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Orthogonal systems of trigonometric polynomials of semi-integer degree with respect to a weight function w ( x ) on [ 0 , 2 π ) have been considered firstly by Turetzkii [A.H. Turetzkii, On quadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (2005) 337–359 (translation in English from Uchenye Zapiski, Vypusk 1(149), Seria Math. Theory of Functions, Collection of papers, Izdatel’stvo Belgosuniversiteta imeni V.I. Lenina, Minsk, (1959) pp. 31–54)]. Such orthogonal systems are connected with quadrature rules with an even maximal trigonometric degree of exactness (with an odd number of nodes), which have application in numerical integration of 2 π -periodic functions. In this paper we study asymptotic behavior of orthogonal trigonometric polynomials of semi-integer degree with respect to a strictly positive weight function satisfying the Lipschitz-Dini condition.

This paper studies the problem of the robustly stability analysis for uncertain neutral systems with discrete and distributed delays, and the uncertainties under considerations are norm-bounded. By constructing an appropriate Lyapunov–Krasovskii functional, some new delay-dependent criteria can be obtained by using the free-weighting matrices approach to estimate the derivative of the Lyapunov functional, which are established in terms of linear matrix inequalities (LMIs). The novelties in this paper are that any bounding technique and any mode transformation method are not utilized, and the decisive variables are much fewer than ones involved in existing literatures. Finally, three numerical examples are presented to illustrate the significant improvement on the conservatism of the delay bound over some reported results in the literature.

Under investigation in this paper are the normalized linearly coupled nonlinear wave equations, which can be used to describe the pulse propagation in the two-core optical fiber. Based on the Hirota’s method and symbolic computation, the bilinear form and soliton solutions for the equations are derived. Furthermore, Backlund transformation (BT) in bilinear forms is given. Through the BT, a type of solutions and Lax pair are derived under certain conditions. Propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis. Results of this paper might be helpful for the future development of some optical devices used in the modern optical communication systems, such as the directional couplers, polarization splitters, interferometers and modulators.

This paper studies linear systems under sustained random perturbations. The stochastic perturbation model is given by a bounded Markov diffusion process, as it appears, e.g., in the description of load or generation uncertainties of power systems. For such systems, the Lyapunov exponents describe necessary and sufficient conditions for almost sure asymptotic stability. The paper presents several numerical methods for the computation of Lyapunov exponents and applies this methodology to linear oscillators in dimension 2 and 3, and to a one machine – infinite bus electric power system.

In this paper, we give a sharp eigenvalue bound for the positive stable block triangular preconditioned matrix presented in a recent paper by Cao [Z.-H. Cao, Positive stable block triangular preconditioners for symmetric saddle point problems, Appl. Numer. Math. 57 (2007) 899–910]. The intervals containing these eigenvalues of the preconditioned matrix remove the origin, which is benefit for further study. Numerical experiments of a model Stokes problem are presented to show the estimate and the effectiveness of the positive stable block triangular preconditioners.

In this paper, we consider a modification of Srivastava–Gupta operators, which is a general sequence of summation-integral operators. We are able to achieve faster convergence for our modified operators over the well known Srivastava–Gupta operators. Our results include some approximation properties, which include rate of convergence and Voronovskaya kind results. In the last section of this paper we give Stancu variant of these operators.

In this paper, we consider a class of fractional neutral control systems governed by abstract nonlinear fractional neutral differential equations. This paper deals with the exact controllability for fractional differential neutral control systems. First, we establish a new set of sufficient conditions for the controllability of nonlinear fractional systems by using a fixed point analysis approach. Further, we extend the result to study the controllability concept with nonlocal conditions. In particular, the controllability of nonlinear systems is established under the natural assumption that the associated linear control system is exactly controllable.

In a recent paper, Hernandez and Salanova proposed an interesting Newton-like method for the solution of nonlinear Fredholm integral equations. The aim of the present paper is to carry over this method to nonlinear operator equations of similar type. By using a functional analytic approach, the presentation becomes much clearer and more general. Apart from this, some new results in the Banach space setting as well as in the applications are obtained.

