Approximation Properties Research Articles

Variational inequalities of multilayer elastic contact systems with interlayer friction: Existence and uniqueness of solution and convergence of numerical solution

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions.

Approximation in quantum calculus of the Phillips operators by using the sequences of q-Appell polynomials

In this paper, we attempt to use the Dunkl analog to study the convergence properties of q-Phillips operators by using the q-Appell polynomials. By applying the new sequences of continuous functions νs,q(z)=(z−12[s]q)ϱ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\ u _{s,q}(z)=\\Bigl(z- \\frac{1}{2[s]_{q}}\\Bigr)_{\\varrho}$\\end{document} on [0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$[0, \\infty )$\\end{document}, we construct an improved version of the q-Phillips operators. We calculate the qualitative outcomes in weighted Korovkin spaces to better understand the Phillips operators’ uniform convergence results. We obtain the approximation properties by use of the modulus of continuity and functions belonging to the Lipschitz class. Moreover, we give some direct theorems for the function belonging to Peetre’s K-functional.

Approximation properties of trigonometric Fourier series in generalized variation classes

Approximation properties of trigonometric Fourier series in generalized variation classes

Approximation by Schurer Type λ-Bernstein–Bézier Basis Function Enhanced by Shifted Knots Properties

In this article, a novel Schurer form of λ-Bernstein operators augmented by Bézier basis functions is presented by utilizing the features of shifted knots. The shifted knots form of Bernstein operators and the Schurer form of the Bézier basis function are used in this article, then, new operators, the Schurer type λ-Bernstein shifted knots operators are constructed in terms of the Bézier basis function. First, the test functions are calculated and the central moments for these operators are obtained. Then, Korovkin’s type approximation properties are studied by the use of a modulus of continuity of orders one and two. Finally, the convergence theorems for these new operators are obtained by using Peetre’s K-functional and Lipschitz continuous functions. In the end, some direct approximation theorems are also obtained.

Approximation by Symmetrized and Perturbed Hyperbolic Tangent Activated Convolution-Type Operators

In this article, for the first time, the univariate symmetrized and perturbed hyperbolic tangent activated convolution-type operators of three kinds are introduced. Their approximation properties are presented, i.e., the quantitative convergence to the unit operator via the modulus of continuity. It follows the global smoothness preservation of these operators. The related iterated approximation as well as the simultaneous approximation and their combinations, are also extensively presented. Including differentiability and fractional differentiability into our research produced higher rates of approximation. Simultaneous global smoothness preservation is also examined.

Approximation properties of torsion classes

AbstractWe strengthen a result of Bagaria and Magidor (Trans. Amer. Math. Soc. 366 (2014), no. 4, 1857–1877) about the relationship between large cardinals and torsion classes of abelian groups, and prove that the Maximum Deconstructibility principle introduced in Cox (J. Pure Appl. Algebra 226 (2022), no. 5) requires large cardinals; it sits, implication‐wise, between Vopěnka's Principle and the existence of an ‐strongly compact cardinal. While deconstructibility of a class of modules always implies the precovering property by Saorín and Šťovíček (Adv. Math. 228 (2011), no. 2, 968–1007), the concepts are (consistently) nonequivalent, even for classes of abelian groups closed under extensions, homomorphic images, and colimits.

Complex Generalized Stancu-Schurer Operators

Abstract In the present study, we determine a Schurer type of complex generalized Stancu operators and examine its some approximation properties. Firstly, we obtain upper quantitative estimates for the convergence and then we get lower estimates from a qualitative Voronovskaja-type result for these specified operators attached to the analytic functions in an open disk centered at the origin. Hence, we present the exact order of simultaneous approximation by using obtained upper and lower estimates.

