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In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform convergence of order O (N − (k − 1 / 2)) on the layer-adapted meshes including Bakhvalov-Shishkin type, and Bakhvalov-type and almost optimal uniform error estimates of order O ((N − 1 ln ⁡ N) (k − 1 / 2)) on Shishkin-type mesh with respect to the perturbation parameter in the energy norm using high-order piecewise discontinuous polynomials of degree k. Here N is the number mesh intervals. We conduct numerical examples to support our theoretical results. [ABSTRACT FROM AUTHOR]

We present a parameter-robust finite difference method for solving a system of weakly coupled singularly perturbed convection-diffusion equations. The diffusion coefficient of each equation is a small distinct positive parameter. Due to this, the solution to the system has, in general, overlapping boundary layers. The problem is discretized using a particular combination of a compact second-order difference scheme and a central difference scheme on a piecewise-uniform Shishkin mesh. The convergence analysis is given, and the method is shown to have almost second-order uniform convergence in the maximum norm with respect to the perturbation parameters. The results of numerical experiments are in agreement with the theoretical outcomes. [ABSTRACT FROM AUTHOR]

The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. Every uniformly closed sublattice of codimension n contains a uniformly closed ideal of codimension at most $ 2n $ 2 n. If the vector lattice is uniformly complete then every ideal of finite codimension is uniformly closed. Results of the paper extend (and are motivated by) results of Abramovich Y.A., Lipecki Z. [On ideals and sublattices in linear lattices and F-lattices. Math Proc Cambridge Philos Soc. 1990;108(1):79–87.; On lattices and algebras of simple functions. Comment Math Univ Carolin. 1990;31(4):627–635.], as well as Kakutani's characterization of closed sublattices of $ C(K) $ C (K) spaces. [ABSTRACT FROM AUTHOR]

A uniformly convergent numerical scheme is developed for solving a singularly perturbed parabolic turning point problem. The properties of continuous solutions and the bounds of the derivatives are discussed. Due to the presence of a small parameter as a multiple of the diffusion coefficient, it causes computational difficulty when applying classical numerical methods. As a result, the scheme is formulated using the Crank-Nicolson method in the temporal discretization and an exponentially fitted finite difference method in the space on a uniform mesh. The existence of a unique discrete solution is guaranteed by the comparison principle. The stability and convergence analysis of the method are investigated. Two numerical examples are considered to validate the applicability of the scheme. The numerical results are displayed in tables and graphs to support the theoretical findings. The scheme converges uniformly with order one in space and two in time. [ABSTRACT FROM AUTHOR]

This paper chiefly takes into account the singularly perturbed delay Volterra-Fredholm integro-differential equations by numerically. In this context, firstly, priori estimates are given and a new discretization is constructed on piecewise-equidistant mesh by using interpolating quadrature rules [2] and composite integration formulas. Then, the convergence analysis and stability bounds of the presented method are discussed. Finally, numerical results are demonstrated with two test problems. [ABSTRACT FROM AUTHOR]

This article introduces a numerical method to solve a singularly perturbed Fredholm integro-differential equation of second order with a discontinuous source term. To effectively handle the problem, we utilized the finite difference method on an adaptive mesh. This adaptive mesh, generated via the grid equidistribution method, significantly improves solution accuracy and resolves issues inherent in traditional numerical methods. Our numerical results provide evidence of ε - uniform convergence rates and prove the effectiveness of our approach. Overall, our scheme offers an efficient solution for singularly perturbed Fredholm integro-differential equations that include discontinuous source terms. [ABSTRACT FROM AUTHOR]

We study the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum. [ABSTRACT FROM AUTHOR]

In this article, we explore the discontinuous Galerkin finite element method for two-parametric singularly perturbed convection-diffusion problems with a discontinuous source term. Due to the discontinuity in the source term, the problem typically shows a weak interior layer. Also, the presence of multiple perturbation parameters in the problem causes boundary layers on both sides of the boundary. In this work, we develop the nonsymmetric discontinuous Galerkin finite element method with interior penalties to handle the layer phenomenon. With the use of a typical Shishkin mesh, the domain is discretized, and a uniform error estimate is obtained. Numerical experiments are conducted to validate the theoretical conclusions.