In this paper, we have developed and discussed a generalized difference method (GDM) for the approximation of a class of nonlinear evolution equations with integral items in one space dimension. The main idea of the paper is to introduce a generalized Volterra projection operator and we design a generalized difference formulation with a new approximate value of initial function, which is of special importance for the error analysis. Then some theories of the generalized Volterra projection operator are analyzed. Finally, we deduce the optimal error estimates. Moreover, some conclusions are obtained and our further research work is proposed.

The aim of this paper is to establish the oscillation theorems, Rayleigh principle, and coercivity results for linear Hamiltonian and symplectic systems with general boundary conditions, i.e., for the case of separated and jointly varying endpoints, and with no controllability (normality) and strong observability assumptions. Our method is to consider the time interval as a time scale and apply suitable time scales techniques to reduce the problem with separated endpoints into a problem with Dirichlet boundary conditions, and the problem with jointly varying endpoints into a problem with separated endpoints. These more general results on time scales then provide new results for the continuous time linear Hamiltonian systems as well as for the discrete symplectic systems. This paper also solves an open problem of deriving the oscillation theorem for problems with periodic boundary conditions. Furthermore, the present work demonstrates the utility and power of the analysis on time scales in obtaining new results especially in the classical continuous and discrete time theories.

The purpose of this paper is to establish fixed point theorems for operators in the Frechet space of continuous functions on the real half-axis. In our considerations we apply the technique of measures of noncompactness in conjunction with the Tikchonov fixed point theorem. The obtained results are applied in the proof of the solvability of a nonlinear functional integral equation with the initial value. Moreover, we show that solutions of that equation are uniformly globally asymptotically attractive. The results presented in the paper allow to improve existence theorems for integral and functional equations obtained earlier in many research papers.

This paper investigates a defect appearing in “Observer-based finite-time control of time-delayed jump systems”, which the observer-based finite-time H ∞ controller via state feedback could not ensure stochastic finite-time boundedness and stochastic finite-time stabilization with the prescribed H ∞ control index of the resulting closed-loop Markov jump systems. The corrected results are given and the improved optimal algorithms and new simulation results are also presented in this paper.

In this paper, we study the following singular eigenvalue problem for a higher order fractional differential equation - D α x ( t ) = λ f ( x ( t ) , D μ 1 x ( t ) , D μ 2 x ( t ) , … , D μ n - 1 x ( t ) ) , 0 t 1 , x ( 0 ) = 0 , D μ i x ( 0 ) = 0 , D μ x ( 1 ) = ∑ j = 1 p - 2 a j D μ x ( ξ j ) , 1 ⩽ i ⩽ n - 1 , where n ≥ 3 , n ∈ N , n - 1 α ⩽ n , n - l - 1 α - μ l n - l , for l = 1 , 2 , … , n - 2 , and μ - μ n - 1 > 0 , α - μ n - 1 ≤ 2 , α - μ > 1 , a j ∈ [ 0 , + ∞ ) , 0 ξ 1 ξ 2 ⋯ ξ p - 2 1 , 0 ∑ j = 1 p - 2 a j ξ j α - μ - 1 1 , D α is the standard Riemann–Liouville derivative, and f : ( 0 , + ∞ ) n → [ 0 , + ∞ ) is continuous. Firstly, we give the Green function and its properties. Then we established an eigenvalue interval for the existence of positive solutions from Schauder’s fixed point theorem and the upper and lower solutions method. The interesting point of this paper is that f may be singular at x i = 0 , for i = 1 , 2 , … , n .

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

Initial boundary value problems for the heat conduction equation with the constant and piecewise constant coefficients are considered in the paper. The Dirac delta function appears in the initial data. The considered problems are mathematical models of the temperature distribution arising from a pulse point heat source in homogeneous or two composite materials. The Fourier series expansion method is applied for solving these problems. The formal solutions are constructed in the form of the infinite series which are not classical functions. We show that these formal solutions are generalized functions (distributions). We prove that the partial sum of the constructed infinite series is a classical function which can be considered as a regularization of the generalized solution. The method of the computation of this regularized solution is suggested in the paper. The computational examples confirm the robustness of the proposed method. (C) 2011 Elsevier Inc. All rights reserved.

