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In a recent paper referred to in the title, the author used the concept of quasi self-adjointness to obtain conservation laws for a system of the ( 2 + 1 ) -dimensional Bogoyavlensky–Konopelchenko equation. Apart from the adjoint system of equations, all the results on the quasi self-adjointness and the subsequent conservation laws obtained are inaccurate. In this comment, we clarify these inaccuracies and also generate conservation laws for a potential form of the underlying equation through Noether theorem.

Corrugated paper is produced by gluing three types of papers of the same breadth. Given a set of orders, we first assign each order to one of the standard breadths, and then sequence those assigned to each standard breadth so that they are continuously manufactured from the three rolls of the specified standard breadth equipped in the machine called corrugator. Here we are asked to achieve multi-goals of minimizing total length of roll papers, total loss of papers caused by the differences between standard breadths and real breadths of the orders, and the number of machine stops needed during production. We use integer programming to assign orders to standard breadths, and then develop a special purpose algorithm to sequence the orders assigned to each standard breadth. This is a first attempt to handle scheduling problems of the corrugator machine.

We propose a new concept which is a generalization of fuzzy soft subset and fuzzy soft equal. Using such notions, we will be able to consider the distributive law of fuzzy soft sets. Using the distributive law of fuzzy soft sets, we point out that the distributive law of trapezoidal fuzzy soft sets as proposed by Xiao et al. (2012) is not true. The correction will further improve further extensions of the results of Xiao et al. (2012). We will also establish the generalized distributive law of trapezoidal fuzzy soft sets along with illustrative examples.

A controllability analysis of a paper machine forming zone in a newsprint mill was performed to determine the maximum allowable variability of the pulp furnished to the process. The goal was to keep the output variables, basis weight and first-pass retention between certain limits in spite of disturbances and paper grade changes. The results indicate that the process is more disturbed at high frequencies, and that the two output variables may be maintained within the interval ±1% of their nominal values if the maximum variability in the consistency and fines content of the thick-stock is kept lower than ±1 and ±2%, respectively.

We analyze the multiauthorship matrix M, defined as the matrix where a cell M (j, @k) denotes the number of times authors with j publications are ranked as the @k^t^h author of an article. We prove that if the distribution of the number of authors per paper follows a power law, then the author rank distribution is approximately equal to this power law (more precisely, equal in Landau's big O sense). We further determine the author rank distribution in the case where authors can be characterized through a seed number; this is the probability of preceding a fixed author in the byline of an article. Such a seed is determined for alphabetical ranking of authors using the standard western alphabet.

Rock-paper-scissors games metaphorically model cyclic dominance in ecology and microbiology. In a static environment, these models are characterized by fixation probabilities obeying two different "laws" in large and small well-mixed populations. Here, we investigate the evolution of these three-species models subject to a randomly switching carrying capacity modeling the endless change between states of resources scarcity and abundance. Focusing mainly on the zero-sum rock-paper-scissors game, equivalent to the cyclic Lotka-Volterra model, we study how the ${\it coupling}$ of demographic and environmental noise influences the fixation properties. More specifically, we investigate which species is the most likely to prevail in a population of fluctuating size and how the outcome depends on the environmental variability. We show that demographic noise coupled with environmental randomness "levels the field" of cyclic competition by balancing the effect of selection. In particular, we show that fast switching effectively reduces the selection intensity proportionally to the variance of the carrying capacity. We determine the conditions under which new fixation scenarios arise, where the most likely species to prevail changes with the rate of switching and the variance of the carrying capacity. Random switching has a limited effect on the mean fixation time that scales linearly with the average population size. Hence, environmental randomness makes the cyclic competition more egalitarian, but does not prolong the species coexistence. We also show how the fixation probabilities of close-to-zero-sum rock-paper-scissors games can be obtained from those of the zero-sum model by rescaling the selection intensity., 31 pages, 15 figures: Main text (18 pages, 9 figures) followed by Supplementary Material (13 pages, 6 figures). Supplementary Information and resources available at https://doi.org/10.6084/m9.figshare.8858273.v1

This paper presents a new method to calculate the feedback control gain for a class of multivariable bilinear system, and also applied this method on the control of two sections of paper-making process with disturbance. The robust H∞ control problem is to design a state feedback controller such that the robust stability and a prescribed H∞ performance of the resulting closed-loop system are ensured. The controller turns out to be robust with respect to the disturbance in the plant. Utilizing the Schur complement and some variable transformations, the stability conditions of the multivariable bilinear systems are formulated in terms of Lyapunov function via the form of linear matrix inequality (LMI). The gain of controller will be calculated via LMI. Finally, the application examples of a headbox section and a dryer section of paper-making process are used to illustrate the applicability of the proposed method.