In this article, we introduce a modified class of Bernstein–Kantorovich operators depending on an integrable function ψα$$ {\psi}_{\alpha } $$ and investigate their approximation properties. By choosing an appropriate function ψα$$ {\psi}_{\alpha } $$, the order of approximation of our operators to a function f$$ f $$ is at least as good as the classical Bernstein–Kantorovich operators on the interval [0,1]$$ \left[0,1\right] $$. We compared the operators defined in this study not only with Bernstein–Kantorovich operators but also with some other Bernstein–Kantorovich type operators. In this paper, we also study the results on the uniform convergence and rate of convergence of these operators in terms of the first‐ and second‐order moduli of continuity, and we prove that our operators have shape‐preserving properties. Finally, some numerical examples which support the results obtained in this paper are provided. [ABSTRACT FROM AUTHOR]

This article is concerned with a class of singularly perturbed semilinear parabolic convection‐diffusion partial differential equations (PDEs) with discontinuous source function. Solutions of these PDEs usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. We begin our study by proving existence of the analytical solution of the considered nonlinear PDE by means of the upper and lower solutions approach; and the ε$$ \varepsilon $$‐uniform stability of the analytical solution is established by using the comparison principle for the continuous nonlinear operator. In order to realize the asymptotic behavior of the analytical solution, we derive a priori bounds of the solution derivatives via decomposition of the solution into the smooth and the layer components. For an efficient numerical solution of the nonlinear PDE, the time‐derivative is approximated by the Crank–Nicolson method on an equidistant mesh, and we approximate the spatial derivative by a finite difference scheme on a suitable layer‐adapted mesh. We establish the comparison principle for the nonlinear difference operator to prove the ε$$ \varepsilon $$‐uniform stability of the discrete solution and further construct a suitable decomposition of the discrete solution for pursuing the convergence analysis. The computational method is proven to be parameter‐robust with second‐order time accuracy in the discrete supremum norm. The theoretical estimate is finally verified by the numerical experiments. [ABSTRACT FROM AUTHOR]

The almost everywhere pointwise and uniform convergences for the generalized KP-II equation are investigated when the initial data is in anisotropic Sobolev space H s 1 , s 2 (R 2) . Firstly, we show that the solution u(x, y, t) converges pointwisely to the initial data f (x , y) ∈ H s 1 , s 2 (R 2) for a.e. (x , y) ∈ R 2 when s 1 ≥ 1 4 , s 2 ≥ 1 4 . The proof relies upon the Strichartz estimate and high-low frequency decomposition. Secondly, We prove that s 1 ≥ 1 4 , s 2 ≥ 1 4 is a necessary condition for the maximal function estimate of the generalized KP-II equation to hold. Finally, by using the Fourier restriction norm method, we establish the nonlinear smoothing estimate to show the uniform convergence of the generalized KP-II equation in H s 1 , s 2 (R 2) with s 1 ≥ 3 2 - α 4 + ϵ , s 2 > 1 2 and α ≥ 4 . [ABSTRACT FROM AUTHOR]

Two-sample multiple testing has a wide range of applications. Most of the literature considers simultaneous tests of equality of parameters. The article takes a different perspective and investigates the null hypotheses that the two support sets are equal. This formulation of the testing problem is motivated by the fact that in many applications where the two parameter vectors being compared are both sparse, one might be more concerned about the detection of differential sparsity structures rather than the difference in parameter magnitudes. Focusing on this type of problem, we develop a general approach, which adapts the newly proposed symmetry data aggregation tool combined with a novel double thresholding (DT) filter. The DT filter first constructs a sequence of pairs of ranking statistics that fulfill global symmetry properties and then chooses two data-driven thresholds along the ranking to simultaneously control the False Discovery Rate (FDR) and maximize the number of rejections. Several applications of the methodology are given including high-dimensional linear models and Gaussian graphical models. We show that the proposed method is able to asymptotically control the FDR and have power guarantee under certain conditions. Numerical results confirm the effectiveness and robustness of DT in FDR control and detection ability. for this article are available online. [ABSTRACT FROM AUTHOR]

