Uniform Convergence Properties Research Articles

The local uniform convergence of positive harmonic function sequence

The Harnack distance on space and its conformal invariance were constructed and studied by Herron. In this paper, we obtain the Harnack distance on domains in . Then, we use this concept to investigate some properties of the positive harmonic function class. These results are obtained in the complex plane, so it is advantageous to take some tools of the complex analysis. The main result of this paper is the property of the local uniform convergence of the positive harmonic sequences on a domain in the complex plane.

The Uniform Convergence Property of Sequence of Fractal Interpolation Functions in Complicated Networks

In order to further research the relationship between fractals and complicated networks in terms of self-similarity, the uniform convergence property of the sequence of fractal interpolation functions which can generate self-similar graphics through iterated function system defined by affine transformation is studied in this paper. The result illustrates that it is can be proved that the sequence of fractal interpolation functions uniformly converges to its limit function and its limit function is continuous and integrable over a closed interval under the uniformly convergent condition of the sequence of fractal interpolation functions. The following two conclusions can be indicated. First, both the number sequence limit operation of the sequence of fractal interpolation functions and the function limit operation of its limit function are exchangeable over a closed interval. Second, the two operations of limit and integral between the sequence of fractal interpolation functions and its limit function are exchangeable over a closed interval.

On Kantorovich variant of Baskakov type operators preserving some functions

This paper deals with a generalization of Kantorovich variant of Baskakov type operators preserving constant function and e-2y. We discuss uniform convergence properties and weighted approximation for this generalized Baskakov-Kantorovich type operators.

Uniform [formula omitted] behavior of the first-order interpolant quadrature scheme for some partial integro-differential equations

We study the time discretization for the solution of a Volterra equation with weakly singular kernel. The time discrete scheme is based on the first-order backward difference method. The memory term is approximated by the interpolating quadrature. We use Laplace transform technique to show that this interpolating quadrature scheme has a convergence rate of O(△t) where △t denotes the time step, for smooth and non-smooth initial data in homogeneous case. Special attention is that the uniform l1 convergence property of the discretization in time is given.

Approximation properties of mixed sampling-Kantorovich operators

In the present paper we study the pointwise and uniform convergence properties of a family of multidimensional sampling Kantorovich type operators. Moreover, besides convergence, quantitative estimates and a Voronovskaja type theorem have been established.

On a Durrmeyer-type modification of the Exponential sampling series

In this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using the usual modulus of continuity for uniformly continuous functions. Some examples are also described.

A Convex Optimization Approach to Dynamic Programming in Continuous State and Action Spaces

In this paper, a convex optimization-based method is proposed for numerically solving dynamic programs in continuous state and action spaces. The key idea is to approximate the output of the Bellman operator at a particular state by the optimal value of a convex program. The approximate Bellman operator has a computational advantage because it involves a convex optimization problem in the case of control-affine systems and convex costs. Using this feature, we propose a simple dynamic programming algorithm to evaluate the approximate value function at pre-specified grid points by solving convex optimization problems in each iteration. We show that the proposed method approximates the optimal value function with a uniform convergence property in the case of convex optimal value functions. We also propose an interpolation-free design method for a control policy, of which performance converges uniformly to the optimum as the grid resolution becomes finer. When a nonlinear control-affine system is considered, the convex optimization approach provides an approximate policy with a provable suboptimality bound. For general cases, the proposed convex formulation of dynamic programming operators can be modified as a nonconvex bi-level program, in which the inner problem is a linear program, without losing uniform convergence properties.

Uniform global asymptotic synchronization of Kuramoto oscillators via hybrid coupling

Using a hybrid framework, we propose a generalized version of the well-known Kuramoto model for interconnected oscillators. The proposed model does not modify the classical model close to the synchronization set, but avoids the typical non-uniform convergence phenomenon. For the two-oscillators case, we prove the uniform global asymptotic stability of the consensus set by using a hybrid Lyapunov function whose generality promises possible extension of the result to higher order dynamics. We comparatively illustrate the achieved uniform convergence properties by simulating both the case with two and multiple oscillators, thus confirming the effectiveness of our approach.

