Class Of Smooth Functions Research Articles

One-parametric class of smoothing functions for (WSOCCP) and its Jacobian consistency

In this paper, we are interested in the study of weighted second-order cone complementarity problem (WSOCCP) and the smoothing function associated with it. A weighted second-order cone complementarity problem is to find a pair of vectors ( x , y ) that belong to the intersection of a manifold and a second-order cone such that the product of the vectors in a certain algebra, x ∘ y equals a given weight vector w. It is obvious to see that if w is zero, we get a second-order cone complementarity problem (SOCCP). Thus, (WSOCCP) provides the essential framework for a broader class of problems arising from real applications (science and engineering) more conveniently. This paper demonstrates the Jacobian consistency of the one-parametric class of smoothing functions. Furthermore, we provide an estimation of the distance between the subgradient of the one-parametric class of the weighted SOC complementarity functions and the gradient of its associated smoothing function. This estimation facilitates to adjust an appropriate parameter in smoothing methods.

Fixed-time adaptive fuzzy control with prescribed tracking performances for flexible-joint manipulators

In this paper, we study the tracking control problem of flexible-joint manipulators with uncertain unstructured dynamics, and develop a prescribed fixed-time solution scheme by combining direct adaptive fuzzy control design with a class of smooth functions. With our scheme, only one adaptive law needs to be calculated online, and also there are no complicated terms, e.g., the partial derivatives of some multivariable functions, involved in virtual controllers and the actual control input, which show that the raised scheme is computationally attractive. Moreover, the boundedness of all closed-loop signals, and the fixed-time convergence of tracking error to a prescribed interval can be established through a rigorous proof, and the obtained results are further confirmed by simulation tests.

Optimal Hermite-Fejér interpolation of algebraic polynomials and the best one-sided approximation on the interval [−1,1

In this paper, we give the relations of the optimal Hermite-Fejér interpolation and the best one-sided approximation to the smooth function classes BC2r,+, r∈N in weighted space L1,ω([−1,1]), with a positive, continuous and integrable weight function ω on (−1,1). We proved that the Hermite-Fejér interpolation based on the set of the zeros of some orthogonal polynomials is optimal respectively in the space L1,ω([−1,1]) and the space L∞([−1,1]) and gave the exact constants of the approximation errors of these Hermite-Fejér interpolation. We also obtained the exact constant of the approximation errors of the optimal Hermite-Fejér interpolation when the endpoints of the interval [−1,1] are included in the considered sets of the nodes.

ON AN OPTIMIZATION PROBLEM FOR A PRESTRESSED PLATE WITH VARIABLE STIFFNESS

Most studies in the field of optimization problems in the mechanics of the deformable solid body are aimed at finding the optimal shape of structural elements, which most often leads to the search for the optimal distributions of geometric characteristics for the considered objects in order to ensure reducing the objects? weight and improve their strength characteristics. At the same time, with the advent of modern materials with variable physical and mechanical properties, new relevant branches arose in the optimization problems, as well as the inverse problems on the reconstruction of non-uniform variable characteristics. In this study, the problem of steady-state oscillations of a prestressed inhomogeneous round elastic plate in an axisymmetric formulation is considered. Variational and weak problem statements are presented. The case of free oscillations of a plate simply supported along the contour is investigated; it is shown that the eigenvalue problem is self-adjoint and quite definite. The problem of finding the optimal distribution of the plates? elastic modulus and the corresponding oscillation shape in order to maximize the first natural frequency is considered. An isoperimetric condition for the variable stiffness function is proposed. Based on the Rayleigh energy ratio, the minimum principle for the first eigenvalue and the corresponding eigenfunctionis formulated. On the basis of the variational principle, using the constructed Rayleigh ratio, the optimality condition is formulated, and an explicit representation is obtained for the functions of the optimal oscillation shape and optimal stiffness, taking into account the initial radial stretching or compression of the plate. The range of possible values for the initial stress is determined. The results of solving the problem via constructing optimal inhomogeneity laws for the stiffness function in the class of smooth functions in the considered range of changes in initial stresses are obtained and analyzed.