We investigate a second-order accurate time-stepping scheme for solving a time-fractional diffusion equation with a Caputo derivative of order α ∈ (0 , 1) . The basic idea of our scheme is based on local integration followed by linear interpolation. It reduces to the standard Crank–Nicolson scheme in the classical diffusion case, that is, as α → 1 . Using a novel approach, we show that the proposed scheme is α -robust and second-order accurate in the L 2 (L 2) -norm, assuming a suitable time-graded mesh. For completeness, we use the Galerkin finite element method for the spatial discretization and discuss the error analysis under reasonable regularity assumptions on the given data. Some numerical results are presented at the end. [ABSTRACT FROM AUTHOR]

A class of two-dimensional singularly perturbed convection–diffusion problems with turning points is studied in this work. Since the turning point takes place inside the domain, the problem usually exhibits a weak interior layer of cusp-type. The inclusion of parabolic boundary layers in the solution is caused by the presence of a small perturbation parameter. Two methods are designed to outperform the error estimation: the non-symmetric discontinuous Galerkin technique with an interior penalty (NIPG) and the symmetric discontinuous Galerkin method with an interior penalty (SIPG). To circumvent the layering effect in domain discretization, the traditional Shishkin mesh is used. Uniform error estimates are derived in the respective norms of order O (N - 1 ln 3 / 2 N) and O (N - 1 ln 2 N) for the NIPG and the SIPG methods, respectively. Numerical experiments are carried out to validate theoretical findings. [ABSTRACT FROM AUTHOR]

The main objective of this article is to introduce a global version of continuity of multifunctions. Such a family of multifunctions is called uniformly continuous. Equivalent characterizations and various interesting properties of uniformly continuous multifunctions are studied in detail. [ABSTRACT FROM AUTHOR]

This study uses numerical methods to solve a specific type of reaction-diffusion problem arising in viscoelasticity (singularly perturbed Fredholm integro-differential equations). These equations are challenging because they exhibit ‘boundary layers’ near the edges of the area of interest. To approximate solutions, a technique known as a second-order scheme is used for derivatives, and the trapezoidal rule is used for integral terms. This is done on non-standard grids known as Shishkin-type meshes. We found that this numerical method and its rate of improvement (or convergence) are both of the second order, which means they improve consistently as the calculations continue. This improvement rate remains consistent even when dealing with small parameters in the equations. In addition, a post-processing method is used to enhance the rate of convergence from second order to almost fourth order, indicating a significant improvement in the speed and accuracy of the solutions. The practical effectiveness of these methods is confirmed through performance testing of the numerical scheme.

In this paper, we focus on investigating a post-processing technique de-signed for one-dimensional singularly perturbed parabolic convection-diffusion problems that demonstrate a regular boundary layer. We use a back-ward Euler numerical approach for time derivatives with uniform mesh in the temporal direction, and a simple upwind scheme is used for spa-tial derivatives with modified graded mesh in the spatial direction. In this study, we demonstrate the effectiveness of the Richardson extrapola-tion technique in enhancing the ε-uniform accuracy of simple upwinding within the discrete supremum norm, as evidenced by an improvement from O(N −1 ln(1/ε) + △θ) to O(N −2 ln2(1/ε) + △θ2). Furthermore, to validate the theoretical findings, computational experiments are conducted for two test examples by applying the proposed technique.

A fitted tension spline numerical scheme for a singularly perturbed parabolic problem (SPPP) with time delay is proposed. The presence of a small parameter ε as a multiple of the diffusion term leads to the suddenly changing behaviors of the solution in the boundary layer region. This results in a challenging duty to solve the problem analytically. Classical numerical methods cause spurious nonphysical oscillations unless an unacceptable number of mesh points is considered, which requires a large computational cost. To overcome this drawback, a numerical method comprising the backward Euler scheme in the time direction and the fitted spline scheme in the space direction on uniform meshes is proposed. To establish the stability and uniform convergence of the proposed method, an extensive amount of analysis is carried out. Three numerical examples are considered to validate the efficiency and applicability of the proposed scheme. It is proved that the proposed scheme is uniformly convergent of order one in both space and time. Further, the boundary layer behaviors of the solutions are given graphically. [ABSTRACT FROM AUTHOR]