Noise-resilient designs and analysis for optical neural networks

Abstract All analog signal processing is fundamentally subject to noise, and this is also the case in next generation implementations of Optical Neural Networks (ONNs). Therefore, we propose the first hardware-based approach to mitigate noise in ONNs. A tree-like and an accordion-like design are constructed from a given Neural Network (NN) that one wishes to implement. Both designs have the capability that the resulting ONNs give outputs close to the desired solution. To establish the latter, we analyze the designs mathematically. Specifically, we investigate a probabilistic framework for the tree-like design that establishes the correctness of the design, i.e., for any feed-forward NN with Lipschitz continuous activation functions, an ONN can be constructed that produces output arbitrarily close to the original. ONNs constructed with the tree-like design thus also inherit the universal approximation property of NNs. For the accordion-like design, we restrict the analysis to NNs with linear activation functions and characterize the ONNs’ output distribution using exact formulas. Finally, we report on numerical experiments with LeNet ONNs that give insight into the number of components required in these designs for certain accuracy gains. The results indicate that adding just a few components and/or adding them only in the first (few) layers in the manner of either design can already be expected to increase the accuracy of ONNs considerably. To illustrate the effect we point to a specific simulation of a LeNet implementation, in which adding one copy of the layers components in each layer reduces the Mean Squared Error (MSE) by 59.1% for the tree-like design and by 51.5% for the accordion-like design. In this scenario, the gap in accuracy of prediction between the noiseless NN and the ONNs reduces even more: 93.3% for the tree-like design and 80% for the accordion-like design.

Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties

Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties

Diophantine approximation by rational numbers of certain parity types

AbstractFor a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd, and odd/even rational numbers. We study algorithms to find best approximations by rational numbers of given parities and compare these algorithms with continued fraction expansions.

Pointwise Convergence of Noncommutative Fourier Series

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 1 pointwise, so that the associated Fourier multipliers on noncommutative L p L_p -spaces satisfy the pointwise convergence for all p > 1 p>1 . In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejér and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.

Observer-Based Fault-Tolerant Finite-Time Control of Nonlinear Multiagent Systems.

In this article, an adaptive neural containment control for a class of nonlinear multiagent systems considering actuator faults is introduced. By using the general approximation property of neural networks, a neuro-adaptive observer is designed to estimate unmeasured states. In addition, in order to reduce the computational burden, a novel event-triggered control law is designed. Furthermore, the finite-time performance function is presented to improve the transient and steady-state performance of the synchronization error. Utilizing the Lyapunov stability theory, it will be shown that the closed-loop system is cooperatively semiglobally uniformly ultimately bounded (CSGUUB), and the followers' outputs reach the convex hull constructed by the leaders. Moreover, it is shown that the containment errors are limited to the prescribed level in a finite time. Eventually, a simulation example is presented to corroborate the capability of the proposed scheme.

Fixed-Time Congestion Control for a Class of Uncertain Multi-Bottleneck TCP/AWM Networks

As network technology continues to advance, network congestion has become an inevitable aspect of network communication. Considering the external interference, unmodeled uncertainty and the interaction between nodes, a multi-bottleneck TCP/AWM network model is established in this paper. A new fixed-time congestion controller was designed by combining a neural network and the backstepping technique. The neural network approximation property is used to eliminate the interference of unmodeled uncertainty and UDP flow in the system. The controller designed in this paper can ensure the stability of the TCP/AWM closed-loop system in a fixed time. Finally, the simulation results demonstrate the effectiveness of the proposed TCP/AWM controller.

Deep Convolutional Tables: Deep Learning Without Convolutions.

We propose a novel formulation of deep networks that do not use dot-product neurons and rely on a hierarchy of voting tables instead, denoted as convolutional tables (CTs), to enable accelerated CPU-based inference. Convolutional layers are the most time-consuming bottleneck in contemporary deep learning techniques, severely limiting their use in the Internet of Things and CPU-based devices. The proposed CT performs a fern operation at each image location: it encodes the location environment into a binary index and uses the index to retrieve the desired local output from a table. The results of multiple tables are combined to derive the final output. The computational complexity of a CT transformation is independent of the patch (filter) size and grows gracefully with the number of channels, outperforming comparable convolutional layers. It is shown to have a better capacity:compute ratio than dot-product neurons, and that deep CT networks exhibit a universal approximation property similar to neural networks. As the transformation involves computing discrete indices, we derive a soft relaxation and gradient-based approach for training the CT hierarchy. Deep CT networks have been experimentally shown to have accuracy comparable to that of CNNs of similar architectures. In the low-compute regime, they enable an error:speed tradeoff superior to alternative efficient CNN architectures.

