. However, in a real drilling process, the actual curvatures and tool-face angles will inevitably deviate from the planned optimal values, thus causing an unexpected increase in the target error. This is a critical challenge that must be overcome for successful practical implementation. Accordingly, this paper introduces a sensitivity function that measures the rate of change in the target error with respect to the curvature and tool-face angle of each turn segment. Based on the sensitivity function, we propose a new optimization problem in which the switching points are adjusted to minimize target error sensitivity subject to continuous state inequality constraints arising from engineering specifications, and an additional constraint specifying the maximum allowable increase in the path length from the optimal value. Our main result shows that the sensitivity function can be evaluated by solving a set of auxiliary dynamic systems. By combining this result with the well-known time-scaling transformation, we obtain an equivalent transformed problem that can be solved using standard nonlinear programming algorithms. Finally, the paper concludes with a numerical example involving a practical path planning problem for a Ci-16-Cp146 well.

In this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative Volterra type operators. In a recent paper of us a variation of constants formula has been given for such systems, and this is particularly suited to handling the studied problems. First, we investigate the boundedness of the solutions. Then some estimates are given for the upper and lower limits of the solutions. By using this, we study the existence of the limit of the solutions, and obtain a formula for the limit. The applicability of our results is illustrated by studying the dependence of limits of the solutions on the choice of inhomogeneities, synchronisation, and Lotka-Volterra type delay functional differential systems.

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2015.08.039 Under the assumption of the known multiplicity of zeros of nonlinear equations, a class of two-point sextic-order multiple-zero finders and their dynamics are investigated in this paper by means of extensive analysis of modified double-Newton type of methods. Wit the introduction of a bivariate weight function dependent on function-to-function and derivative-to-derivative ratios, higher-order convergence is obtained. Additional investigation is carried out for extraneous fixed points of the iterative maps associated with the proposed methods along with a comparison with typically selected cases. Through a variety of test equations, numerical experiments strongly support the theory developed in this paper. In addition, relevant dynamics of the proposed methods is successfully explored for various polynomials with a number of illustrative basins of attraction. National Research Foundation of Korea Ministry of Education, Science and Technology under the research grant (Project Number: 2015-R1D1A3A-01020808

In this paper we introduce a discontinuous Galerkin method on polygonal meshes. This method arises from the discontinuous Galerkin composite finite element method (DGFEM) for source problems on domains with micro-structures. In the context of the present paper, the flexibility of DGFEM is applied to handle polygonal meshes. We prove the a priori convergence of the method for both eigenvalues and eigenfunctions for elliptic eigenvalue problems. Numerical experiments highlighting the performance of the proposed method for problems with discontinuous coefficients and on convex and non-convex polygonal meshes are presented.

In this paper, to analyze the effect of the cross patch infection between different groups to the spread of gonorrhea in a community, we establish the complete global dynamics of a multi-group SIS epidemic model with varying total population size by a threshold parameter. In the proof, we use special Lyapunov functional techniques, not only one proposed by the paper Pruss et?al., 2006, but also the other one for a varying total population size with some ideas specified to our model and no longer need a grouping technique derived from the graph theory which is commonly used for the global stability analysis of multi-group epidemic models.

A time-dependent Hermite-Galerkin spectral method (THGSM) is investigated in this paper for the nonlinear convection-diffusion equations in the unbounded domains. The time-dependent scaling factor and translating factor are introduced in the definition of the generalized Hermite functions (GHF). As a consequence, the THGSM based on these GHF has many advantages, not only in theoretical proofs, but also in numerical implementations. The stability and spectral convergence of our proposed method have been established in this paper. The Korteweg-de Vries-Burgers (KdVB) equation and its special cases, including the heat equation and the Burgers' equation, as the examples, have been numerically solved by our method. The numerical results are presented, and it surpasses the existing methods in accuracy. Our theoretical proof of the spectral convergence has been supported by the numerical results.

