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In the work the almost everywhere (a.e.) convergence (absolute convergence) of series by the general Haar and Franklin systems corresponding to weakly regular division of the segment are compared. It is proved that if a series by the general Haar system diverges (absolutely diverges) on a set , then the series by the general Franklin system with the same coefficients diverges (absolutely diverges) a.e. in . As a consequence, it is obtained that if a sequence is not a Weyl multiplier for unconditional a.e. convergence of series by the general Haar system, then it is not a Weyl multiplier for unconditional a.e. convergence of series by the general Franklin series. [ABSTRACT FROM AUTHOR]

In the paper, the sufficient conditions are found for the convergence, summability by the Cesàro methods and unconditional convergence almost everywhere of orthogonal series, which are equivalent to the well-known theorems of Menshov–Redemacher, Menshov, and Orlich. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyze a second order unconditionally convergent mixed finite element method for solving incompressible magnetohydrodynamic problem based on magnetic vector potential. We utilize the second-order backward difference formula within the framework of the mixed finite element method and linearize the nonlinear terms by using the extrapolated technique. The finite element approximation is proposed with the fluid equations are discretized by Taylor–Hood type elements and the magnetic vector potential equation by edge elements. As a result, the divergence-free condition on the magnetic induction preserves exactly on the discrete level. Unconditional error estimates for the velocity and magnetic vector potential are established in the sense that no any restrictions are imposed on the relationship between the time-step size and the spatial size. Finally, several three-dimensional numerical simulations are carried out to illustrate the developed scheme, including convergence tests and some benchmark problems, such as the driven cavity problems and hydromagnetic Kelvin–Helmholtz instability [ABSTRACT FROM AUTHOR]

Dynamic soil-water coupling analyses, based on the u-p formulation, are inapplicable to highly permeable soils, causing numerical instability. In this study, it is demonstrated that theoretical solutions to the u-p formulation itself certainly exhibit unconditional convergence regardless of the permeability coefficient. This suggests that the instability is only numerical and can be observed in a temporally discretized system. Firstly, the linearized governing equation for the u-p formulation was proven to be reduced to a damped wave equation under a one-dimensional condition, similar to the Full formulation. Secondly, theoretical solutions for the u-p formulation were derived and their unconditional convergence was confirmed. Then, the essential characteristics of the u-p theoretical solutions, that is, the underestimation of permeability, overestimation of compression wave celerity, and occurrence of negative pore water pressure against positive load application, were described and compared with theoretical solutions for the Full formulation.

The paper considers the issues of convergence and summability of Fourier series of the functions of class for general orthonormal systems (GOSs). The sufficient conditions for the GOS functions are found that make the Fourier series in this system of each function from the class converging almost everywhere, unconditionally converging, or being summable by the method , . The sharpness of some results is proved. [ABSTRACT FROM AUTHOR]

This paper is concerned with the unconditionally optimal H1-error estimate of a fast second-order scheme for solving nonlinear subdiffusion equations on the nonuniform mesh. We use the Galerkin finite element method (FEM) to discretize the spacial direction, the Newton linearization method to approximate the nonlinear term and the sum-of-exponentials (SOE) approximation to speed up the evaluation of Caputo derivative. Our analysis of the unconditionally optimal H1-error estimate involves the temporal-spatial error splitting approach, an improved discrete fractional Grönwall inequality and error convolution structure. In order to find a suitable test function to estimate H1-error, we here consider two cases: linear and high-order FEM space, using the time-discrete operator and Laplace operator in the test function respectively. Numerical tests are provided demonstrate the effectiveness and the unconditionally optimal H1-error convergence of our scheme. [ABSTRACT FROM AUTHOR]

This paper is concerned with a linearized second-order finite difference scheme for solving the nonlinear time-fractional Schrödinger equation in d (d = 1,2,3) dimensions. Under a weak assumption on the nonlinearity, the optimal error estimate of the numerical solution is established without any restriction on the grid ratio. Besides the standard energy method, the key tools for analysis include the mathematical induction method, several inverse Sobolev inequalities, and a discrete fractional Gronwall-type inequality. The convergence rate of the proposed scheme is of O(τ2 + h2) with time step τ and mesh size h. Numerical results are carried out to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