We introduce an algorithm for approximating a 2-manifold 3D mesh by a set of developable surfaces. Each developable surface is a generalized cylinder represented as a strip of triangles not necessarily taken from the original mesh. Our algorithm is automatic, creates easy-to-assemble pieces, and provides L"@K global error bounds. The algorithm consists of three stages. In the first stage, the mesh is segmented into meaningful components. The second stage approximates each of the mesh components in 3D by a set of triangle strips with smooth (non-jagged) boundaries that guarantee a maximal user-defined error relative to the original mesh. Finally, these strips are unfolded to the plane, resulting in flat patterns, which may be cut and assembled from paper. The approximation quality is controlled by a user-supplied parameter specifying the allowed Hausdorff distance between the input mesh and its piecewise-developable approximation. The strips generated by our algorithm may be parameterized to conform with the parameterization of the original mesh, if given, to facilitate texture mapping. We demonstrate this by physically assembling paper-craft models from the strips generated by our algorithm when run on several polygonal 3D mesh data sets.

In this note, we point out an error in the recently published article [R.K. Mohanty, M.K. Jain, D. Dhall, A cubic spline approximation and application of TAGE iterative method for the solution of two-point boundary value problems with forcing function in integral form, Appl. Math. Model. 35 (2011) 3036–3047] and then correct it.

In a recent paper (Computer Aided Geometric Design 23 (3), 243-260), Yang presented a novel kind of nonlinear and geometry driven subdivision scheme for curve interpolation. The author declared that this scheme was shape preserving when all free parameters were explicitly selected by some specific rules. In this note, we demonstrate that the author (a) ignored the consistency of the definition to straight edges, and also (b) did not give a perfect scheme and proof for the shape preserving subdivision.

The paper addresses ranking of factors with a use of the pairwise comparison method in sense of the analytic hierarchy process (AHP). The aim of this study is to show that the logarithmic least squares method used to approximate judgement matrix in order to calculate the ranking of factors does not cause the reduction of weight of a new alternative as the eigenvector method does.

Yang (the second author of this paper) et al. (2009) [3] established the distributive law of interval-valued fuzzy soft sets. In this article, it is pointed that it is not true in general. But, in order to establish the distributive law of interval-valued fuzzy soft sets, more general notions than interval-valued fuzzy soft sets and interval-valued fuzzy soft equal are considered. Using such notions, the generalized distributive law of interval-valued fuzzy soft sets is established.

Ahn, Lee, Park and Yoo proved that the best constrained degree reduction of a polynomial f in L"2-norm equals the best weighted Euclidean approximation of the Bernstein-Bezier coefficients of f in a paper, published in the journal, Computer Aided Geometric Design 21 (2) (2004) 181-191. In this note, we point out an error in their paper and give the correct result.

In a recently published article in EJOR, Christofides et al. (CAT) present a depth-first search, branch-and-bound solution procedure for the multiple-resource constrained, single project scheduling problem. While there are many important contributions in this paper, we show by counterexample that if the branching strategy described by the authors is used, the optimal solution might not result. Computational experience on a set of test problems appearing in the open literature is reported both with the original branching strategy suggested by the authors and a modified branching strategy that we propose. The modified strategy guarantees the determination of the optimal solution in all instances of the problem at the expense of an increase in node evaluations and average CPU time. Computational results using the revised procedure and a hybrid, breadth-first search procedure also investigated by CAT are reported.

In recent years, many results on oscillation criteria of second-order nonlinear neutral delay differential equations with distributed deviating argument have been obtained, but some of these criteria are incorrect due to a mistake in a crucial auxiliary result. We give two counterexamples.

A nonoscillation criterion for a second-order linear difference equation is established correcting a result in [1].

Two augmentation block triangular preconditioners were introduced by Shen et?al. (2012) for the regularized saddle point problem. However, the spectral analysis of the preconditioner based on the augmentation of the ( 2 , 2 ) block was not throughly derived there. In this short paper, we give a detailed spectral analysis on this preconditioner. A numerical example is employed to validate the efficiency of the presented theoretical results.