This paper deals with a concept and description of a RC network as an electro-analog model of diffusion process which enables to simulate heat dissipation under different initial and boundary conditions. It is based on well-known analogy between heat and electrical conduction. In the paper are compared analytical solution together with numerical solution and experimentally measured data. For the first time a fractional order model of diffusion process and its modeling via lumped RC network has been used. Simple examples of simulations, measurements and their comparison are shown.

In this paper, the author presents a graphical tool that allows to study the real dynamics of iterative methods whose iterations depends on one parameter in an easy and compact way. This tool gives the information as previous tools such as Feigenbaum diagrams and Lyapunov exponents for every initial point. The convergence plane can be used, inter alia, to find the elements of a family that have good convergence properties, to see how the basins of attraction changes along the elements of the family, to study two-point methods such as Secant method or even to study two-parameter families of iterative methods. To show the applicability of the tool an example of the dynamics of the Damped Newton's method applied to a cubic polynomial is presented in this paper.

Meshless Local Petrov Galerkin (MLPG) methods are pure meshless techniques for solving Partial Differential Equations. They have been successfully used for solving many real-life problems. Their efficiency is downgraded by the requirement of performing numerical multi-dimensional integration of tricky, non-polynomial factors. Recently, Direct MLPG (DMLPG) methods have been proposed. DMLPG techniques require lower computational costs with respect to their MLPG counterparts. The DMLPG accuracy has been initially analyzed in few papers, but its performance is quite unexplored. In this paper, we perform numerical comparisons between MLPG and DMLPG accuracy and efficiency in solving anisotropic diffusion problems. In particular, we set different boundary conditions, in order to check if and when MLPG and/or DMLPG suffer locking effects.

The uniqueness of sparsest solutions of underdetermined linear systems plays a fundamental role in the newly developed compressed sensing theory. Several new algebraic concepts, including the sub-mutual coherence, scaled mutual coherence, coherence rank, and sub-coherence rank, are introduced in this paper in order to develop new and improved sufficient conditions for the uniqueness of sparsest solutions. The coherence rank of a matrix with normalized columns is the maximum number of absolute entries in a row of its Gram matrix that are equal to the mutual coherence. The main result of this paper claims that when the coherence rank of a matrix is low, the mutual-coherence-based uniqueness conditions for the sparsest solution of a linear system can be improved. Furthermore, we prove that the Babel-function-based uniqueness can be also improved by the so-called sub-Babel function. Moreover, we show that the scaled-coherence-based uniqueness conditions can be developed, and that the right-hand-side vector $b$ of a linear system, the support overlap of solutions, the orthogonal matrix out of the singular value decomposition of a matrix, and the range property of a transposed matrix can be also integrated into the criteria for the uniqueness of the sparsest solution of an underdetermined linear system.

Dynamic systems described by fc(z) = z2 + c is called Mandelbrot set (M-set), which is important for fractal and chaos theories due to its simple expression and complex structure. fc(z) = zk + c is called generalized M set (k–M set). This paper proposes a new theory to compute the higher and lower bounds of generalized M set while exponent k is rational, and proves relevant properties, such as that generalized M set could cover whole complex number plane when k < 1, and that boundary of generalized M set ranges from complex number plane to circle with radius 1 when k ranges from 1 to infinite large. This paper explores fractal characteristics of generalized M set, such as that the boundary of k–M set is determined by k, when k = p/q, where p and q are irreducible integers, (GCD(p, q) = 1, k > 1), and that k–M set can be divided into |p–q| isomorphic parts.

This paper introduces a computational approach for solving non-linear optimal control problems in which the objective function is a discontinuous function of the state. We illustrate this approach using a dynamic model of shrimp farming in which shrimp are harvested at several intermediate times during the production cycle. The problem is to choose the optimal harvesting times and corresponding optimal harvesting fractions (the percentage of shrimp stock extracted) to maximize total revenue. The main difficulty with this problem is that the selling price of shrimp is modelled as a piecewise constant function of the average shrimp weight and thus the revenue function is discontinuous. By performing a time-scaling transformation and introducing a set of auxiliary binary variables, we convert the shrimp harvesting problem into an equivalent optimization problem that has a smooth objective function. We then use an exact penalty method to solve this equivalent problem. We conclude the paper with a numerical example.