If { x n } n ∈ N is a frame for a Hilbert space H, then there exists a canonical dual frame { x ~ n } n ∈ N such that for every x ∈ H we have x = ∑ ⟨ x , x ~ n ⟩ x n , with unconditional convergence of this series. However, if the frame is not a Riesz basis, then there exist alternative duals { y n } n ∈ N and synthesis pseudo-duals { z n } n ∈ N such that x = ∑ ⟨ x , y n ⟩ x n and x = ∑ ⟨ x , x n ⟩ z n for every x. We characterize the frames for which the frame series x = ∑ ⟨ x , y n ⟩ x n converges unconditionally for every x for every alternative dual, and similarly for synthesis pseudo-duals. In particular, we prove that if { x n } n ∈ N does not contain infinitely many zeros then the frame series converge unconditionally for every alternative dual (or synthesis pseudo-duals) if and only if { x n } n ∈ N is a near-Riesz basis. We also prove that all alternative duals and synthesis pseudo-duals have the same excess as their associated frame. [ABSTRACT FROM AUTHOR]

We define a topological ring $R$ to be Hirsch, if for any unconditionally convergent series $\sum_{n\in\omega} x_i$ in $R$ and any neighborhood $U$ of the additive identity $0$ of $R$ there exists a neighborhood $V\subseteq R$ of $0$ such that $\sum_{n\in F} a_n x_n\in U$ for any finite set $F\subset\omega$ and any sequence $(a_n)_{n\in F}\in V^F$. We recognize Hirsch rings in certain known classes of topological rings. For this purpose we introduce and develop the technique of seminorms on actogroups. We prove, in particular, that a topological ring $R$ is Hirsch provided $R$ is locally compact or $R$ has a base at the zero consisting of open ideals or $R$ is a closed subring of the Banach ring $C(K)$, where $K$ is a compact Hausdorff space. This implies that the Banach ring $\ell_\infty$ and its subrings $c_0$ and $c$ are Hirsch. Applying a recent result of Banakh and Kadets, we prove that for a real number $p\ge 1$ the commutative Banach ring $\ell_p$ is Hirsch if and only if $p\le 2$. Also for any $p\in (1,\infty)$, the (noncommutative) Banach ring $L(\ell_p)$ of continuous endomorphisms of the Banach ring $\ell_p$ is not Hirsch.

We consider problems of unconditional convergence of Fourier series of functions with respect to general orthonormal systems (ONSs). Sufficient conditions on the functions of an ONS are found under which the Fourier series of every function with respect to this system converges unconditionally. We show that some of the obtained results are sharp. We also prove that from any ONS one can extract a subsequence with respect to which the Fourier series of every function converges unconditionally. [ABSTRACT FROM AUTHOR]

S. Banach proved that for a function, even with good differentiability properties, the convergence of its Fourier series with respect to the general orthonormal systems (ONS) is not guaranteed. In the present paper, we find conditions on the functions of an ONS, under for which the Fourier series of differentiable functions are a.e. unconditionally convergent. The obtained results are best possible. We also prove that any ONS contains a subsystem such that the Fourier series of any function with derivatives of bounded variation are unconditionally convergent a.e. on [0, 1]. [ABSTRACT FROM AUTHOR]

In this study, we investigate the numerical discretization of two-dimensional fractional Laplacian diffusion equations. After discretizing the space derivative, we split the discrete fractional Laplacian into two parts. One is the τ preconditioner multiplying a constant, solved using the leap-frog scheme, while the other part and the remaining nonlinear term are treated with an explicit scheme. This results in what is termed the stiff-cut leap-frog scheme, which is more efficient than the traditional leap-frog scheme. The main advantage of the proposed scheme is that the coefficient matrices of the generated linear systems belong to τ algebra, which can be diagonalized by fast discrete sine transform with the complexity of O (N log N) , where N is the freedom of unknowns. Hence, no iterative method is required at each time integration. Moreover, we theoretically prove that the proposed scheme is unconditionally stable and converges in second order. Ultimately, numerical experiments are conducted to demonstrate the precision and effectiveness of the stiff-cut leap-frog scheme. [ABSTRACT FROM AUTHOR]