To safeguard analytical tractability and the concavity of objective functions, the vast majority of models belonging to oligopoly theory relies on the restrictive assumption of linear demand functions. Here we lay out the analytical solution of a differential Cournot game with hyperbolic inverse demand, where firms accumulate capacity over time à la Ramsey. The subgame perfect equilibrium is characterized via the Hamilton–Jacobi–Bellman equations solved in closed form both on infinite and on finite horizon setups. To illustrate the applicability of our model and its implications, we analyze the feasibility of horizontal mergers in both static and dynamic settings, and find appropriate conditions for their profitability under both circumstances. Static profitability of a merger implies dynamic profitability of the same merger. It appears that such a demand structure makes mergers more likely to occur than they would on the basis of the standard linear inverse demand.

The effect of inhibition on the firing variability is examined in this paper using the biologically-inspired temporal noisy-leaky integrator (TNLI) neuron model. The TNLI incorporates hyperpolarising inhibition with negative current pulses of controlled shapes and it also separates dendritic from somatic integration. The firing variability is observed by looking at the coefficient of variation (CV) (standard deviation/mean interspike interval) as a function of the mean interspike interval of firing (ΔtM) and by comparing the results with the theoretical curve for random spike trains, as well as looking at the interspike interval (ISI) histogram distributions. The results show that with 80% inhibition, firing at high rates (up to 200 Hz) is nearly consistent with a Poisson-type variability, which complies with the analysis of cortical neuron firing recordings by Softky and Koch [1993, J. Neurosci. 13(1) 334-530]. We also demonstrate that the mechanism by which inhibition increases the CV values is by introducing more short intervals in the firing pattern as indicated by a small initial hump at the beginning of the ISI histogram distribution. The use of stochastic inputs and the separation of the dendritic and somatic integration which we model in TNLI, also affect the high firing, near Poisson-type (explained in the paper) variability produced. We have also found that partial dendritic reset increases slightly the firing variability especially at short ISIs. Copyright © 2000 Elsevier Science Ireland Ltd. 58 1-3 41 48 Cited By :18

This paper studies a two-machine cross-docking flow shop scheduling problem in which a job at the second machine can be processed only after the processing of some jobs at the first machine has been completed. The objective is to minimize the makespan. We first show that the problem is strongly NP-hard. Some polynomially solvable special cases are provided. We then develop a polynomial approximation algorithm with an error-bound analysis. A branch-and-bound algorithm is also constructed. Computational results show that the branch-and-bound algorithm can optimally solve problems with up to 60 jobs within a reasonable amount of time.

This paper studies the two-machine permutation flowshop scheduling problem with anticipatory setup times and an availability constraint imposed only on the first machine. The objective is to minimize the makespan. Under the assumption that interrupted jobs can resume their operations, we present a polynomial-time approximation scheme for this problem.

In this paper, we consider single machine scheduling problem in which job processing times are controllable variables with linear costs. We concentrate on two goals separately, namely, minimizing a cost function containing total completion time, total absolute differences in completion times and total compression cost; minimizing a cost function containing total waiting time, total absolute differences in waiting times and total compression cost. The problem is modelled as an assignment problem, and thus can be solved with the well-known algorithms. For the case where all the jobs have a common difference between normal and crash processing time and an equal unit compression penalty, we present an O(n log n) algorithm to obtain the optimal solution.

A variable time delay compensator based on the Smith predictor is proposed. Finite difference and cinematic techniques are employed for a numerical approximation of the variable delay. The resulting multivariable controller is analyzed for robustness using the structured singular value. The technique is demonstrated through two simulated systems: a single-input, single-output numerical example, and a cross-direction paper machine control problem.

A method of tuning for PI controllers with one-way lead/lag decoupling tuning is demonstrated for 2×2 input-output systems, based on finite frequency response data. The frequency response estimates are obtained from closed loop tests using an identification technique based on bandpass filters. The open-loop frequency response is calculated and used to tune PI controllers and to fit lead-lag compensators for decoupling by weighted least-squares. Separate optimization of the PI controllers and the lead-lag compensator is preferred. The tuning takes into account violations of desired multivariable robustness criteria. The method is illustrated by application to a paper machine headbox simulation and by experiments in distillation column temperature control.

The definition of the analytic blossom introduced in (Morin and Goldman, 2001) attributes a special role to the value zero. Hans-Peter Seidel (private communication) pointed out that this definition is somehow troublesome, since the underlying domain is affine, and therefore all points in the domain should play the same role. In this short paper, we address this issue and propose a more general definition of the blossom of an analytic function that is not linked with the zero of the coordinate system.