The Drazin inverse solutions of the matrix equations ${\rm {\bf A}}{\rm {\bf X}} = {\rm {\bf B}}$, ${\rm {\bf X}}{\rm {\bf A}} = {\rm {\bf B}}$ and ${\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}} $ are considered in this paper. We use both the determinantal representations of the Drazin inverse obtained earlier by the author and in the paper. We get analogs of the Cramer rule for the Drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations, ${\bf X}'+ {\bf A}{\bf X}={\bf B}$ and ${\bf X}'+{\bf X}{\bf A}={\bf B}$, where the matrix ${\bf A}$ is singular., 20 pages

The fundamental construct of numerical algebraic geometry is the representation of an irreducible algebraic set, A, by a witness set, which consists of a polynomial system, F, for which A is an irreducible component of V(F), a generic linear space L of complementary dimension to A, and a numerical approximation to the set of witness points, [email protected]?A. Given F, methods exist for computing a numerical irreducible decomposition, which consists of a collection of witness sets, one for each irreducible component of V(F). This paper concerns the more refined question of finding a numerical irreducible decomposition of the intersection [email protected]?B of two irreducible algebraic sets, A and B, given a witness set for each. An existing algorithm, the diagonal homotopy, computes witness point supersets for [email protected]?B, but this does not complete the numerical irreducible decomposition. In this paper, we use the theory of isosingular sets to complete the process of computing the numerical irreducible decomposition of the intersection.

The problem of finding a rank-one solution to a system of linear matrix equations arises from many practical applications. Given a system of linear matrix equations, however, such a low-rank solution does not always exist. In this paper, we aim at developing some sufficient conditions for the existence of a rank-one solution to the system of homogeneous linear matrix equations (HLME) over the positive semidefinite cone. First, we prove that an existence condition of a rank-one solution can be established by a homotopy invariance theorem. The derived condition is closely related to the so-called P"@A property of the function defined by quadratic transformations. Second, we prove that the existence condition for a rank-one solution can be also established through the maximum rank of the (positive semidefinite) linear combination of given matrices. It is shown that an upper bound for the rank of the solution to a system of HLME over the positive semidefinite cone can be obtained efficiently by solving a semidefinite programming (SDP) problem. Moreover, a sufficient condition for the nonexistence of a rank-one solution to the system of HLME is also established in this paper.

In this paper we introduce a new Jacobi–Davidson type eigenvalue solver for a set of commuting matrices, called JDCOMM, used for the global optimization of so-called Minkowski-norm dominated polynomials in several variables. The Stetter–Möller matrix method yields such a set of real non-symmetric commuting matrices since it reformulates the optimization problem as an eigenvalue problem. A drawback of this approach is that the matrix most relevant for computing the global optimum of the polynomial under investigation is usually large and only moderately sparse. However, the other matrices are generally much sparser and have the same eigenvectors because of the commutativity. This fact is used to design the JDCOMM method for this problem: the most relevant matrix is used only in the outer loop and the sparser matrices are exploited in the solution of the correction equation in the inner loop to greatly improve the efficiency of the method. Some numerical examples demonstrate that the method proposed in this paper is more efficient than approaches that work on the main matrix (standard Jacobi–Davidson and implicitly restarted Arnoldi), as well as conventional solvers for computing the global optimum, i.e., SOSTOOLS, GloptiPoly, and PHCpack. Keywords: Multivariate polynomial optimization; Global optimization; Solving systems of polynomial equations; Stetter–Möller matrix method; Commuting matrices; Jacobi–Davidson; Correction equation; Arnoldi method