In this paper, the unconditionally optimal H 1 -norm error estimate for a two-step fast L1 scheme is considered for a nonlinear subdiffusion equation, which involves the considerations of both the initial singularity of the solution and the computational complexity to evaluate the Caputo derivative. For the fully discrete scheme, we use the fast L1 scheme developed in [28] to speed up the evaluation of the Caputo derivative on nonuniform time step, employ the Galerkin finite element (FE) method to discretize the spatial direction, and take the Newton linearized method to approximate the nonlinear term. While the theory of the L 2 -norm error estimate is well studied, there are few works on the unconditional convergence of H 1 -norm error estimate of the linearized Galerkin FE scheme for nonlinear subdiffusion equations. The main reason is that it requires to find a suitable test function in the FE space. To this end, we here respectively take the time-discrete operator and Laplace operator as the suitable test function in the FE space, and combine an improved discrete fractional Grönwall inequality to achieve the unconditional error estimate in H 1 -norm. Numerical results are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Labour productivity growth in manufacturing is decomposed into technical efficiency change (movement towards or away from the frontier), technological progress (shifts in the production frontier), and capital accumulation (movement along the frontier) using a nonparametric production frontier method. The results suggest that labour productivity growth is primarily driven by capital accumulation and to a lesser extent technological progress, while technical efficiency is deteriorating over the period 1995–2014. We find that technological progress appears to be non-neutral, as the world manufacturing production frontier expands only at higher capital intensities, benefiting highly industrialized nations. We also find evidence of unconditional convergence in labour productivity for the manufacturing sector. Capital accumulation is the main driver of the observed unconditional convergence, whereas technological change is contributing to divergence rather than convergence. The findings suggest that expanding manufacturing activities through capital accumulation is essential for developing countries to catch-up with developed countries. [ABSTRACT FROM AUTHOR]

Throughout this paper, K denotes a complete, non-trivially valued, ultrametric (or non-archimedean) field. Entries of infinite matrices, sequences and infinite series are in K. In this paper, following ([4], [5]), we prove some more Steinhaus type theorems over K. [ABSTRACT FROM AUTHOR]

• It is difficult to develop efficient conservative difference schemes for space fractional Klein-Gordon-Zakharov system. • New auxiliary equations are introduced to transform the original fractional system into the equivalent system of equations. • Two novel linearly implicit conservative difference schemes for the transformed system are developed and analyzed. • The discrete energy conservation properties, unconditional convergence of the suggested schemes are proved rigidly. • Numerical results are exhibited to study the physical behavior of the wave propagation and confirm efficiency of schemes. In this article, novel linearized implicit difference schemes with energy conservation property for fractional Klein-Gordon-Zakharov system are constructed and analyzed. The important feature of the article is that new auxiliary equations ∂ u ∂ t = − v and ∂ ϕ ∂ t = ∂ 2 ψ ∂ x 2 are introduced to transform the original fractional Klein-Gordon-Zakharov system into an equivalent system of equations exactly. Especially, two kinds of efficacious difference operators, the leap-frog and modified Crank-Nicolson methods are respectively utilized to establish the linearized implicit difference schemes with energy conservation property for simulating the propagation of transformed equations. And above all, by employing the discrete energy method, we have proven that the constructed difference algorithms enjoy the convergence order of O (Δ t 2 + h 2) and O (Δ t 2 + h 4) in L ∞ - and L 2 -norms, without imposing any restrictive conditions on the grid ratio compared with the existing literature. Two numerical examples are carried out to investigate the physical behaviors of the wave propagation and substantiate the effectiveness of the suggested schemes. [ABSTRACT FROM AUTHOR]