Kernel-type nonparametric estimates of Poisson regression function are considered. We establish the conditions of uniform consistency and the limit theorems for continuous functionals connected with this function on C [ a , 1 - a ] , 0 < a < 1 2 . [ABSTRACT FROM AUTHOR]

In this work we deal with the numerical solution of one dimensional semilinear parabolic singularly perturbed systems of convection-diffusion type. We assume that the coupling in the convection terms is weak and also that the coupling reaction terms are nonlinear. In the case of considering different small diffusion parameters at each equation with different orders of magnitude, the exact solution usually shows overlapping boundary layers on the outflow of the spatial domain. To approximate it, we construct a numerical scheme which combines the upwind finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize in space and a linearized version of the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. Then, the fully discrete method is uniformly convergent with respect to both diffusion parameters, having first order in time and almost first order in space. The choice of this time integrator provokes that only tridiagonal linear systems must be solved at each time step; in this way, the computational cost of the algorithm is considerably lower than this one associated to classical implicit methods. The numerical results obtained for different test problems corroborate in practice the good performance of the numerical algorithm. [ABSTRACT FROM AUTHOR]

In this paper, singularly perturbed pseudo-parabolic initial-boundary value problems with time-delay parameter are considered by numerically. Initially, the asymptotic properties of the analytical solution are investigated. Then, a discretization with exponential coefficient is suggested on a uniform mesh. The error approximations and uniform convergence of the presented method are estimated in the discrete energy norm. Finally, some numerical experiments are given to clarify the theory. [ABSTRACT FROM AUTHOR]

In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the $ L_\infty $ L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper. [ABSTRACT FROM AUTHOR]

A weak Galerkin finite element method is applied to singularly perturbed delay reaction-diffusion problems. A robust uniform convergence has been proved both in the energy and balanced norms using higher-order piecewise discontinuous polynomials on Shishkin meshes. The error analysis for singularly perturbed reaction-diffusion problems with negative or positive shift in the balanced norm has appeared for the first time. The proposed method uses piecewise polynomials of order $ k\geq ~1 $ k ≥ 1 on interior of each element and piecewise constant polynomials on the end points of each element. By the Schur complement technique, the interior degrees of foredoom (DOF) can be eliminated from the discrete system resulting from the numerical scheme, and thus the degrees of freedom of the proposed method comparable with the classical finite element methods, and it is remarkably less than that of the discontinuous Galerkin method. Finally, we give various numerical experiments to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

This study uses numerical methods to solve a specific type of reaction-diffusion problem arising in viscoelasticity (singularly perturbed Fredholm integro-differential equations). These equations are challenging because they exhibit 'boundary layers' near the edges of the area of interest. To approximate solutions, a technique known as a second-order scheme is used for derivatives, and the trapezoidal rule is used for integral terms. This is done on non-standard grids known as Shishkin-type meshes. We found that this numerical method and its rate of improvement (or convergence) are both of the second order, which means they improve consistently as the calculations continue. This improvement rate remains consistent even when dealing with small parameters in the equations. In addition, a post-processing method is used to enhance the rate of convergence from second order to almost fourth order, indicating a significant improvement in the speed and accuracy of the solutions. The practical effectiveness of these methods is confirmed through performance testing of the numerical scheme. [ABSTRACT FROM AUTHOR]

A time dependent singularly perturbed convection diffusion equation involving shift parameters are considered. The terms containing the shifts are treated using Taylor series approximation up to second order terms. Classical numerical methods developed for solving regular problems fail to give good approximate solution and become unstable while applied for the considered problem. In this paper, numerical scheme is developed using $-method for semi-discretizing in time derivative; in spatial discrtization fitted operator finite difference method is applied by inducing exponential fitting parameter. To accelerate the convergence of the scheme, Richardson extrapolation technique is applied in spatial discretization. Existence of unique discrete solution is guaranteed by establishing the discrete comparison principle. The proposed scheme is stable for all values of the perturbation parameter. A uniformly convergent solution is obtained by Richardson extrapolation method which provides second order accuracy under some condition. Test examples are considered for validating the theoretical results numerically. [ABSTRACT FROM AUTHOR]