In this paper, we consider the initial-boundary value problem of the advection–diffusion equation with rapidly oscillating coefficients in both time and space domain. A multiscale asymptotic expansion of the solution for this kind of problem is presented. A full discrete finite element method for computing above mentioned problem is introduced. The proposed method not only can greatly save computer memory and CPU time, but also has high precision for approximating the solution. The numerical results show that the method presented in this paper is effective and reliable.

In this paper, the exact solution of the quadratic mixed-parity Helmholtz–Duffing oscillator is derived by using Jacobi elliptic functions. It is also shown that the exact period of oscillation is given as a function of the complete elliptic integral of the first kind. At the end of the paper, we examine the stability of the system and determine the regions for which periodic and unbounded motions take place.

In this paper, we have formulated a second-order mixed symmetric dual programs for a class of nondifferentiable multiobjective programming problem. Weak, strong and converse duality theorems are then proved for the aforementioned pair using the notion of second-order F-convexity/pseudoconvexity assumptions. Further, special cases are discussed to show that this paper extends some known results of the literature.

The nonlinear complementarity problems (denoted by NCPs) usually are reformulated as the solution of a nonsmooth system of equations. In this paper, we will present a regularized smoothing Newton method for solving nonlinear complementarity problems with P 0 -function ( P 0 -NCPs) based on the generalized smoothing Fischer–Burmeister NCP-function ϕ p ( μ , a , b ) with p > 1, where μ is smoothing parameter. Without requiring strict complementarity assumption at the P 0 -NCPs solution, the proposed algorithm is proved to be globally and superlinearly convergent under suitable assumptions. Furthermore, the algorithm is locally quadratic convergent under mild conditions. Numerical experiments indicate that the proposed method is quite effective. In addition, in this paper, the regularization parameter e in our algorithm is viewed as an independent variable, hence, our algorithm seems to be simpler and more easily implemented compared to many existing literatures.

In the paper [A. Rababah, S. Mann, Iterative process for G 2 -multi degree reduction of Bezier curves, Applied Mathematics and Computation 217 (2011) 8126–8133], Rababah and Mann proposed an iterative method for multi-degree reduction of Bezier curves with C 1 and G 2 -continuity at the endpoints. In this paper, we provide a theoretical proof for the existence of the unique solution in the first step of the iterative process, while the proof in their paper applies only in some special cases. Also, we give a complete convergence proof for the iterative method. We solve the problem by using convex quadratic optimization.

In this paper, we study the existence and uniqueness of mild solutions of neutral stochastic evolution equations with infinite delay and Poisson jumps in real separable Hilbert spaces. We study the continuous dependence of solutions on the initial value. The nonlinear term in our equations are not assumed to Lipschitz continuous. The results of this paper generalize and improve some known results.

In this paper, we investigate exponential stability for stochastic BAM networks with mixed delays. The mixed delays include discrete and distributed time-delays. The purpose of this paper is to establish some criteria to ensure the delayed stochastic BAM neural networks are exponential stable in the mean square. A sufficient condition is established by consructing suitable Lyapunov functionals. The condition is expressed in terms of the feasibility to a couple LMIs. Therefore, the exponential stability of the stochastic BAM networks with discrete and distributed delays can be easily checked by using the numerically efficient Matlab LMI toobox. A simple example is given to demonstrate the usefulness of the derived LMI-based stability conditions.

In this paper, we give a hybrid extragradient iterative method for finding the approximate element of the common set of solutions of a generalized equilibrium problem, a system of variational inequality problems, a variational inequality problem and a fixed point problem for a strictly pseudocontractive mapping in a real Hilbert space. Further we establish a strong convergence theorem based on this method. The results presented in this paper improves and generalizes the results given in Yao et al. [36] and Ceng et al. [7] , and some known corresponding results in the literature.

The paper concerns a problem of optimal reinsurance and investment in order to minimizing the probability of ruin. In the whole paper, the cedent’s surplus is allowed to invest in a risk-free asset and a risky asset and the company’s risk is reduced through proportional reinsurance, while in addition the claim process is assumed to follow a Brownian motion with drift. By solving the corresponding Hamilton–Jacobi–Bellman equations, the optimal reinsurance–investment strategy is derived. The presented results generalize those by Taksar [1] .