Approximation by max-product sampling Kantorovich operators with generalized kernels

In a recent paper, for max-product sampling operators based on general kernels with bounded generalized absolute moments, we have obtained several pointwise and uniform convergence properties on bounded intervals or on the whole real axis, including a Jackson-type estimate in terms of the first uniform modulus of continuity. In this paper, first, we prove that for the Kantorovich variants of these max-product sampling operators, under the same assumptions on the kernels, these convergence properties remain valid. Here, we also establish the [Formula: see text] convergence, and quantitative estimates with respect to the [Formula: see text] norm, [Formula: see text]-functionals and [Formula: see text]-modulus of continuity as well. The results are tested on several examples of kernels and possible extensions to higher dimensions are suggested.

The Picard and Gauss–Weierstrass Singular Integrals in (p, q)-Calculus

The vast development of the techniques in both the quantum calculus and the post-quantum calculus leads to a significant increase in activities in approximation theory due to applications in computational science and engineering. Herein, we introduce (p, q)-Picard and (p, q)-Gauss–Weierstrass integral operators in terms of the (p, q)-Gamma integral. We give a general formula for the monomials under both (p, q)-Picard and (p, q)-Gauss–Weierstrass operators as well as some special cases. We discuss the uniform convergence properties of them. We show that both operators have optimal global smoothness preservation property via usual modulus of continuity. Finally, we establish the rate of approximation using the weighted modulus of smoothness. Depending on the choices of parameters p and q in the integrals, we are able to obtain better error estimation than classical ones.

Consistency rates and asymptotic normality of the high risk conditional for functional data

Abstract The maximum of the conditional hazard function is a parameter of great importance in seismicity studies, because it constitutes the maximum risk of occurrence of an earthquake in a given interval of time. Using the kernel nonparametric estimates of the first derivative of the conditional hazard function, we establish uniform convergence properties and asymptotic normality of an estimate of the maximum in the context of independence data.

Towards essential improvement for the Parareal-TR and Parareal-Gauss4 algorithms

Parareal is an iterative algorithm and is characterized by two propagators G and F, which are respectively associated with large step size ΔT and small step size Δt, where ΔT=JΔt and J≥2 is an integer. For symmetric positive definite (SPD) system u′(t)+Au(t)=g(t) arising from semi-discretizing time-dependent PDEs, if we fix the G-propagator to the Backward-Euler method and choose for F some L-stable time-integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy ρ≈13, ∀J≥2 and ∀σ(A)⊂[0,+∞), where σ(A) is the spectrum of the matrix A. However, this result does not hold when time-integrators that lack L-stability, such as the Trapezoidal rule and the 4th-order Gauss RK method, are chosen as the F-propagator. The parareal algorithms using these two methods for the F-propagator are denoted by Parareal-TR and Parareal-Gauss4. In this paper, we propose a strategy to let these two parareal algorithms possess such a uniform convergence property. The idea is to choose an L-stable propagator F˜ and on each coarse time-interval [Tn,Tn+1] we perform first two steps of F˜, then followed by J−2 steps of F. Precisely, for the Trapezoidal rule we select the 2nd-order SDIRK method as the F˜-propagator, and for the 4th-order Gauss RK method we select the 4th-order Lobatto III-C method as the F˜-propagator. Numerical results are given to support our theoretical conclusions.

Uniform integrability of fuzzy variable sequences

A fuzzy variable is a measurable function from a possibility measure space to a set of real numbers. It is a useful tool to describe the fuzziness of phenomena in a decision making process. In this paper, we focus on investigating the properties of sequences of fuzzy variables. The concepts of uniform integrability, uniform absolute continuity and uniform boundedness are introduced for fuzzy variable sequences, and then the relations among them are discussed. As the applications of these concepts, we also present several convergence theorems for sequences of fuzzy variables by using uniform integrability. Specifically, the relations between uniform integrability and convergence properties, such as convergence in measure, convergence in mean, and convergence almost surely, are discussed.