Phase transitions in nonparametric regressions

When the unknown regression function of a single variable is known to have derivatives up to the (γ+1)th order bounded in absolute values by a common constant everywhere or a.e. (i.e., (γ+1)th degree of smoothness), the minimax optimal rate of the mean integrated squared error (MISE) is stated as 1n2γ+22γ+3 in the literature. This paper shows that: (i) if n≤γ+12γ+3, the minimax optimal MISE rate is lognnlog(logn) and the optimal degree of smoothness to exploit is roughly maxlogn2loglogn,1; (ii) if n>γ+12γ+3, the minimax optimal MISE rate is 1n2γ+22γ+3 and the optimal degree of smoothness to exploit is γ+1.The fundamental contribution of this paper is a set of metric entropy bounds we develop for smooth function classes. Some of our bounds are original, and some of them improve and/or generalize the ones in the literature (e.g., (Kolmogorov and Tikhomirov, 1959)). Our metric entropy bounds allow us to show phase transitions in the minimax optimal MISE rates associated with some commonly seen smoothness classes as well as non-standard smoothness classes, and can also be of independent interest outside the nonparametric regression problems.

A bootstrap functional central limit theorem for time-varying linear processes

We provide a functional central limit theorem for a broad class of smooth functions for possibly non-causal multivariate linear processes with time-varying coefficients. Since the limiting processes depend on unknown quantities, we propose a local block bootstrap procedure to circumvent this inconvenience in practical applications. In particular, we prove bootstrap validity for a very large class of processes. Our results are illustrated by some numerical examples.

Contraction property of certain classes of log-[formula omitted]-subharmonic functions in the unit ball

We prove a contraction property for certain classes of smooth functions, whose absolute values of elements are log-subharmonic functions in the unit ball, thus extending the results of Kulikov to higher-dimensional space (GAFA (2022)). Moreover, by applying those results we get some new results for harmonic mappings in the complex plane.

Infinite Dimensional Widths and Optimal Recovery of a Function Class in Orlicz Spaces in L(R) Metric

In this paper, we study the infinite dimensional widths and optimal recovery of Wiener–Sobolev smooth function classes WM,1(Pr(D)) determined by the r-th differential operator Pr(D) in Orlicz spaces with L(R) metric. Using tools such as the Hölder inequality, we give the exact values of the infinite dimensional Kolmogorov width and linear width of WM,1(Pr(D)) in L(R) metric. We also study the related optimal recovery problem.

Approximation bounds for norm constrained neural networks with applications to regression and GANs

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.

On generalized definitions of ultradifferentiable classes

We show that the ultradifferentiable-like classes of smooth functions introduced and studied by S. Pilipović, N. Teofanov and F. Tomić are special cases of the general framework of spaces of ultradifferentiable functions defined in terms of weight matrices in the sense of A. Rainer and the third author. We study classes “beyond geometric growth factors” defined in terms of a weight sequence and an exponent sequence, prove that these new types admit a weight matrix representation and transfer known results from the matrix-type to such a non-standard ultradifferentiable setting.

Acceleration of Convergence of Fourier Series Using the Phenomenon of Over-Convergence

In recent publications of the author, the phenomenon of over-convergence was discovered, and a spectral method has been presented for accelerating the convergence of truncated Fourier series for smooth functions. On this basis, a certain parametric system that is biorthogonal to the corresponding segment of the Fourier system turned out to be unusually effective. This article reconsiders some approaches and makes some adjustments to previous publications. As a result, two improved schemes for the recovery of a function based on a finite set of its Fourier coefficients are proposed. Numerical experiments confirm a significant increase in the efficiency of corresponding algorithms in typical classes of smooth functions. In conclusion, some prospects for the development and generalization of the above approaches are discussed.