In this paper, we consider a hybrid projection method for finding a common element in the set of fixed points of a infinite family of asymptotically quasi-ϕ-nonexpansive mappings and in the set of solutions of a generalized mixed equilibrium problem. Some strong convergence theorems of common elements are established in a uniformly smooth and strictly convex Banach space which has the Kadec–Klee property. The results presented in the paper improve and extend some recent results.

The purpose of this paper is to establish some new matching theorems in G -convex spaces and, as applications, to obtain some new fixed point theorems, section theorems and a minimax theorem in G -convex spaces. The results presented in this paper improve and generalize the corresponding results in [1] , [2] , [3] , [4] , [5] , [7] , [8] , [9] , [10] , [11] , [12] .

By incorporating the Legendre multiwavelet into the discontinuous Galerkin (DG) method, this paper presents a novel approach for solving Poisson’s equation with Dirichlet boundary, which is known as the discontinuous Legendre multiwavelet element (DLWE) method, derive an adaptive algorithm for the method, and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. Furthermore, this paper generalizes the DLWE method to the general elliptic equations defined on a bounded domain and describes the possibilities of constructing optimal adaptive algorithm. The proposed method and its generalizations are also applicable to some other kinds of partial differential equations.

A stochastic predator–prey model is studied. Firstly, we prove the existence, uniqueness and positivity of the solution. Then, we show the upper bounds for moments and growth rate of population. In some cases, the growth rate is negative and the population dies out rapidly. The paper ends with some reviews for the paper [13] .

In this paper, we explain some well known experimental observations in fluid solid interaction from a thermodynamic perspective. In particular we use the extremum of the rate of entropy production to establish the stability of specific patterns observed in single and multiparticle sedimentation in an infinite fluid and the sedimentation of spheres in the presence of walls. While these phenomena have been explained numerically, there is no known rigorous theoretical argument to establish the stability of the observed configurations. We provide a very convincing theoretical basis using entropy based arguments that are considered by several scientists as the underlying theme of nature, life and evolution. In the absence of many rigorous examples for the entropy production principle, our paper advances this argument and lends it much credibility. In addition to looking at the rate of entropy production, we also put forth a very plausible heuristic argument based on the thermal gradients in the systems being studied, which could be the underlying causal principle for many known patterns in nature.

This paper deals with the blow-up of positive solutions for a nonlinear reaction–diffusion equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in finite time. Moreover, an upper bound of the blow-up time, an upper estimate of the blow-up rate, and an upper estimate of the global solutions are given. At last we give two examples to which the theorems obtained in the paper may be applied.

In this paper, we derive an alternative more condensed Cramer rule for the unique solution of some restricted left and right systems of quaternion linear equations. The findings of this paper extend some known results in the literature.

The paper presents a method to find the solution of the constant coefficient matrix differential Riccati differential in terms of solutions of algebraic Riccati and Lyapunov equations, and the state transition matrix (matrix exponential) of the corresponding linear dynamic system. The method presented represents an improved method of Potter–Anderson–Moore since the solution is obtained under milder assumptions than the original algorithm of Potter–Anderson–Moore. An aircraft and satellite examples done in the paper demonstrate the advantages of the improved algorithm.

A discrete nonlinear N-species cooperation system with time delays and feedback controls is considered in this paper. Sufficient conditions which ensure the permanence of the system are obtained. It is shown that these conditions are weaker than those of Chen [F.D. Chen, Permanence of a discrete N-species cooperation system with time delays and feedback controls, Appl. Math. Comput. 186(2007) 23–29], that is, our investigation shows that the additional condition in Chen’s paper is not necessary.

