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This paper proposes a dead-reckoning (DR) method for vehicles using Lie theory. This approach treats the pose (position and attitude) and velocity of the vehicle as elements of the Lie group SE 2 (3) and follows the computations based on Lie theory. Previously employed DR methods, which have been widely used, suffer from cumulative errors over time due to inaccuracies in the calculated changes from velocity during the motion of the vehicle or small errors in modeling assumptions. Consequently, this results in significant discrepancies between the estimated and actual positions over time. However, by treating the pose and velocity of the vehicle as elements of the Lie group, the proposed method allows for accurate solutions without the errors introduced by linearization. The incremental updates for pose and velocity in the DR computation are represented in the Lie algebra. Experimental results confirm that the proposed method improves the accuracy of DR. In particular, as the motion prediction time interval of the vehicle increases, the proposed method demonstrates a more pronounced improvement in positional accuracy. [ABSTRACT FROM AUTHOR]

An adaptive fuzzy sliding control (AFSMC) approach is adopted in this paper to address the problem of angular position control and vibration suppression of rotary flexible joint systems. AFSMC consists of fuzzy-based singleton control action and switching control law. By adjusting fuzzy parameters with the self-learning ability to discard system nonlinearities and uncertainties, singleton control based on fuzzy approximation theory can estimate the perfect control law of feedback linearization. To enhance robustness, an additional switching control law is incorporated to reduce the approximation error between the derived singleton control action and the perfect control law of feedback linearization. AFSMC's closed-loop stability will be demonstrated via sliding surface and Lyapunov function analysis of error function. In order to demonstrate the effectiveness of the AFSMC in tracking performance as well as its ability to respond to model uncertainties and external perturbations, simulations are carried out using Simulink and Matlab in order to demonstrate how well it adapts to these situations. Based on these results, it can be concluded that the AFSMC performs satisfactorily in terms of tracking. [ABSTRACT FROM AUTHOR]

In this paper, as a generalization of Szasz operators, a brand-new sequence of operators including the Appell polynomials of class A 2 is introduced. First, the convergence of this new sequence of operators is obtained, and then, some approximation results are presented by using the tools of approximation theory. In addition, an explicit example for this kind of sequence of operators containing Gould–Hopper polynomials is introduced. The error of the approximation of this new sequence of operators to a function is established. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new framework for studying continued fractions (CFs) by means of the backwards continued fraction (BCF). We develop an approximation theory for BCFs based on taking expansions of a fixed length, show the correspondence between continued fractions and their BCFs counterpart, and illustrate a rich approximation theory for continued fractions based off the methods of the approximation theory for the backwards case. In particular, we construct explicit functions that are sharp bounds for the BCF or CF error infinitely often over any BCF or CF cylinder set, and work out the details to pass seamlessly between the BCF and CF expansion of any real number. [ABSTRACT FROM AUTHOR]

This paper studies the approximation capacity of ReLU neural networks with norm constraint on the weights. We prove upper and lower bounds on the approximation error of these networks for smooth function classes. The lower bound is derived through the Rademacher complexity of neural networks, which may be of independent interest. We apply these approximation bounds to analyze the convergences of regression using norm constrained neural networks and distribution estimation by GANs. In particular, we obtain convergence rates for over-parameterized neural networks. It is also shown that GANs can achieve optimal rate of learning probability distributions, when the discriminator is a properly chosen norm constrained neural network. [ABSTRACT FROM AUTHOR]

The Chirp-Fourier transform is one of the most important tools of the modern signal processing. It has been widely used in the fields of ultrasound imaging, parameter estimation, and so on. The key to its application lies in the sampling and fast algorithms. In practical applications, nonuniform sampling can be caused by sampling equipment and other reasons. For the nonuniform sampling, we utilized function approximation and interpolation theory to construct different approximation forms of Chirp-Fourier transform kernel function, and proposed three fast nonuniform Chirp-Fourier transform algorithms. By analyzing the approximation error and the computational complexity of these algorithms, the effectiveness of the proposed algorithms was proved. [ABSTRACT FROM AUTHOR]