Sup-norm adaptive drift estimation for multivariate nonreversible diffusions

We consider the question of estimating the drift for a large class of ergodic multivariate and possibly nonreversible diffusion processes, based on continuous observations, in sup-norm loss. Nonparametric classes of smooth functions of unknown order are considered, and we suggest an adaptive approach which allows to construct drift estimators attaining optimal sup-norm rates of convergence. Reversibility structures and related functional inequalities are known to be key tools for these estimation problems. We can discard such restrictions by making use of mixing properties which are satisfied for the very general class of processes under consideration. Analysing diffusions, the scalar case is very distinct from the general multivariate setting. Therefore, we treat scalar and multivariate processes separately which leads to in several aspects improved univariate results. While we consider drift estimation on bounded domains for exponentially β-mixing multivariate processes, for scalar diffusion processes we work under minimal assumptions that allow estimation of unbounded drift terms over the entire real line, and we provide classical minimax results (including lower bounds) which cannot be obtained under state-of-the-art conditions in the multivariate case. In addition, we prove a Donsker theorem for the classical kernel estimator of the invariant density in the scalar setting and establish its semiparametric efficiency.

Causal functional calculus

We construct a new topology on the space of stopped paths and introduce a calculus for causal functionals on generic domains of this space. We propose a generic approach to pathwise integration without any assumption on the variation index of a path and obtain functional change of variable formulae which extend the results of Föllmer [Séminaire de probabilités 15 (1981), 143–150] and Cont and Fournié [J. Funct. Anal. 259 (2010), no. 4, 1043–1072] to a larger class of functionals, including Föllmer's pathwise integrals. We show that a class of smooth functionals possess a pathwise analogue of the martingale property. For paths that possess finite quadratic variation, our approach extends the Föllmer–Ito calculus and removes previous restriction on the time partition sequence. We introduce a foliation structure on this path space and show that harmonic functionals may be represented as pathwise integrals of closed 1-forms.

Polynomial inequalities, o-minimality and Denjoy–Carleman classes

We study several intimately related problems in the theory of multivariate polynomial inequalities. Firstly, given a map h in certain quasianalytic Denjoy–Carleman classes, we show how to decide whether the image under h of a set satisfying Markov's (resp. Nikolskii's) inequality satisfies Markov's (resp. Nikolskii's) inequality. Our approach relies heavily on the theory of o-minimal structures, particularly on the work of Rolin, Speissegger and Wilkie. Secondly, we establish the relation between Markov's inequality and Nikolskii's inequality (both in general setting and in o-minimal setting). Thirdly, we prove that each compact, definable (in an o-minimal structure) set satisfying Markov's inequality is fat. In particular, this solves in the o-minimal category the well-known problem of nonpluripolarity of sets satisfying Markov's inequality. And lastly, we develop a unified method of polynomial approximation for various classes of smooth functions of several variables.

Fast inertial dynamic algorithm with smoothing method for nonsmooth convex optimization

In order to solve the minimization of a nonsmooth convex function, we design an inertial second-order dynamic algorithm, which is obtained by approximating the nonsmooth function by a class of smooth functions. By studying the asymptotic behavior of the dynamic algorithm, we prove that each trajectory of it weakly converges to an optimal solution under some appropriate conditions on the smoothing parameters, and the convergence rate of the objective function values is $$o\left( t^{-2}\right)$$ . We also show that the algorithm is stable, that is, this dynamic algorithm with a perturbation term owns the same convergence properties when the perturbation term satisfies certain conditions. Finally, we verify the theoretical results by some numerical experiments.