The purpose of this paper is by using the hybrid iterative method to prove some strong convergence theorems for approximating a common element of the set of solutions to a system of generalized mixed equilibrium problems and the set of common fixed points for two countable families of closed and asymptotically relatively nonexpansive mappings in Banach space. The results presented in the paper improve and extend the corresponding results of Su et al. [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890–3906], Li and Su [H.Y. Li, Y.F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72 (2) (2010) 847–855], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. TMA 73 (2010) 2260–2270], Kang et al. [J. Kang, Y. Su, X. Zhang, Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications, Nonlinear Anal. HS 4 (4) (2010) 755–765], Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory 134 (2005) 257–266], Tan et al. [J.F. Tan, S.S. Chang, M. Liu, J.I. Liu, Strong convergence theorems of a hybrid projection algorithm for a family of quasi-ϕ-asymptotically nonexpansive mappings, Opuscula Math. 30 (3) (2010) 341–348], Takahashia and Zembayashi [W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009) 45–57] and Wattanawitoon and Kumam [K. Wattanawitoon, P. Kumam, Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal. Hybrid Systems 3 (2009) 11–20] and others.

This paper is concerned with the dissipativity analysis problem for singular systems with time-varying delays. A delay-dependent criterion is established to guarantee the dissipativity of the underlying systems using the delay partitioning technique. All the results given in this paper are not only dependent upon the time delay, but also dependent upon the number of delay partitions. The effectiveness and the reduced conservatism of the derived results are demonstrated by two illustrative examples.

In this paper we propose a new method for pricing double-barrier options with moving barriers under the Black-Scholes and the CEV models. First of all, by applying a variational technique typical of the boundary element method, we derive an integral representation of the double-barrier option price in which two of the integrand functions are not given explicitly but must be obtained solving a system of Volterra integral equations of the first kind. Second, we develop an ad hoc numerical method to regularize and solve the system of integral equations obtained. Several numerical experiments are carried out showing that the overall algorithm is extraordinarily fast and accurate, even if the barriers are not differentiable functions. Moreover the numerical method presented in this paper performs significantly better than the finite difference approach. © 2011 Elsevier Inc. All rights reserved.

In this paper, we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts. Similar boundary value problems are encountered while simulating several real life processes for instance, first exit time problem in the modelling of neuronal variability. A rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem. In the development of numerical methods for constructing an approximation to the solution of the problem, a special type of mesh is generated to tackle the delay term along with the turning point. Then, to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term, an exponential fitting parameter is used. Several numerical examples are presented to support the theory developed in the paper.

This paper proposes a method for comparison of payoff configurations in the framework of characteristic function form games with non-transferable utility. The proposed method in this paper is a relation derived from objections and counter-objections. Some examples which show how the proposed method works are given. This paper presents propositions which show that the proposed method satisfies the properties, called strict partial order and independence from linear transformation and parallel shift. An example verifies that the proposed method does not satisfy the property called monotonicity.

This paper deals with the second order nonlinear neutral delay system of difference equations Δ [ a n Δ ( x n + b n x n - τ ) ] + f ( n , x h 1 n , … , x h kn , y w 1 n , … , y w kn ) = c n , n ⩾ n 0 , Δ [ p n Δ ( y n + q n y n - σ ) ] + g ( n , x s 1 n , … , x s kn , y t 1 n , … , y t kn ) = r n , n ⩾ n 0 . By virtue of the Banach fixed point theorem and some natural modifications of the techniques in the literature, we prove the existence results of uncountably many bounded positive solutions for the system of difference equations. Seven examples are constructed to explain the main results presented in this paper.

We in this paper consider the bisymmetric nonnegative definite solution with extremal ranks and inertias to a system of quaternion matrix equations AX = C , XB = D . We derive the extremal ranks and inertias of the common bisymmetric nonnegative definite solution to the system. The general expressions of the bisymmetric nonnegative definite solution with extremal ranks and inertias to the system mentioned above are also presented. In addition, we give a numerical example to illustrate the results of this paper.

Let ψ be a holomorphic function on the open unit disk D and φ a holomorphic self-map of D . Let C φ , M ψ and D denote the composition, multiplication and differentiation operator, respectively. We find an asymptotic expression for the essential norm of products of these operators on weighted Bergman spaces on the unit disk. This paper is a continuation of our recent paper concerning the boundedness of these operators on weighted Bergman spaces.