Features of resonant operator equations are discussed, and existence results presented for nonlinear resonant integral equations.

This short paper extends the results of the paper on ‘Fitting continuous-time and discrete-time models using discrete-time data and their application’ (Zhu J., Shieh, L. S. and Yates, R. E.: Appl. Math. Modelling 1985, 9, 93) to systems with multiple poles.

We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Theorem. The mass matrix is computed through an approximate L 2 polynomial projection, whose properties are carefully investigated in the paper. We prove the well-posedness of the resulting scheme in two different ways that reveal the contractive nature of the VEM and its connection with the minimization of quadratic functionals. The convergence analysis requires the existence of a nonnegative quasi-interpolation operator, whose construction is also discussed in the paper. The variational crime introduced by the virtual element setting produces five error terms that we control by estimating a suitable upper bound. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time on three different mesh families including distorted squares, nonconvex elements, and Voronoi tesselations.

The paper presents a method for determining maintenance and spare provisioning costs, for systems containing a non exponential component. The input data of the model are R ( t ) - the reliability function with time of the component, T the duration of the life cycle, cost of a repair, cost of a spare part and the turnaround time. The model is applicable for any form of R ( t ) : Weibull, Normal, expressed by an explicit or implicit set of equations and even in the form of discrete frequency table. The calculations can be performed by a simple Excel worksheet and do not require expensive tools (e.g. a flexible Monte Carlo package): The algorithm requires a few iterations of the dot product of two matrices. The entries of the matrices are calculated from the input values and the results of previous iterations, at M equidistant time points along the life cycle. M determines the accuracy of the calculations. The paper indicates sufficient conditions for M , which ensure a desired accuracy. Numerical examples performed on an Excel worksheet are presented. Since the model is applicable for any R ( t ) , it was applied, for verification purposes for R ( t ) = e − λ t , which has well known accurate results. The comparison confirmed the accuracy of the approximations.

Funding Information: We wish to thank the reviewers for useful suggestions and comments, which greatly improved our paper. Publisher Copyright: © 2021 The Authors Our paper focuses on a robustness analysis of efficiency scores in the context of Data Envelopment Analysis (DEA) assuming interval scale data, as defined in A. Dehnokhalaji, P. J. Korhonen, M. Köksalan, N. Nasrabadi and J. Wallenius, “Efficiency Analysis to incorporate interval scale data”, European Journal of Operational Research 207 (2), 2010, pp. 1116–1121. We first show that the definition of the efficiency score used in our paper is a well-defined measure according to Aparicio and Pastor (J. Aparicio and J. T. Pastor, “A well-defined efficiency measure for dealing with closest targets in DEA”, Applied Mathematics and Computation 219 (17), 2013, pp. 9142–9154.). Next, we characterize how robust the efficiency scores are with respect to improvements and deteriorations of inputs and outputs. We illustrate our analysis with two examples: a simple numerical example and a more complex example using real-world data.

This paper is concerned with fast, parallel and numerically accurate pricing of two-asset American options under the Merton jump-diffusion model, which gives rise to a two-dimensional partial integro-differential complementarity problem (PIDCP) with a nonlocal two-dimensional integral term. Following method-of-lines approach, the solution to the PIDCP can be computed quite accurately by a robust numerical technique that combines Ikonen–Toivanen splitting with an alternating direction implicit scheme. However, we observed that computing the numerical solution with this technique becomes extremely time consuming, mainly due to the handling of the integral term. In this paper we parallelize this technique by applying a parallel fast Fourier transformation algorithm to all matrix-vector multiplications involving the huge and dense integral approximation matrix by exploiting its block Toeplitz with Toeplitz block structure. We also parallelize other computationally intensive steps of this technique by applying a recently developed parallel cyclic reduction algorithm for pentadiagonal systems. Our solutions computed on a graphics processing unit (GPU) using CUDA® platform are compared for accuracy with those available in the literature. It is observed that by solving the PIDCP parallelly we could bring down the computational times from several hours to a few seconds in certain cases in our experiments.