Approximation properties of Gaussian-binary restricted Boltzmann machines and Gaussian-binary deep belief networks

Despite the successful use of Gaussian-binary restricted Boltzmann machines (GB-RBMs) and Gaussian-binary deep belief networks (GB-DBNs), little is known about their theoretical approximation capabilities to represent distributions of continuous random variables. In this paper, we address the expressive properties of GB-RBMs and GB-DBNs, contributing theoretical insights to the optimal number of hidden variables. We first treat the GB-RBM’s unnormalized log-likelihood as a sum of a special two-layer feedforward neural network and a negative quadratic term. Then, a series of simulation results are established, which can be used to relate GB-RBMs to general two-layer feedforward neural networks whose expressive properties are much better understood. On this basis, we show that a two-layer ReLU network with all weights in the second layer being 1, along with a negative quadratic term, can approximate all continuous functions. In addition, we provide qualified lower bounds for the number of hidden variables of GB-RBMs required to approximate distributions whose log-likelihood are given by some classes of smooth functions. Moreover, we further study the universal approximation of GB-DBNs with two hidden layers by providing a sufficient number of hidden variables O(ɛ−2) that are guaranteed to approximate any given strictly positive continuous distribution within a given error ɛ. Finally, numerical experiments are carried out to verify some of the proposed theoretical results.

Resilient Adaptive Neural Control for Uncertain Nonlinear Systems With Infinite Number of Time-Varying Actuator Failures.

Existing studies on adaptive fault-tolerant control for uncertain nonlinear systems with actuator failures are restricted to a common result that only system stability is established. Such a result of not being asymptotically stable is a tradeoff paid for reducing the number of online learning parameters. In this article, we aim to obviate such restrictions and improve the bounded error control to asymptotic control. Toward this end, a resilient adaptive neural control scheme is newly proposed based on a new design of the Lyapunov function candidates, a projection-associated tuning functions method, and an alternative class of smooth functions. It is proved that the system stability is guaranteed for the case of an infinite number of failures and when the number of failures is finite, asymptotic tracking performance can be automatically recovered, and besides, an explicit bound for the tracking error in terms of L2 norm is established. Illustrative examples demonstrate the methods developed.

On the Solvability of Some Boundary Value Problems for the Nonlocal Poisson Equation with Boundary Operators of Fractional Order

In this paper, in the class of smooth functions, integration and differentiation operators connected with fractional conformable derivatives are introduced. The mutual reversibility of these operators is proved, and the properties of these operators in the class of smooth functions are studied. Using transformations generalizing involutive transformations, a nonlocal analogue of the Laplace operator is introduced. For the corresponding nonlocal analogue of the Poisson equation, the solvability of some boundary value problems with fractional conformable derivatives is studied. For the problems under consideration, theorems on the existence and uniqueness of solutions are proved. Necessary and sufficient conditions for solvability of the studied problems are obtained, and integral representations of solutions are given.

Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian

We state and prove that a certain class of smooth functions said to be BH-separable is a meagre subset for the Fr\'echet topology. Because these functions are the only admissible Hamiltonians for Arnold-Liouville systems admitting a bi-Hamiltonian structure, we get that, generically, Arnold-Liouville systems cannot be bi-Hamiltonian. At the end of the paper, we determine, both as a concrete representation of our general result and as an illustrative list, which polynomial Hamiltonians $H$ of the form $H(x,y)=xy+ax^3+bx^2y+cxy^2+dy^3$ are BH-separable.

A Nonmonotone Smoothing Newton Algorithm for Weighted Complementarity Problem

The weighted complementarity problem (denoted by WCP) significantly extends the general complementarity problem and can be used for modeling a larger class of problems from science and engineering. In this paper, by introducing a one-parametric class of smoothing functions which includes the weight vector, we propose a smoothing Newton algorithm with nonmonotone line search to solve WCP. We show that any accumulation point of the iterates generated by this algorithm, if exists, is a solution of the considered WCP. Moreover, when the solution set of WCP is nonempty, under assumptions weaker than the Jacobian nonsingularity assumption, we prove that the iteration sequence generated by our algorithm is bounded and converges to one solution of WCP with local superlinear or quadratic convergence rate. Promising numerical results are also reported.