In this article, we explore the discontinuous Galerkin finite element method for two-parametric singularly perturbed convection-diffusion problems with a discontinuous source term. Due to the discontinuity in the source term, the problem typically shows a weak interior layer. Also, the presence of multiple perturbation parameters in the problem causes boundary layers on both sides of the boundary. In this work, we develop the nonsymmetric discontinuous Galerkin finite element method with interior penalties to handle the layer phenomenon. With the use of a typical Shishkin mesh, the domain is discretized, and a uniform error estimate is obtained. Numerical experiments are conducted to validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

This paper deals the design and analysis of a parameter uniform finite difference algorithm for a class of parabolic singularly perturbed convection-diffusion problems. The discrete nearby problem is designed using the backward Euler scheme in time and upwind scheme on H (ℓ) mesh in space. We present the convergence analysis of the algorithm, which proves that it is parameter uniform in L ∞ norm. The numerical experiments support the theoretical estimates and demonstrates the efficiency of H (ℓ) mesh with the existing meshes, like Shishkin and Bakhvalov type meshes. [ABSTRACT FROM AUTHOR]

This paper solves a class of Fredholm integro-differential equations involving a small parameter with integral boundary conditions numerically. The solution to the problem possesses boundary layers at both end boundaries. A central difference scheme is used for approximating the derivatives. In contrast, the trapezoidal rule is used for the integral term, provided an appropriately adapted mesh is considered, namely Shishkin and Bakhvalov-Shishkin meshes. The proposed numerical method presents a uniform second-order convergence rate regardless of the perturbation parameter. Furthermore, using a post-processing technique, we have significantly improved the convergence from second to fourth order. The effectiveness of the proposed approach is validated through some numerical examples. [ABSTRACT FROM AUTHOR]

In this work, we propose a numerical method for solving a singularly perturbed convection-diffusion problem that involves a time delay term. A priori bounds and properties of the continuous solution are discussed. Using the backward Euler method for the time derivative term, the problem is approximated by a set of singularly perturbed boundary value problems. Then, using a higher-order finite difference method, the boundary value problem is approximated on a piecewise uniform Shishkin mesh. The stability analysis of the method is studied using the comparison principle and discrete solution bounds. We proved that the proposed scheme is uniformly convergent, with an order of convergence of almost two in space and one in time. Two numerical examples are considered to validate the applicability of the proposed scheme. The proposed scheme has better accuracy than some schemes in the literature. [ABSTRACT FROM AUTHOR]

We prove that a uniformly convergent trigonometric series may not satisfy the permutation sign convergence condition, hence it may not satisfy the Rademacher condition as well. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method for a class of singularly perturbed parabolic partial differential equations with integral boundary conditions (IBC). The solution to the considered problem exhibits pronounced boundary layers on both the left and right sides of the spatial domain. To address this challenging problem, we propose the use of the implicit Euler method for time discretization and a finite difference method on a well-designed piecewise uniform Shishkin mesh for spatial discretization. The integral boundary condition is approximated using Simpson's 13rule. The presented method demonstrates almost second-order uniform convergence in the discretization of the spatial derivative and first-order convergence in the discretization of the time derivative. To validate the applicability and accuracy of the proposed method, two illustrative examples are employed. The computational results not only accurately reflect the theoretical estimations but also highlight the method's effectiveness in capturing the intricate features of singularly perturbed parabolic partial differential equations with integral boundary conditions. [ABSTRACT FROM AUTHOR]

The Voronovskaja formula for Aldaz–Kounchev–Render operators was established in terms of pointwise convergence. For suitable functions we prove it with uniform convergence. Moreover, we establish the Voronovskaja formula of second order with uniform convergence. [ABSTRACT FROM AUTHOR]

The paper proposes a new method for obtaining a generalized solution to the mixed boundary value problem for a parabolic equation with boundary conditions of the third kind and a continuous initial condition. Generalized functions are understood in the sense of the sequential approach. The representative of the class of sequences, which is a generalized function, is obtained using the function interpolation operator, constructed using solutions to the Cauchy problem. The solution is obtained in the form of a series that converges uniformly inside the domain of the solution. [ABSTRACT FROM AUTHOR]