This paper is concerned with unconditionally optimal error estimates of linearized leap-frog Galerkin finite element methods (FEMs) to numerically solve the d-dimensional (d = 2 , 3) nonlinear Klein–Gordon–Schrödinger (KGS) equation. The proposed FEMs not only conserve the mass and energy in the given discrete norm but also are efficient in implementation because only two linear systems need to be solved at each time step. Meanwhile, an optimal error estimate for the proposed methods is derived by using the temporal-spatial error splitting techniques, which split the error between the exact solution and the numerical solution into two parts, i.e., the temporal error and the spatial error. Since the spatial error is τ -independent, the boundedness of the numerical solution in L ∞ -norm follows an inverse inequality immediately without any restriction on the grid ratios. Then, the optimal L 2 error estimates for r-order FEMs are derived unconditionally. Numerical results in both two and three dimensional spaces are given to confirm the theoretical predictions and demonstrate the efficiency of the methods. [ABSTRACT FROM AUTHOR]

It is shown that the Laurent series of a holomorphic function smooth up to the boundary on a Reinhardt domain in C n converges unconditionally to the function in the Fréchet topology of the space of functions smooth up to the boundary. [ABSTRACT FROM AUTHOR]

In this paper, we establish unconditionally optimal error estimates for linearized backward Euler Galerkin finite element methods (FEMs) applied to nonlinear Schrödinger-Helmholtz equations. By using the temporal-spatial error splitting techniques, we split the error between the exact solution and the numerical solution into two parts which are called the temporal error and the spatial error. First, by introducing a time-discrete system, we prove the uniform boundedness for the solution of this time-discrete system in some strong norms and derive error estimates in temporal direction. Second, by the above achievements, we obtain the boundedness of the numerical solution in L ∞ -norm. Then, the optimal L2 error estimates for r-order FEMs are derived without any restriction on the time step size. Numerical results in both two- and three-dimensional spaces are provided to illustrate the theoretical predictions and demonstrate the efficiency of the methods. [ABSTRACT FROM AUTHOR]

Dynamic soil-water coupling analyses, based on the u - p formulation, are inapplicable to highly permeable soils, causing numerical instability. In this study, it is demonstrated that theoretical solutions to the u - p formulation itself certainly exhibit unconditional convergence regardless of the permeability coefficient. This suggests that the instability is only numerical and can be observed in a temporally discretized system. Firstly, the linearized governing equation for the u - p formulation was proven to be reduced to a damped wave equation under a one-dimensional condition, similar to the Full formulation. Secondly, theoretical solutions for the u - p formulation were derived and their unconditional convergence was confirmed. Then, the essential characteristics of the u - p theoretical solutions, that is, the underestimation of permeability, overestimation of compression wave celerity, and occurrence of negative pore water pressure against positive load application, were described and compared with theoretical solutions for the Full formulation. [ABSTRACT FROM AUTHOR]

A linearized orthogonal spline collocation (OSC) method with C1 splines of degree ≥3 on a suitable graded mesh is formulated and analyzed for approximate solution of an initial‐boundary‐value problem of semilinear subdiffusion equations with nonsmooth solutions in time. The sharp error estimate in the L2 norm is established without any restriction on the relative temporal and spatial mesh sizes. Such unconditional convergence results are proved by including the typical singularity of the solution near the time t = 0. Results of numerical experiments support the analytical results. [ABSTRACT FROM AUTHOR]