Decision making in engineering is becoming increasingly complex due to the large number of alternatives and multiple conflicting goals. Powerful decision-support expert systems powered by suitable software are increasingly necessary. In this paper, the multiple attribute decision method known as analytical hierarchy process (AHP), which uses pairwise comparisons with numerical judgments, is considered. Since judgments may lack a minimum level of consistency, mechanisms to improve consistency are necessary. A method to achieve consistency through optimisation is described in this paper. This method has the major advantage of depending on just n decision variables – the number of compared elements – and so is less computationally expensive than other optimisation methods, and can be easily implemented in virtually any existing computer environment. The proposed approach is exemplified by considering a simplified version of one of the most important problems faced by water supply managers, namely, the minimisation of water loss. 2012 Elsevier Inc. All rights reserved., This work has been performed under the support of project IDAWAS, DPI2009-11591 of the Direccionn General de Investigacion del Ministerio de Educacion y Ciencia (Spain) and ACOMP/2011/188 of the Conselleria de Educacion de la Generalitat Valenciana. The first author was supported by Spanish project MTM2010-18539. The third author is also indebted to the Universitat Politecnica de Valencia for the sabbatical leave granted during the first semester of 2011. The use of English in this paper was revised by John Rawlins.

In this paper, we look at the improvement of our knowledge on a family of tilings of the hyperbolic plane which is brought in by the use of Sergeyev's numeral system based on grossone. It appears that the information we can get by using this new numeral system depends on the way we look at the tilings. The ways are significantly different but they confirm some results which were obtained in the traditional but constructive frame and allow us to obtain an additional precision with respect to this information., 25 pages, 13 figures. The paper will appear in the journal Applied Mathematics and Computations, see below at DOI

This article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p ( t ) and q in the linear parabolic equation u t ( x , t ) = u xx + qu x ( x , t ) + p ( t ) u ( x , t ), with Dirichlet boundary conditions u (0, t ) = ψ 0 , u (1, t ) = ψ 1 . The main purpose of this paper is to investigate the distinguishability of the input–output mapping Φ [ · ] : P → H 1 , 2 [ 0 , T ] , via semigroup theory. In this paper, it is shown that if the nullspace of the semigroup T ( t ) consists of only zero function, then the input–output mapping Φ [·] has the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of the mapping. Moreover, under the light of the measured output data u x (0, t ) = f ( t ) the unknown parameter p ( t ) at ( x , t ) = (0, 0) and the unknown coefficient q are determined via the input data. Furthermore, it is shown that measured output data f ( t ) can be determined analytically by an integral representation. Hence the input–output mapping Φ [ · ] : P → H 1 , 2 [ 0 , T ] is given explicitly interms of the semigroup.

This paper concerns the memoryless quasi-Newton method, that is precisely the quasi-Newton method for which the approximation to the inverse of Hessian, at each step, is updated from the identity matrix. Hence its search direction can be computed without the storage of matrices. In this paper, a scaled memoryless symmetric rank one (SR1) method for solving large-scale unconstrained optimization problems is developed. The basic idea is to incorporate the SR1 update within the framework of the memoryless quasi-Newton method. However, it is well-known that the SR1 update may not preserve positive definiteness even when updated from a positive definite matrix. Therefore we propose the memoryless SR1 method, which is updated from a positive scaled of the identity, where the scaling factor is derived in such a way that positive definiteness of the updating matrices are preserved and at the same time improves the condition of the scaled memoryless SR1 update. Under very mild conditions it is shown that, for strictly convex objective functions, the method is globally convergent with a linear rate of convergence. Numerical results show that the optimally scaled memoryless SR1 method is very encouraging.

This paper studies the global convergence properties of a class of neutral-type neural networks with discrete time delays. This class of neutral systems includes Cohen-Grossberg neural networks, Hopfield neural networks and cellular neural networks. Based on the Lyapunov stability theorems, some delay independent sufficient conditions for the global asymptotic stability of the equilibrium point for this class of neutral-type systems are derived. It is shown that the results presented in this paper for neutral-type delayed neural networks are the generalization of a recently reported stability result. A numerical example is also given to demonstrate the applicability of our proposed stability criteria. (C) 2009 Elsevier Inc. All rights reserved.