In this article, we develop an efficient numerical method for one-dimensional time-delayed singularly perturbed parabolic problems. The proposed nu-merical approach comprises an upwind difference scheme with modified graded mesh in the spatial direction and a backward Euler scheme on uni-form mesh in the temporal direction. In order to capture the local behavior of the solutions, stability and error estimations are obtained with respect to the maximum norm. The proposed numerical method converges uniformly with first-order up to logarithm in the spatial variable and also first-order in the temporal variable. Finally, the outcomes of the numerical experiments are included for two test problems to validate the theoretical findings.

Objectives: In this paper, a uniformly convergent numerical scheme is proposed for solving a singularly perturbed Fredholm integro-differential equation with an integral initial condition. The equation involves a left boundary layer which makes it difficult to solve it using the standard numerical methods. A fitted operator finite difference method is used to approximate the differential part of the equation and the composite Simpson 1 3 rule is used for the integral parts of the equation and the initial condition. Result: The stability bound and error estimation of the approximated solution are performed, to show the uniform convergence of the scheme with order one in the maximum norm. Numerical test examples are provided to calculate the maximum absolute errors, thrgence, and the uniform error for a couple of examples to support theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper develops the estimation method of mean and covariance functions of functional data with additional covariate information. With the strength of both local linear smoothing modeling and general weighing scheme, we are able to explicitly characterize the mean and covariance functions with incorporating covariate for irregularly spaced and sparsely observed longitudinal data, as typically encountered in engineering technology or biomedical studies, as well as for functional data which are densely measured. Theoretically, we establish the uniform convergence rates of the estimators in the general weighing scheme. Monte Carlo simulation is conducted to investigate the finite-sample performance of the proposed approach. Two applications including the children growth data and white matter tract dataset obtained from Alzheimer's Disease Neuroimaging Initiative study are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we focus on investigating a post-processing technique designed for one-dimensional singularly perturbed parabolic convection-diffusion problems that demonstrate a regular boundary layer. We use a backward Euler numerical approach for time derivatives with uniform mesh in the temporal direction, and a simple upwind scheme is used for spatial derivatives with modified graded mesh in the spatial direction. In this study, we demonstrate the effectiveness of the Richardson extrapolation technique in enhancing the ε-uniform accuracy of simple upwinding within the discrete supremum norm, as evidenced by an improvement from O(N-1 ln(1/ε) +Δθ) to O(N-2 ln²(1/ε) +Δθ²). Furthermore, to validate the theoretical findings, computational experiments are conducted for two test examples by applying the proposed technique. [ABSTRACT FROM AUTHOR]

In this article, we develop an efficient numerical method for one-dimensional time-delayed singularly perturbed parabolic problems. The proposed numerical approach comprises an upwind difference scheme with modified graded mesh in the spatial direction and a backward Euler scheme on uniform mesh in the temporal direction. In order to capture the local behavior of the solutions, stability and error estimations are obtained with respect to the maximum norm. The proposed numerical method converges uniformly with first-order up to logarithm in the spatial variable and also first-order in the temporal variable. Finally, the outcomes of the numerical experiments are included for two test problems to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

Abstract. Necessary and sufficient conditions for the uniform convergence of trigonometric Fourier integrals are well-established when admissible monotone or general monotone functions are considered. In this paper, we generalize these main results by giving such conditions for the uniform convergence of sine and cosine integrals ∫∞ 0 f1 (x) sin (uxp) dx and ∫∞ 0 f2(x) cos (uxp) dx in case of admissible general monotone functions f1 and f2. Moreover, we give necessary and sufficient conditions for the Lq -integrability with the power weights of these integrals when non-negative functions f1 and f2 belong to the class GMpθ. [ABSTRACT FROM AUTHOR]