In a previous paper based on the interpolation finite difference method, a calculation system was shown for calculating 1D (one-dimensional) Laplace’s equation and Poisson’s equation using high-order difference schemes. Finite difference schemes, from the usual second-order to tenth-order differences, including odd number order differences, were systematically and instantaneously derived over equally/unequally spaced grid points based on the Lagrange interpolation function. Using the direct method with the band diagonal matrix algorithm, 1D Poisson equations were numerically calculated under double precision floating arithmetic, but it became clear that high accurate calculations could not be secured in high-order differences because “digit-loss errors” caused by the finite precision of computations occurred in the calculations when using the high-order differences. The double precision calculation corresponds to 15 (significant) digit calculation. In this paper, we systematically investigate how the calculation accuracy changes by high precision calculations (30-digit, and 45-digit calculations). Under 45-digit calculation, where the digit-loss error can be almost ignored, the high-order differences enable extremely high-accurate calculations.

Large deviations principles characterize the exponential decay rates of the probabilities of rare events. Cerrai and Rockner (2004) proved that systems of stochastic reaction–diffusion equations satisfy a large deviations principle that is uniform over bounded sets of initial data. This paper proves uniform large deviations results for a system of stochastic reaction–diffusion equations in a more general setting than Cerrai and Rockner. Furthermore, this paper identifies two common situations where the large deviations principle is uniform over unbounded sets of initial data, enabling the characterization of Freidlin–Wentzell exit time and exit shape asymptotics from unbounded sets.

Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem − ∇ 2 u = f in Ω ∖ Γ with Dirichlet boundary condition such that f is smooth in Ω ∖ Γ and the jump functions [ u ] and [ ∇ u ⋅ n → ] across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of − ∇ 2 u = f + g δ Γ in Ω as a special case. Because the source term f is possibly discontinuous across the interface curve Γ and contains a delta function singularity along the curve Γ, both the solution u of the Poisson interface problem and its flux ∇ u ⋅ n → are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Γ splits Ω into two disjoint subregions Ω + and Ω − . The coefficient matrix A in the resulting linear system A x = b , following from the proposed scheme, is independent of any source term f, jump condition g δ Γ , interface curve Γ and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in Ω + or Ω − . The constant coefficient matrix A facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.

All Italian consonants affected by gemination, that is affricates, fricatives, liquids, nasals, and stops, were analyzed within a project named GEMMA that lasted over a span of about 25 years. Results of the analysis on stops, as published in (Esposito, A., and Di Benedetto, M. G. (1999). “Acoustic and Perceptual Study of Gemination in Italian Stops,” The Journal of the Acoustical Society of America, ASA, Vol. 30, pp. 175-185) showed that the main acoustic cue to gemination in Italian was closure duration, while frequency and energy domain parameters were not significantly affected by gemination. This paper - the first of a set of two covering all remaining consonants - addresses nasals and liquids; its companion paper addresses affricates and fricatives. Results on nasals and liquids confirm the findings on stops, in particular that the primary acoustic cue to gemination in Italian is durational in nature and corresponds to a lengthened consonant duration. Results also show an inverse correlation between consonant and pre-consonant vowel durations which is, however, also present when considering singleton vs. geminate word sets separately, indicating a sort of duration compensation between these segments to eventually preserve rhythmical structures; this inverse correlation is reinforced when considering singleton and geminate sets combined. Classification tests of singleton vs. geminate consonants show that, for both nasals and liquids, best classification scores are obtained when consonant duration is used as a classification parameter. Although slightly less performing, the ratio between consonant and pre-consonant vowel durations is also a potential good candidate for automatic classification of geminate vs singleton nasals and liquids in Italian.

The post-buckling of a clamped column made of nonlinear elastic material and subject to axial compression is investigated in this paper. The Ramberg−Osgood type constitutive relation is adopted and it is expanded by using Taylor series. Based on Euler−Bernoulli beam theory, the exact governing equations are expressed in terms of rotation angle. Because the equations contain strongly nonlinear terms that are functions of the rotation angle, analytical solutions are virtually impossible. In this respect, this paper is focused on presenting an alternative method to construct concise yet accurate analytical approximate solutions for post-buckling of the Ramberg−Osgood column that is related to large rotation amplitude. The improved harmonic balance method is used to solve the nonlinear governing equations which are simplified via the Maclaurin series expansion and orthogonal Chebyshev polynomials. In addition, numerical solutions by applying the shooting method on the governing equations are obtained for comparison. The second-order analytical approximate solutions presented in this paper show excellent accuracy by comparing with numerical solutions. The analytical approximate method and numerical result presented in this paper can be applied as design guidelines for designing engineering structures that sustain large deformation, such as slender nonlinear compression of aluminum alloy columns, rods or braces.