Using slope enclosures may provide sharper bounds for the range of a function than using enclosures of the derivative. Hence, slope enclosures may be useful in verifying the assumptions for existence tests or in algorithms for global optimization. Previous papers by Kolev and Ratz show how to compute slope enclosures for convex and concave functions. In this paper, we generalize these formulas and show how to obtain slope enclosures for a function that has exactly one point of inflection or whose derivative has exactly one point of inflection.

In this paper, two anticontrol algorithms for synthesis of discrete chaos are introduced. In these algorithms, the control parameter of a discrete dynamical system is switched, either randomly or in a deterministic way, between two or more values corresponding to periodic motions, the result being chaotic behavior. These algorithms require no knowledge of specific mathematical properties of the underlying map modeling the system. The existence of chaos is demonstrated using various tools including graphical iteration, histogram, Lyapunov exponent and surrogate tests. In this paper, these very simple and implementable chaotifiers are applied to the logistic map.

This paper explores an idea to extend Newton–Raphson power flow problem to handle power system transmission line flow limits, by means of generation redispatch and phase shifters. We extend and reformulate the power flow so that it includes a variety of flow limits (thermal, small-signal stability, voltage difference), generation redispatch, and phase shifters. The novelty of the approach is three step procedure (in case any limit violations exist in the system): run ordinary power flow (and identify flow limits violated), solve a set of linear equations using extended power flow Jacobian by adding a new column and a new raw that characterize particular limit, and resolve ordinary power flow with initial solution obtained after the correction made by solution of linear equations. The use of ordinary power flow Jacobian and minimal extensions to it in the case of limits identified makes this approach an attractive alternative for practical use. A simple numerical example and the examples using an approximate model of real-life European Interconnected Power System are included in the paper to illustrate the concept.

Focus in the paper is on the definition of linear bilevel programming problems, the existence of optimal solutions and necessary as well as sufficient optimality conditions. In the papers [C. Shi, G. Zhang, J. Lu, An extended Kuhn–Tucker approach for linear bilevel programming, Appl. Math. Comput. 162 (2005) 51–63] and [C. Shi, G. Zhang, J. Lu, On the definition of linear bilevel programming solution, Appl. Math. Comput. 160 (2005) 169–176], the authors claim to suggest a refined definition of linear bilevel programming problems and related optimality conditions. Mainly their attempt reduces to shifting upper level constraints involving both the upper and the lower level variables into the lower level. We investigate such a shift in more details and show that it is not allowed in general. We show that an optimal solution of the bilevel program exists under the conditions in [C. Shi, G. Zhang, J. Lu, On the definition of linear bilevel programming solution, Appl. Math. Comput. 160 (2005) 169–176] if we add the assumption that the inducible region is not empty. The necessary optimality condition reduces to check optimality in one linear programming problem. Optimality of one feasible point for a certain number of linear programs implies optimality for the bilevel problem.

In this paper, we extend TOPSIS (Technique for Order Preference by Similarity Ideal Solution) for solving Large Scale Multiple Objective Programming problems involving fuzzy parameters. These fuzzy parameters are characterized as fuzzy numbers. For such problems, the ?-Pareto optimality is introduced by extending the ordinary Pareto optimality on the basis of the ?-Level sets of fuzzy numbers. An interactive fuzzy decision making algorithm for generating ?-Pareto optimal solution through TOPSIS approach is provided, where a decision maker (DM) is asked to specify the degree ? and the relative importance of objectives. Finally, a numerical example is given to clarify the main results developed in the paper. This article has been retracted. Link to the retraction 10.2298/YJOR141008034U