In this study, we design and analyse an efficient and accurate Fourier pseudo-spectral method for solving the nonlinear Schrödinger equation with wave operator (NLSW). In this method, a modified leap-frog finite difference method is adopted for time discretization and a Fourier pseudo-spectral method is employed for spatial discretization. A new type of discrete energy functional is defined in a recursive way. It is then proved that the proposed method preserves the total energy in the discrete sense. Thanks to the FFT algorithm, the proposed method can be explicitly computed in the practical computation. Besides the standard energy method, the key tools used in our numerical analysis are a mathematical induction argument and a lifting technique. Without any restriction on the grid ratio, we prove that the proposed method is of spectral accuracy in space and second-order accuracy in time. Numerical results are reported to verify the error estimate and energy conservation of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we establish the unconditionally optimal error estimates of linearized second-order backward difference formula (BDF2) Galerkin finite element methods (FEMs) for the Landau-Lifshitz equation describing the magnetic behavior in ferromagnetic materials. By using the temporal-spatial error splitting techniques, we split the error between the exact solution and the numerical solution into two parts which are called the temporal error and the spatial error. First, we analyze the temporal error by introducing a time-discrete system and derive some regularity of the solution of the time-discrete system. Second, by the above achievements, we obtain the τ -independent spatial error and the boundedness of the numerical solution in L ∞ -norm. Then, the optimal L 2 and H 1 error estimates for r -th order FEMs (r = 1 , 2) are derived without any restriction on the time step size. Numerical results in both two and three dimensional spaces are presented to confirm the theoretical predictions and demonstrate the efficiency of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonstationary magnetohydrodynamic coupled heat equation through the well-known Boussinesq approximation. The Crank-Nicolson extrapolation scheme is used for time derivative terms, and the mixed finite method is used for spatial discretization. We employ the Taylor-Hood finite elements to approximate Navier-Stokes equations, Nédélec edge element for the magnetic induction and the equal order Lagrange elements for the thermal equation. This fully discrete scheme only needs to solve a linear system at each time step, and the system is unique solvable. We prove the proposed scheme is unconditionally energy stable. Under a weak regularity hypothesis on the exact solution, we present optimal error estimates for the velocity, magnetic induction and temperature. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme. [ABSTRACT FROM AUTHOR]

• Unconditionally convergent energy dissipative numerical methods are developed and analyzed for nonlinear wave equations. • Two novel efficient energy dissipative difference schemes for the wave equations are first set forth and analyzed. • The discrete energy dissipation properties, solvability, unconditional convergence and stability results are proven rigidly. • Some numerical results are provided to illustrate the computational accuracy and efficiency of the proposed schemes. In this paper, two new efficient energy dissipative difference schemes for the strongly coupled nonlinear damped space fractional wave equations are first set forth and analyzed, which involve a two-level nonlinear difference scheme, and a three-level linear difference scheme based on invariant energy quadratization formulation. Then the discrete energy dissipation properties, solvability, unconditional convergence and stability of the proposed schemes are exhibited rigidly. By the discrete energy analysis method, it is rigidly shown that the proposed schemes achieve the unconditional convergence rates of O (Δ t 2 + h 2) in the discrete L ∞ -norm for the associated numerical solutions. At last, some numerical results are provided to illustrate the dynamical behaviors of the damping terms and unconditional energy stability of the suggested schemes, and testify the efficiency of theoretical results. [ABSTRACT FROM AUTHOR]

This study examines the convergence of growth of per capita income across the districts of Madhya Pradesh first estimating the basic crosssection equations for unconditional and conditional convergence then provides the estimates from spatial data model over a period from 2004/05 to 2016/17. There is existence of unconditional beta (ß) convergence and conditional beta convergence when controlled for more conditioning variables, sigma convergence also confirm the decline in disparity in distribution of per capita income. From the spatial data analysis a significant spatial autocorrelation is revealed in the initial level of per capita income and majority of other conditioning variables. The significant spatial econometric models implies that the growth rate of per capita income of districts depends on how better their neighbours are doing. This has important policy making implications in Madhya Pradesh. [ABSTRACT FROM AUTHOR]