In this paper, we consider an inverse problem for time-fractional advection-dispersion equation in a multi-layer composite medium. The main goal of our paper is to approximate the initial information, which is inaccessible for measurement, from the observation data at a certain point in second layer by constructing a regularized solution using a filter regularization method. Under appropriate regularity assumptions of the exact solution, we give convergence rate of the error between the reconstructed solution and the exact one. A numerical example is provided to illustrate the main results.

The use of constant order differential equations to describe the evolution of complex systems is often unable to describe some of the changing characteristics of the systems accurately. Variable order fractional derivatives provide us with new tools to solve such problems. In this paper, the accumulation and derivative orders of the classic grey model are expanded from constants to functions, and a variable order fractional grey model is established to describe the evolution process of complex systems. Firstly, this paper defines the variable order fractional accumulation generation sequence. On the basis of this sequence, a variable order fractional derivative grey model is established, the parameters of the model are estimated using the least square method, and the quantum particle swarm optimization algorithm is used to solve the order of fractional derivative and accumulation. Sadik transform and Laplace transform are adopted to obtain the analytical solution of the new model. Lastly, the effectiveness of the new model is verified through four cases. Compared with other models, the variable order fractional model can describe the development process of complex systems more accurately.

In this paper we intend to construct a structure preserving difference scheme for two-dimensional space-fractional nonlinear Schrodinger (2D SFNS) equation with the integral fractional Laplacian. The temporal direction is discretized by the modified Crank-Nicolson method, and the spatial variable is approximated by a novel fractional central difference method. The mass and energy conservations and the convergence are rigorously proved for the proposed scheme. For 1D SFNS equation, the convergence relies heavily on the L ∞ -norm boundness of the numerical solution of the proposed scheme. However, we cannot obtain the L ∞ -norm boundness of the numerical solution by using the similar process for the 2D SFNS equation. One of the major significance of this paper is that we first obtain the L ∞ -norm boundness of the numerical solution and L 2 -norm error estimate via the popular “cut-off” function for the 2D SFNS equation. Further, we reveal that the spatial discretization generates a block-Toeplitz coefficient matrix, and it will be ill-conditioned as the spatial grid mesh number M and the fractional order α increase. Thus, we exploit an linearized iteration algorithm for the nonlinear system, so that it can be efficiently solved by the Krylov subspace solver with a suitable preconditioner, where the 2D fast Fourier transform (2D FFT) is applied in the solver to accelerate the matrix-vector product, and the standard orthogonal projection approach is used to eliminate the drift of mass and energy. Extensive numerical results are reported to confirm the theoretical analysis and high efficiency of the proposed algorithm.

Charnes, Cooper, Seiford, and Stutz (1982, 1983) and Banker and Maindiratta (1986) suggested multiplicative radial measures for efficiency gauging. More recently, Peyrache and Coelli (2009) presented a multiplicative directional distance function that was elaborated by Mehdiloozad, Sahoo, and Roshdi (2014). In this paper, we extend these studies and propose an exponential approach based upon a new exponential distance function endowed with a multiplicative production structure. The main purposes of this paper are twofold: one is to provide a general production theoretic basis for the approach and the other is to extend it to a nonparametric framework. The first purpose is accomplished as follows: (1) the exponential distance function is formally defined and its properties are established; (2) it is shown how the exponential distance function is characterized under a Napierian technology; (3) a duality relationship between Napierian profit and the exponential distance functions is established; (4) shadow prices of inputs and outputs are derived based on the Napierian technology. The second purpose is accomplished by providing nonparametric programming extensions, which include data envelopment analysis (DEA) models, productivity indexes and returns to scale models. Here, the efficacy of our nonparametric theoretical results is demonstrated by applying DEA to the data on accommodation establishments in OECD.

In this paper, two linearized Galerkin finite element methods, which are based on the L1 approximation and the WSGD operator, respectively, are proposed to solve the nonlinear fractional Rayleigh-Stokes problem. In order to obtain the unconditionally optimal error estimate, we firstly introduce a time-discrete elliptic equation, and derive the unconditional error estimate between the exact solution and the solution of the time-discrete system in H 2 -norm. Secondly, we obtain the boundedness of the fully discrete finite element solution in L ∞ -norm through the more detailed study of the error equation. Then, the optimal L 2 -norm error estimate is derived for the fully discrete system without any restriction conditions on the time step size. Finally, some numerical experiments are presented to confirm the theoretical results, showing that the two linearized schemes given in this paper are efficient and reliable.