This paper presents a finite difference solution method for the incompressible Navier-Stokes equations, in primitive variables form. To provide the necessary coupling between continuity and momentum penalty method is introduced into continuity equation. Differential terms arising in Navier-Stokes equations are discretized by stabilized finite differences of the SU type (streamline upwind) to calculate the convective terms. The use of a penalty function method in a finite difference method of staggered grids, allows to eliminate the pressure and the result is a system of equations of the velocities arises. The aim of this paper is to describe the techniques used in the approximations to calculate each term of the discretized model. In particular, time integration techniques, solution of non-linearities associated to convective terms and constitutive laws, stabilization procedures and discretization of Navier-Stokes equation via finite differences are explained. Solving three standard fluid mechanics problems validates the method: the flow of a Newtonian fluid in a sudden contraction, the developing non-Newtonian fluid in a circular tube and a transient laminar axisymmetric flow trough a tube with a moderate local area reduction. Comparison with available experimental results is used to examine the accuracy of the method, reported a close agreement between the numerical prediction and the experimental data.

The main contribution in this paper is the introduction of a methodology for approximately characterizing the superposition process of N>=2 arbitrary (and possibly heterogeneous) discrete-time Markov Renewal Processes (MRP). In this model, the superposition process is characterized by a MRP with a state space that grows exponentially with N. We consider an on/off traffic source model, where the distribution of the on and off periods is arbitrary, as a special case of the general MRP. Subsequently, a queueing model for a FIFO finite-buffer multiplexer with arbitrary on/off input sources is analyzed. We provide numerical results for testing the algorithms introduced in the paper. We also study the effect of some of the statistical properties of on/off input sources on the multiplexer's performance.

Numerical approaches for the solution of vector fields (differential equations defined on a manifold) have attracted wide attention over the past few years. This paper first reviews the various numerical approaches available in the literature for the solution of vector fields namely, Parameterization approach, Constraint Stabilization approach, and Perturbation approach (PA). In the process, the paper also makes the following useful contributions: an expanded analysis and a new perturbation scheme for the PA; and a new way of choosing integration error tolerances for the parameterization approach. A comparison of all the approaches is carried out, both by means of a crude cost analysis as well as by studying their numerical performance on examples of Vector fields arising from constrained mechanical systems (CMS). Based on this comparison, recommendations are made for a proper choice of a suitable approach. Overall, the PA performs 'better' than the other approaches.

In this paper, we study symmetric integrators for solving second-order ordinary differential equations on the basis of the notion of continuous-stage Runge–Kutta–Nystrom methods. The construction of such methods heavily relies on the Legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions. New families of symmetric integrators as illustrative examples are presented. For comparing the numerical behaviors of the presented methods, some numerical experiments are also reported.

In this article, families of non-linear subdivision schemes are presented that are based on univariate polynomials up to degree three. Theses families of schemes are constructed by using dynamic iterative re-weighed least squares method. These schemes are suitable for fitting scattered data with noise and outliers. Although these schemes are non-interpolatory, but have the ability to preserve the shape of the initial polygon in case of non-noisy initial data. The numerical examples illustrate that the schemes constructed by non-linear polynomials give better performance than the schemes that are constructed by linear polynomials (Computer-Aided Design, 58, 189-199). Moreover, the numerical examples show that these schemes have the ability to reproduce polynomials and do not cause over and under fitting of the data. Furthermore, families of non-linear bivariate subdivision schemes are also presented that are based on linear and non-linear bivariate polynomials., Comment: There are 98 figures and 41 pages in this paper

In the classic 2-species Lotka–Volterra competition model, and more general competitive planar Kolmogorov models, there is a continuous curve called the carrying simplex that links all non-zero steady states and attracts all non-zero population densities. This curve is where the opposing processes of population growth and decline balance. In this paper, we use stability analysis and index theory to show that such a curve also exists when the interactions between two species are more general, such as co-operative or predator-prey, provided that reasonable biologically motivated conditions hold. For example, both species experience intraspecific competition and all population densities remain bounded for all time. We consider systems where there is at most one co-existence steady state. The ‘balance manifold’ is formed of heteroclinic orbits and attracts all non-zero population densities, but unlike its competitive analogue, the curve is no longer necessarily continuously differentiable.