Let A,X,Y be Banach spaces and A×X→Y, (a,x)↦ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence (an)n∈ω in A and unconditionally convergent series ∑n∈ωxn in X, the series ∑n∈ωanxn is unconditionally convergent in Y. We prove that a Banach action A×X→Y preserves unconditional convergence if and only if for any linear functional y*∈Y* the operator Dy*:X→A*, Dy*(x)(a)=y*(ax) is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from ℓ1 to ℓ2, we prove that a Banach action A×X→Y preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis (en)n∈ω such that for every x∈X, the series ∑n∈ωenx is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers p,q,r∈[1,∞] with 1r≤1p+1q, the coordinatewise multiplication ℓp×ℓq→ℓr preserves unconditional convergence if and only if one of the following conditions holds: (i) p≤2 and q≤r, (ii) 2

In this paper, we prove that the unconditional constants of the g-frame expansion in a Hilbert space are bounded by B / A , where A , B are the frame bounds of the g-frames. It follows that tight g-frames have unconditional constant one. Then we generalize this to a classification of such g-frames by showing that a g-Bessel sequence has unconditional constant one if it is an orthogonal sum of g-tight frames. We also obtain a new result under which a g-Bessel sequence is a g-frame from the view of unconditional constant. Finally, we prove similar results for cross g-frame expansions as long as the cross g-frame expansions stay uniformly bounded away from zero. [ABSTRACT FROM AUTHOR]

To recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time‐fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 < α < 2 and present a global consistency analysis under some reasonable regularity assumptions. The temporal nonuniform L2 formula is then utilized to develop a linearized difference scheme for a time‐fractional Benjamin–Bona–Mahony‐type equation. The unconditional convergence of our scheme on both uniform and nonuniform (graded) time meshes are proven with respect to the discrete H1‐norm. Numerical examples are provided to justify the accuracy. [ABSTRACT FROM AUTHOR]

An extended regularized Hermitian and skew-Hermitian splitting iteration method, for solving the double saddle point linear system arising from PDE optimal control problem, is proposed. The unconditional convergence of the method is proved. Furthermore, some spectral properties of the preconditioned matrix are investigated. Numerical experiments show that the iteration method is feasible and efficient. [ABSTRACT FROM AUTHOR]

In this paper, we put emphasis on discussing a full discrete finite element scheme for the Korteweg-de Vries equation, where nonlinear term is dealt with a semi-implicit scheme and temporal term is discreted by the Euler scheme. Theoretical analysis is based on error splitting technique, i.e., error function is split as temporal error function plus spatial error function, and then unconditionally optimal error estimates of the considered full discrete scheme are obtained. Numerical results are provided to confirm our theoretical analysis, which show that no time-step condition is needed. [ABSTRACT FROM AUTHOR]

Studies of economic convergence among countries usually focus on per capita GDP or on a particular sector, such as labor productivity in the manufacturing sector. Using value-added data from the United Nations Statistics Division's National Accounts and employment data from the International Labour Organization (ILO), we compile labor productivity data for 160 countries for the period 1991–2016 in order to examine labor productivity convergence for broad sectors and for the aggregate economy. The results demonstrate the emergence of convergence in labor productivity among countries. The rate of convergence increases significantly when we consider only 141 countries, excluding the low-income countries. The sectoral convergence test reveals that the service sector is the leading force behind the recently observed catching-up in labor productivity among countries, followed by manufacturing. Reallocation of labor across sectors seems to have increased aggregate productivity in developing countries while reducing it in developed countries. [ABSTRACT FROM AUTHOR]

• A linearized transformed L 1 -Galerkin finite element method is presented for numerically solving the semi-linear time-fractional parabolic problems. • The numerical schemes are developed by taking the weak singularity of the solutions into account. • By the temporal-spatial error splitting argument, unconditionally optimal error estimates of the proposed schemes are proved. A linearized Galerkin finite element method is presented for numerically solving the semi-linear time-fractional parabolic problems, whose solutions always display a initial weak singularity. The transformed L 1 scheme based on a change of variable is used to approximate Caputo derivatives and the finite element approximations to the spatial variables. By the temporal-spatial error splitting argument, unconditionally optimal error estimates of the proposed schemes are proved. Finally, several numerical experiments are given to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