A stabilized mixed finite element method is proposed for solving the time-harmonic Maxwell’s equations, with the divergence constraint imposed by the multiplier in a weak sense. By a grad-div stabilization, for some lowest-order edge elements on nonaffine quadrilateral, hexahedral and prismatic grids, we prove a type of uniform convergence for the zero-frequency Maxwell’s equations, then prove the well-posedness and the convergence for the time-harmonic Maxwell’s equations. Numerical results confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, a time delay parabolic convection-reaction-diffusion singularly perturbed problem with two small parameters is considered. We investigate the layer behavior of the solution for both smooth and non-smooth data. A numerical method to solve the problems described is developed using the Crank-Nicolson scheme to discretize the time-variable on a uniform mesh while a hybrid finite difference is applied for the space-variable. The hybrid scheme is a combination of the central, upwind and mid-point differencing on a piecewise uniform mesh of Shishkin type. The convergence analysis shows that the proposed method is uniformly convergent of second order in both space and time. Numerical experiments conducted on some test examples confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In the present paper we study the perturbed sampling Kantorovich operators in the general context of the modular spaces. After proving a convergence result for continuous functions with compact support, by using both a modular inequality and a density approach, we establish the main result of modular convergence for these operators. Further, we show several instances of modular spaces in which these results can be applied. In particular, we show some applications in Musielak–Orlicz spaces and in Orlicz spaces and we also consider the case of a modular functional that does not have an integral representation generating a space, which can not be reduced to previous mentioned ones. [ABSTRACT FROM AUTHOR]

AbstractIn the big data era, subsampling or sub-data selection techniques are often adopted to extract a fraction of informative individuals from the massive data. Existing subsampling algorithms focus mainly on obtaining a representative subset to achieve the best estimation accuracy under a given class of models. In this article, we consider a semi-supervised setting wherein a small or moderate sized “labeled” data is available in addition to a much larger sized “unlabeled” data. The goal is to sample from the unlabeled data with a given budget to obtain informative individuals that are characterized by their unobserved responses. We propose an optimal subsampling procedure that is able to maximize the diversity of the selected subsample and control the false selection rate (FSR) simultaneously, allowing us to explore reliable information as much as possible. The key ingredients of our method are the use of predictive inference for quantifying the uncertainty of response predictions and a reformulation of the objective into a constrained optimization problem. We show that the proposed method is asymptotically optimal in the sense that the diversity of the subsample converges to its oracle counterpart with FSR control. Numerical simulations and a real-data example validate the superior performance of the proposed strategy. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

The present paper aims to study the uniform convergence of a sequence of functions via the μ − statistical uniform convergence where μ is a finitely additive set function defined on a field of subsets of positive integers. In particular we deal with an analog of Dini's theorem. It turns out that μ − statistical uniform convergence of a sequence of functions is characterized by that of another decreasing sequence of functions. [ABSTRACT FROM AUTHOR]

This paper deals with the study of a higher‐order numerical approximation for a class of singularly perturbed convection‐diffusion problems with time delay. The method combines a higher‐Order Difference with an Identity Expansion (HODIE) scheme over a piece‐wise uniform mesh in the spatial direction and the backward Euler method on a uniform mesh for discretization in the temporal direction. A priori bounds for the continuous solution and its derivatives are derived by splitting the solution into regular and singular components. These bounds are useful in the error analysis of the proposed scheme. The present scheme converges ε$$ \varepsilon $$‐uniformly with the order of convergence one in time and almost second‐order in space direction. Further, to increase the rate of convergence in the time variable, we implemented the Richardson extrapolation technique. Thus, finally, the resultant scheme with Richardson extrapolation in time becomes almost second‐order ε$$ \varepsilon $$‐uniformly convergent in both the space and time variable. The detailed stability and convergence analysis have been done using the derived a priori estimates. We consider three test problems to validate the predicted theory and show that numerical results are in good agreement with our theoretical findings. [ABSTRACT FROM AUTHOR]

In the paper, we present a necessary and sufficient condition for the existence of a sequence of measurable functions with finite values, which converge to any given essential bounded function in the topology of essential supremum in a Banach space. A new convergence method is proposed, which allows for the discovery of an essential bounded function F that is valued in a Banach space. Generally speaking, there exists a Banach-valued essential bounded function F which Fn can't converge to F in the topology of essential supremum for any sequence of finite-valued measurable function. [ABSTRACT FROM AUTHOR]