Lipschitz-free spaces over properly metrizable spaces and approximation properties

Let T T be a topological space admitting a compatible proper metric, that is, a locally compact, separable and metrizable space. Let M T \mathcal {M}^T be the non-empty set of all proper metrics d d on T T compatible with its topology, and equip M T \mathcal {M}^T with the topology of uniform convergence, where the metrics are regarded as functions on T 2 T^2 . We prove that the set A T , 1 \mathcal {A}^{T,1} of metrics d ∈ M T d\in \mathcal {M}^T for which the Lipschitz-free space F ( T , d ) \mathcal {F}(T,d) has the metric approximation property is a dense set in M T \mathcal {M}^T , and is furthermore residual in M T \mathcal {M}^T when T T is zero-dimensional. We also prove that if T T is uncountable then the set A f T \mathcal {A}^T_f of metrics d ∈ M T d\in \mathcal {M}^T for which F ( T , d ) \mathcal {F}(T,d) fails the approximation property is dense in M T \mathcal {M}^T . Combining the last statement with a result of Dalet, we conclude that for any ‘properly metrizable’ space T T , A f T \mathcal {A}^T_f is either empty or dense in M T \mathcal {M}^T .

New Kantorovich-type Szász–Mirakjan Operators

In this paper, we present a Kantorovich-type Szász–Mirakjan operators. Initially, we establish the recurrence relationship for the moments of these operators and provide the central moments up to the fourth degree. Subsequently, we analyze the local approximation properties of these operators using Peetre’s K-function. We investigate the rate of convergence, by utilizing the ordinary modulus of continuity and Lipschitz-type maximal functions. Additionally, we prove weighted approximation theorems and Voronoskaja-type theorems specific to these new operators. Following this, we introduce bivariate extension of these operators and investigate some approximation properties. Lastly, we include several numerical illustrative examples.

The approximation property for the predual of weighted spaces of holomorphic mappings on Banach spaces

In this paper, we investigate the approximation property for the predual of the weighted space of holomorphic mappings, where is a countable family of weights defined on a balanced open subset U of a Banach space E. We introduce a locally convex topology on and show that is topologically isomorphic to for a suitably restricted family . As a consequence, certain characterizations of the approximation property for the space are proved.

Physics-informed quantum neural network for solving forward and inverse problems of partial differential equations

Recently, physics-informed neural networks (PINNs) have aroused an upsurge in the field of scientific computing including solving partial differential equations (PDEs), which convert the task of solving PDEs into an optimization challenge by adopting governing equations and definite conditions or observation data as loss functions. Essentially, the underlying logic of PINNs is based on the universal approximation and differentiability properties of classical neural networks (NNs). Recent research has revealed that quantum neural networks (QNNs), known as parameterized quantum circuits, also exhibit universal approximation and differentiability properties. This observation naturally suggests the application of PINNs to QNNs. In this work, we introduce a physics-informed quantum neural network (PI-QNN) by employing the QNN as the function approximator for solving forward and inverse problems of PDEs. The performance of the proposed PI-QNN is evaluated by various forward and inverse PDE problems. Numerical results indicate that PI-QNN demonstrates superior convergence over PINN when solving PDEs with exact solutions that are strongly correlated with trigonometric functions. Moreover, its accuracy surpasses that of PINN by two to three orders of magnitude, while requiring fewer trainable parameters. However, the computational time of PI-QNN exceeds that of PINN due to its operation on classical computers. This limitation may improve with the advent of commercial quantum computers in the future. Furthermore, we briefly investigate the impact of network architecture on PI-QNN performance by examining two different QNN architectures. The results suggest that increasing the number of trainable network layers can enhance the expressiveness of PI-QNN. However, an excessive number of data encoding layers significantly increases computational time, rendering the marginal gains in performance insufficient to compensate for the shortcomings in computational efficiency.

On Saphar’s Approximation Property of Order p and the $$w_p$$-Approximation Property

On Saphar’s Approximation Property of Order p and the $$w_p$$-Approximation Property

PDE-Based Boundary Adaptive Consensus Control of Multiagent Systems With Input Constraints.

The leader-follower adaptive consensus control problem is addressed for partial differential equations (PDEs) multiagent systems (MASs), and these agents are composed of flexible manipulator systems with input nonlinearity, boundary uncertainties, and time-varying disturbances. Because of the spatial variables in the model, the design of adaptive protocols is more difficult than that of ordinary differential equation (ODE) MASs. By designing the Lyapunov function, a novel distributed boundary control (BC) protocol is constructed, which not only ensures the consensus of angular positions but also suppresses the boundary vibration of each agent. The hybrid effects of dead zones and input saturation on flexible manipulator systems are addressed using the approximation properties of neural networks (NNs). In addition, the disturbance adaptive laws are proposed to provide a control solution for bounded and time-varying disturbances. Furthermore, by applying the Lyapunov stability theory, the uniformly bounded stability of the multiflexible manipulator can be ensured. Finally, the feasibility of the presented control approach is verified using numerical examples.