An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems

AbstractIn this paper, we propose an uniformly convergent adaptive finite element method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation (BEC) and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations for the problem are reviewed to confirm the asymptotic behaviors of the solutions in the boundary/interior layer regions. By using the normalized gradient flow, we propose an adaptive finite element with hybrid basis to solve the singularly perturbed nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively to the small parameter ε. Extensive numerical results are reported to show the uniform convergence property of our method. We also apply the AFEM-HB to compute the ground and excited states of BEC with box/harmonic/optical lattice potential in the semiclassical regime (0 <ε≪C 1). In addition, we give a detailed error analysis of our AFEM-HB to a simpler singularly perturbed two point boundary value problem, show that our method has a minimum uniform convergence order

Consistency rates and asymptotic normality of the high risk conditional for functional data

Abstract The maximum of the conditional hazard function is a parameter of great importance in seismicity studies, because it constitutes the maximum risk of occurrence of an earthquake in a given interval of time. Using the kernel nonparametric estimates of the first derivative of the conditional hazard function, we establish uniform convergence properties and asymptotic normality of an estimate of the maximum in the context of independence data.

A two-grid method for an elliptic equation with boundary layers on a Shishkin mesh

A linear elliptic equation with regular and parabolic boundary layers is considered. It is solved by using an upwind difference scheme on a Shishkin mesh with the property of uniform convergence with respect to a small parameter. The problem of reducing the number of arithmetic operations required to implement the difference scheme on the basis of a two-grid method is investigated. The application of the Richardson extrapolation method to increase the accuracy of the difference scheme is studied. Results of numerical experiments are presented.

Parameter-dependent integral on time scales

In this paper, we focus on the parameter-dependent integral on time scales. By combining multiple integration theory on time scales and usual parameter-dependent integration theory in classical mathematical analysis, both normal parameter-dependent integral on time scales and improper parameter-dependent integral on time scales are investigated efficiently. Many criteria for uniform convergence and important analytical properties are developed, respectively. Finally, a lot of examples are also presented to substantiate the theoretical results.

Generalization performance of bipartite ranking algorithms with convex losses

Previous works describing the generalization performance of bipartite ranking algorithms are usually based on the assumption of (0–1) loss or the area under the receiver operating characteristic (ROC) curve. In this paper we go far beyond this classical framework by investigating the generalization performance of bipartite ranking algorithms with convex losses over reproducing kernel Hilbert spaces. Based on the McDiarmid inequality and Rademacher complexity, we establish the upper bound on the generalization error for a bipartite ranking algorithm. The theoretical analysis is different from the previous results on error analysis and shows the attractive uniform convergence property of regularized bipartite ranking algorithms.

Robustness of Global Attractors for Reversible Gray–Scott Systems

Global asymptotic dynamics of a typical cubic-autocatalytic reaction-diffusion system, the reversible Gray–Scott system of three components, are investigated. The upper semicontinuity (robustness) of the global attractors in the H1 product space for the solution semiflows with respect to the reverse reaction rates as they converge to zero is proved. Through an approach of transformative decomposition, the hurdle of the perturbed singularity between the reversible and non-reversible systems is overcome by showing the uniform dissipation, the uniformly bounded evolution of the union of global attractors, and the uniform convergence property of the bundle of reversible and non-reversible semiflows.

On uniform consistent estimators for convex regression

A new nonparametric estimator of a convex regression function in any dimension is proposed and its uniform convergence properties are studied. We start by using any estimator of the regression function and convexify it by taking the convex envelope of a sample of the approximation obtained. We prove that the uniform rate of convergence of the estimator is maintained after the convexification is applied. The finite-sample properties of the new estimator are investigated by means of a simulation study and the application of the new method is demonstrated in examples.