Approximation Theorem Research Articles

On the Auslander–Bridger–Yoshino theory for complexes of finitely generated projective modules

Let R be a two-sided noetherian ring. Auslander and Bridger developed a theory of projective stabilization of the category of finitely generated R-modules, which is called the stable module theory. Recently, Yoshino established a stable “complex” theory, i.e., a theory of a certain stabilization of the homotopy category of complexes of finitely generated projective R-modules. We introduce higher versions of several notions introduced by Yoshino, such as ⁎torsionfreeness and ⁎reflexivity. Also, we prove the Auslander–Bridger approximation theorem for complexes of finitely generated projective R-modules.

Reaction barriers at metal surfaces computed using the random phase approximation: Can we beat DFT in the generalized gradient approximation?

Reaction barriers for molecules dissociating on metal surfaces (as relevant to heterogeneous catalysis) are often difficult to predict accurately with density functional theory (DFT). Although the results obtained for several dissociative chemisorption reactions via DFT in the generalized gradient approximation (GGA), in meta-GGA, and for GGA exchange + van der Waals correlation scatter around the true reaction barrier, there is an entire class of dissociative chemisorption reactions for which GGA-type functionals collectively underestimate the reaction barrier. Little is known why GGA-DFT collectively fails in some cases and not in others, and we do not know whether other methods suffer from the same inconsistency. Here, we present barrier heights for dissociative chemisorption reactions obtained from the random phase approximation in the adiabatic-connection fluctuation-dissipation theorem (ACFDT-RPA) and from hybrid functionals with different amounts of exact exchange. By comparing the results obtained for the dissociative chemisorption reaction of H2 on Al(110) (where GGA-DFT collectively underestimates the barrier) and H2 on Cu(111) (where GGA-DFT scatters around the true barrier), we can gauge whether the inconsistent description of the systems persists for hybrid functionals and ACFDT-RPA. We find hybrid functionals to improve the relative description of the two systems, but to fall short of chemical accuracy. ACFDT-RPA improves the results further and leads to chemically accurate barriers for both systems. Together with an analysis of the density of states and the results from selected GGA, meta-GGA, and GGA exchange + van der Waals correlation functionals, these results allow us to discuss possible origins for the inconsistent behavior of GGA-based functionals for molecule-metal reaction barriers.

A note on Kronecker’s approximation theorem

We improve on Gonek-Montgomery’s quantitative version of Kronecker’s approximation theorem.

The GroupMax Neural Network Approximation of Convex Functions.

We present a new neural network to approximate convex functions. This network has the particularity to approximate the function with cuts which is, for example, a necessary feature to approximate Bellman values when solving linear stochastic optimization problems. The network can be easily adapted to partial convexity. We give an universal approximation theorem in the full convex case and give many numerical results proving its efficiency. The network is competitive with the most efficient convexity-preserving neural networks and can be used to approximate functions in high dimensions.

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Unstability problem of real analytic maps

AbstractAs well known, the stability of proper maps is characterized by the infinitesimal stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal stability does not imply stability; for instance, a Whitney umbrella is not stable. A main tool for the proof is a relative version of Whitney's analytic approximation theorem that is shown by using H. Cartan's Theorems A and B.

Neural network Kantorovich operators activated by smooth ramp functions

In the present article, we introduce a Kantorovich variant of the neural network interpolation operators activated by smooth ramp functions proposed by Qian and Yu (2022). We discuss the convergence of these operators in the spaces, and , and establish some direct approximation theorems. Further, we derive the converse results by means of Berens–Lorentz lemma and Peetre's K‐functional. We present a multivariate version of the aforementioned Kantorovich neural network interpolation operators and investigate the direct and converse results in the continuous and , spaces.

Approximation of discontinuous signals by exponential‐type generalized sampling Kantorovich series

In the present paper, we analyze the behavior of the exponential‐type generalized sampling Kantorovich operators when discontinuous signals are considered. We present a proposition for the series , and we prove using this proposition certain approximation theorems for discontinuous functions. Furthermore, we give several examples of kernels satisfying the assumptions of the present theory. Finally, some numerical computations are performed to verify the approximation of discontinuous functions by .

Approximation theorem for the Kawahara operator and its application in the control theory

AbstractControl properties of the Kawahara equation are considered when the equation is posed on an unbounded domain. Precisely, the paper's main results are related to an approximation theorem that ensures the exact (internal) controllability in . Following [23], the problem is reduced to prove an approximate theorem which is achieved thanks to a global Carleman estimate for the Kawahara operator.

Approximation by Stancu variant of [formula omitted]-Bernstein shifted knots operators associated by Bézier basis function

The current paper presents the λ-Bernstein operators through the use of newly developed variant of Stancu-type shifted knots polynomials associated by Bézier basis functions. Initially, we design the proposed Stancu generated λ-Bernstein operators by means of Bézier basis functions then investigate the local and global approximation results by using the Ditzian–Totik uniform modulus of smoothness of step weight function. Finally we establish convergence theorem for Lipschitz generated maximal continuous functions and obtain some direct theorems of Peetre’s K-functional. In addition, we establish a quantitative Voronovskaja-type approximation theorem.

Neural Projection Filter: Learning Unknown Dynamics Driven by Noisy Observations.