Numerical integration formulas in n -dimensional Euclidean space of degree three are discussed. In this paper, for the product regions a method is presented to construct numerical integration formulas of degree three with 2 n real points and positive weights. The presented problem is a little different from those dealt with by other authors. All the corresponding one-dimensional integrals can be different from each other and they are also nonsymmetrical. In this paper an n -dimensional numerical integration problem is turned into n one-dimensional moment problems, which simplifies the construction process. Some explicit numerical formulas are given. Furthermore, a more generalized numerical integration problem is considered, which will shed light on the final solution to the third degree numerical integration problem.

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.

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials and the generalized Cesaro polynomials. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.

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.

An improved meshless local Petrov–Galerkin method (MLPG) for stress analysis of two-dimensional solids is presented in this paper. The MLPG method based on the moving least-squares approximation is one of the recent meshless approaches. However, accurate imposition of essential boundary conditions in the MLPG method often presents difficulties because the MLPG shape functions does not possess the Kronecker delta property. In order to eliminate this shortcoming, this approach uses the moving Kriging interpolation instead of the traditional moving least-square approximation to construct the MLPG shape functions, and then, the Heaviside step function is used as the test function over a local sub-domain. In this method, the essential boundary conditions can be enforced as the FEM, no domain integration is needed and only regular boundary integration is involved. In addition, the sensitivity of several important parameters of the present method is mainly studied and discussed. Comparing with the original meshless local Petrov–Galerkin method, the present method has simpler numerical procedures and lower computation cost. The effectiveness of the present method for two-dimensional solids problem is investigated by numerical examples in this paper.

The stabilization of a class of Hamiltonian systems with state time-delay and input saturation is addressed in this paper. Sufficient conditions are derived by using Lyapunov–Krasovskii functional theorem to guarantee the systems as well as the resulted closed-loop systems by output feedback to be asymptotically stable when input saturation effectively occurs. A numerical example is presented to illustrate the effectiveness of the obtained results in this paper.

Usually, a reliability function is defined by a failure rate which is a real function taking the non-negative real values. In this paper the failure rate is assumed to be a stochastic process with non-negative and right continuous trajectories. The reliability function is defined as an expectation of a function of that random process. Particularly, the failure rate defined by the semi-Markov processes is considered here. The theorems dealing with the renewal equations for the conditional reliability functions with a semi-Markov process as a failure rate are presented in this paper. A system of that kind of equations for the discrete state space semi-Markov process is applied for calculating the reliability function for the 3-states semi-Markov random walk. Using the introduced system of renewal equations for the countable state space, the reliability function for the Furry–Yule failure rate process is obtained.

Asian options are popular path-dependent financial derivatives. This paper uses lattices to price fixed-strike European-style Asian options that are discretely monitored. The algorithm proposed can also be applied to floating-strike Asian options as well because fixed-strike and floating-strike Asian options are related through an equation. The discretely monitored version is usually found in practice instead of the continuously monitored version usually encountered in the literature. This paper presents the first provably quadratic-time convergent lattice algorithm for pricing fixed-strike European-style discretely monitored Asian options. It is the most efficient lattice algorithm with convergence guarantees. The algorithm relies on the Lagrange multipliers to choose the number of states for each node of the lattice. Extensive numerical experiments and comparisons with many existing numerical methods confirm the performance claims and the competitiveness of our algorithm. This result places fixed-strike European-style discretely monitored Asian options in the same complexity class as vanilla options.

A particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors, namely FF † and F † F , determined by a complex square matrix F and its Moore–Penrose inverse F † . Various functions of the projectors are considered from the point of view of their nonsingularity, idempotency, nilpotency, or their relation to the known classes of matrices, such as EP, bi-EP, GP, DR, or SR. This part of the paper was inspired by Benitez and Rakocevic [J. Benitez, V. Rakocevic, Matrices A such that AA † − A † A are nonsingular, Appl. Math. Comput. 217 (2010) 3493–3503]. Further characteristics of FF † and F † F , with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues, are derived as well. Besides establishing selected exemplary results dealing with FF † and F † F , the paper develops a general approach whose applicability extends far beyond the characteristics provided therein.