To make better decisions on approximation, we may need to increase reliable and useful information on different aspects of approximation. To enhance information about the quality and certainty of approximating the solution of an Apollonius-type quadratic functional equation, we need to measure both the quality and the certainty of the approximation and the maximum errors. To measure the quality of it, we use fuzzy sets, and to achieve its certainty, we use the probability distribution function. To formulate the above problem, we apply the concept of Z-numbers and introduce a special matrix of the form diag (A , B , C) (named the generalized Z-number) where A is a fuzzy time-stamped set, B is the probability distribution function, and C is a degree of reliability of A that is described as a value of A ∗ B . Using generalized Z-numbers, we define a novel control function to investigate H–U–R stability to approximate the solution of an Apollonius-type quadratic functional equation with quality and certainty of the approximation. [ABSTRACT FROM AUTHOR]

Pairwise learning is widely employed in ranking, similarity and metric learning, area under the ROC curve (AUC) maximization, and many other learning tasks involving sample pairs. Pairwise learning with deep neural networks was considered for ranking, but enough theoretical understanding about this topic is lacking. In this letter, we apply symmetric deep neural networks to pairwise learning for ranking with a hinge loss ϕ h and carry out generalization analysis for this algorithm. A key step in our analysis is to characterize a function that minimizes the risk. This motivates us to first find the minimizer of ϕ h -risk and then design our two-part deep neural networks with shared weights, which induces the antisymmetric property of the networks. We present convergence rates of the approximation error in terms of function smoothness and a noise condition and give an excess generalization error bound by means of properties of the hypothesis space generated by deep neural networks. Our analysis is based on tools from U-statistics and approximation theory. [ABSTRACT FROM AUTHOR]

We consider the spectral approximation of flow optimal control problems with state constraint. The main contribution of this work is to derive an a posteriori error estimator, and show the upper and lower bounds for the approximation error. Numerical experiment shows the efficiency of the error indicator, which will be used to construct reliable adaptive hp spectral element approximation for flow control in the future work. [ABSTRACT FROM AUTHOR]

We study the expressivity of deep neural networks. Measuring a network's complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a given complexity decays at a certain rate when increasing the complexity budget. Using results from classical approximation theory, we show that this class can be endowed with a (quasi)-norm that makes it a linear function space, called approximation space. We establish that allowing the networks to have certain types of "skip connections" does not change the resulting approximation spaces. We also discuss the role of the network's nonlinearity (also known as activation function) on the resulting spaces, as well as the role of depth. For the popular ReLU nonlinearity and its powers, we relate the newly constructed spaces to classical Besov spaces. The established embeddings highlight that some functions of very low Besov smoothness can nevertheless be well approximated by neural networks, if these networks are sufficiently deep. [ABSTRACT FROM AUTHOR]

In this article, we investigate the stability analysis of a polynomial-fuzzy-model-based control system by employing a new form of approximate membership functions called Chebyshev membership functions (CMFs) based on the Lyapunov stability theory. The membership-function-dependent method shows that the approximate functions can introduce useful information to relax the conservativeness of stability analysis. Therefore, in the case of the same order and operating domain, how to design the approximate functions to introduce more information to relax stability is a concern. In this article, given the Chebyshev norm relationship between the approximate membership functions and the original ones, CMFs can be obtained by the Chebyshev approximation theory and the Remez/Remez-type iterative algorithm, which own the minimum of the maximum absolute approximation error. Together with CMFs, the boundary information of premise variables on the overall operating domain is considered to reduce the conservativeness of stability analysis. Furthermore, to further relax the stability analysis without increasing the order of CMFs, the operating domain is divided into subdomains, and piecewise Chebyshev membership functions (PCMFs) are proposed to facilitate the stability analysis. Together with PCMFs, the subdomain boundary information of premise variables is introduced into stability conditions. A simulation example is given to demonstrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

We present an algorithm for solving unconstrained optimization problems based on the q-gradient vector. The main idea used in the algorithm construction is the approximation of the classical gradient by a q-gradient vector. For a convex objective function, the quasi-Fejér convergence of the algorithm is proved. The proposed method does not require the boundedness assumption on any level set. Further, numerical experiments are reported to show the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this study, we present a link between approximation theory and summability methods by constructing bivariate Bernstein-Kantorovich type operators on an extended domain with reparametrized knots. We use a statistical convergence type and power series method to obtain certain Korovkin type theorems, and we study certain rates of convergences related to these summability methods. Furthermore, we numerically analyze the theoretical results and provide some computer graphics to emphasize the importance of this study. [ABSTRACT FROM AUTHOR]