In this article, we propose the novel neural stochastic differential equations (SDEs) driven by noisy sequential observations called neural projection filter (NPF) under the continuous state-space models (SSMs) framework. The contributions of this work are both theoretical and algorithmic. On the one hand, we investigate the approximation capacity of the NPF, i.e., the universal approximation theorem for NPF. More explicitly, under some natural assumptions, we prove that the solution of the SDE driven by the semimartingale can be well approximated by the solution of the NPF. In particular, the explicit estimation bound is given. On the other hand, as an important application of this result, we develop a novel data-driven filter based on NPF. Also, under certain condition, we prove the algorithm convergence; i.e., the dynamics of NPF converges to the target dynamics. At last, we systematically compare the NPF with the existing filters. We verify the convergence theorem in linear case and experimentally demonstrate that the NPF outperforms existing filters in nonlinear case with robustness and efficiency. Furthermore, NPF could handle high-dimensional systems in real-time manner, even for the 100-D cubic sensor, while the state-of-the-art (SOTA) filter fails to do it.

Approximation behaviour of generalized Baskakov-Durrmeyer-Schurer operators

The goal of this manuscript is to introduce a new sequence of generalized-Baskakov Durrmeyer-Schurer Operators. Further, basic estimates are calculated. In the subsection sequence, rapidity of convergence and order of approximation are studied in terms of first and second-order modulus of continuity. We prove a Korovkin-type approximation theorem and obtain the rate of convergence of these operators. Moreover, local and global approximation properties are discussed in different functional spaces. Lastly, A-statistical approximation results are presented.

Approximation properties by shifted knots type of α-Bernstein–Kantorovich–Stancu operators

Through the real polynomials of the shifted knots, the α-Bernstein–Kantorovich operators are studied in their Stancu form, and the approximation properties are obtained. We obtain some direct approximation theorem in terms of Lipschitz type maximum function and Peetre’s K-functional, as well as Korovkin’s theorem. Eventually, the modulus of continuity is used to compute the upper bound error estimation.

Constructing Approximations to Bivariate Piecewise-Smooth Functions

This paper demonstrates that the space of piecewise-smooth bivariate functions can be well-approximated by the space of the functions defined by a set of simple (non-linear) operations on smooth uniform tensor product splines. The examples include bivariate functions with jump discontinuities or normal discontinuities across curves, and even across more involved geometries such as a three-corner discontinuity. The provided data may be uniform or non-uniform, and noisy, and the approximation procedure involves non-linear least-squares minimization. Also included is a basic approximation theorem for functions with jump discontinuity across a smooth curve.

Modeling the measurement accuracy of one-dimensional boundary subsets in digital image correlation

Boundary subsets, which contain unwanted pixels from background image or other regions, generally occur at areas of vital mechanical importance. In this work, we establish a theoretical model characterizing the measurement accuracy of 1-dimensional boundary subsets, which explicitly expresses the retrieved displacement in terms of subset size, invalid pixels, shape function order, and underlying deformation field. Particularly, the transfer function of digital image correlation at boundary subsets is derived and verified, indicating possible amplification of amplitude and the existence of phase shift, which are quite different from standard full square subsets. Finally, the retrieved deformation near boundary is theoretically analyzed, and closed-form formulae are deduced for polynomial underlying displacements in consideration of the Weierstrass approximation theorem. To the author's knowledge, this is the first general mathematical model for boundary subsets, and paves the way for enhanced measurement accuracy.

Approximation theorems via Pp -statistical convergence on weighted spaces

Abstract In this paper, we obtain some Korovkin type approximation theorems for double sequences of positive linear operators on two-dimensional weighted spaces via statistical type convergence method with respect to power series method. Additionally, we calculate the rate of convergence. As an application, we provide an approximation using the generalization of Gadjiev-Ibragimov operators for Pp -statistical convergence. Our results are meaningful and stronger than those previously given for two-dimensional weighted spaces.

Best Approximation and Inverse Results for Neural Network Operators

In the present paper we considered the problems of studying the best approximation order and inverse approximation theorems for families of neural network (NN) operators. Both the cases of classical and Kantorovich type NN operators have been considered. As a remarkable achievement, we provide a characterization of the well-known Lipschitz classes in terms of the order of approximation of the considered NN operators. The latter result has inspired a conjecture concerning the saturation order of the considered families of approximation operators. Finally, several noteworthy examples have been discussed in detail

Theoretical and experimental study of unsteady behavior of marine diesel engine under fluctuating back pressure condition

In order to meet the increasingly urgent emission reduction requirements, underwater exhaust has emerged as a promising technology for marine diesel engines and has received widespread attention in recent years. However, under fluctuating high back pressure conditions, the diesel engine as a whole operates in an unsteady state, with significant fluctuations in output power and exhaust temperature, posing a serious threat to their stability and safety. This paper focuses on the unsteady operating characteristics of diesel engines and its impact on the engine output power, efficiency and safety under fluctuating high back pressure conditions. Firstly, based on model order reduction and the first Lyapunov approximation theorem, the theoretical derivation reveals that the filling and emptying effects in the intake and exhaust pipes and the shaft moment of inertia of the turbocharger are the main causes of the unsteady of the engine under fluctuating high back pressure conditions. The factors influencing the strength of engine unsteadiness, including wave period, wave amplitude, mean back pressure and engine properties, are identified. Then, diesel engine experiments under different fluctuating high back pressure conditions are performed and the conclusions derived from the theoretical analysis are validated through experiments. Based on the research results mentioned above, this paper proposes a criterion to determine the unsteadiness of diesel engines. Through comparison with experimental data, it is found that this criterion effectively reflects the strength of engine unsteadiness, with a correlation coefficient of 0.9169 with the unsteady parameter.

Direct, Inverse Theorems of Approximation by Linear Combinations of Weighted Baskakov–Durrmeyer Operators in Orlicz Spaces

Direct, Inverse Theorems of Approximation by Linear Combinations of Weighted Baskakov–Durrmeyer Operators in Orlicz Spaces

Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation Δh + h = 0 on Rd, over the natural length scale O(λ−1/2) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces Hd(κ). As the dimension of the space of bound states of these operators tends to infinity as κ ↘ 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ > 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on Hd(κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different.