A Newton linearized Galerkin finite element method is proposed to solve nonlinear time fractional parabolic problems with non-smooth solutions in time direction. Iterative processes or corrected schemes become dispensable by the use of the Newton linearized method and graded meshes in the temporal direction. The optimal error estimate in the L 2 -norm is obtained without any time step restrictions dependent on the spatial mesh size. Such unconditional convergence results are proved by including the initial time singularity into concern, while previous unconditional convergent results always require continuity and boundedness of the temporal derivative of the exact solution. Numerical experiments are conducted to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations or small time-steps because it significantly reduces the computational cost O (M N 2) and storage O(MN) for the standard L1 formula to O (M N log N) and O (M log N) , respectively, for M grid points in space and N levels in time. The nonuniform time mesh would be graded to handle the typical singularity of the solution near the time t = 0 , and Newton linearization is used to approximate the nonlinearity term. Our analysis relies on three tools: a recently developed discrete fractional Grönwall inequality, a global consistency analysis and a discrete H 2 energy method. A sharp error estimate reflecting the regularity of solution is established without any restriction on the relative diameters of the temporal and spatial mesh sizes. Numerical examples are provided to demonstrate the effectiveness of our approach and the sharpness of error analysis. [ABSTRACT FROM AUTHOR]

Unconditional convergence across countries worldwide is typically rejected in terms of GDP per capita. But when focusing on a specific internationally competitive industry, such as manufacturing, rather than the overall economy, unconditional convergence has been found to hold. As the epitome of competition and globalization, this paper uses the performance of national soccer teams as a further test case. We rely on data of more than 25,000 games between 1950 and 2014 and find clear evidence of unconditional - and -convergence in national team performance, as measured either by win percentages or goal difference. We argue that transfer of technologies, skills and best practices fosters this catch-up process. But there are limits: we show that good teams from Africa and Asia are failing to close the gap with top European or South American teams for reasons that are analogous to the 'middle income trap'. Lessons for other sectors include the virtues of internationally transferable human capital as well as the mixed blessings of regional integration for worldwide convergence. [ABSTRACT FROM AUTHOR]

In this paper, a conservative compact difference scheme is proposed for the two‐dimensional nonlinear Zakharov equation with periodic boundary condition and initial condition. The proposed scheme not only conserve the mass and energy in the discrete level but also are efficient in practical experiments because the Fast Fourier transform (FFT) can be used to speed up the numerical computation. By using the standard energy method and induction argument, we can establish rigorously the unconditional and optimal H2‐error estimates. Some numerical examples are provided to support our theoretical results and show the accuracy and efficiency of the new scheme. [ABSTRACT FROM AUTHOR]

The optimal error estimates of the semi-implicit BDF2-FEM were studied for cubic Schrödinger equations. First, an error estimate was divided into 2 parts: the temporal-discretization and the spatial-discretization. Through introduction of a temporal-discretization equation, the uniform boundedness of the solution and the temporal error estimate were obtained. The unconditionally optimal error estimates of the 2nd-order backward difference (BDF2-FEM) semi-implicit scheme for cubic Schrdinger equations were given. Finally, numerical examples verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

Two numerical methods are presented for the approximation of the Zakharov-Rubenchik equations (ZRE). The first one is the finite difference integrator Fourier pseudospectral method (FFP), which is implicit and of the optimal convergent rate at the order of O(N−r + τ2) in the discrete L2 norm without any restrictions on the grid ratio. The second one is to use the Fourier pseudospectral approach for spatial discretization and exponential wave integrator for temporal integration. Fast Fourier transform is applied to the discrete nonlinear system to speed up the numerical computation. Numerical examples are given to show the efficiency and accuracy of the new methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a conservative compact finite difference scheme is presented for solving the N-coupled nonlinear Schrödinger–Boussinesq equations. By using the discrete energy method, it is proved that our scheme is unconditionally convergent in the maximum norm and the convergent rate is at O (τ 2 + h 4) with time step τ and mesh size h. Numerical results including the comparisons with other numerical methods are reported to demonstrate the accuracy and efficiency of the method and to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]