Approximation Theorem Research Articles

Approximating Long Cycle Above Dirac’s Guarantee

Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit “natural” guarantees bringing to algorithmic questions whether a better solution (above the guarantee) could be obtained efficiently. For example, for every boolean CNF formula on m clauses, there is an assignment that satisfies at least m/2 clauses. How difficult is it to decide whether there is an assignment satisfying more than m/2+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m/2 +k$$\\end{document} clauses? Or, if an n-vertex graph has a perfect matching, then its vertex cover is at least n/2. Is there a vertex cover of size at least n/2+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n/2 +k$$\\end{document} for some k≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 1$$\\end{document} and how difficult is it to find such a vertex cover? The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on well-studied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac’s theorem from 1952. Dirac’s theorem provides the following guarantee on the length of the longest cycle: for every 2-connected n-vertex graph G with minimum degree δ(G)≤n/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta (G)\\le n/2$$\\end{document}, the length of a longest cycle L is at least 2δ(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\delta (G)$$\\end{document}. Thus the “essential” part in finding the longest cycle is in approximating the “offset” k=L-2δ(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k = L - 2 \\delta (G)$$\\end{document}. The main result of this paper is the above-guarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably well-behaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length 2δ(G)+Ω(f(k))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\delta (G)+\\Omega (f(k))$$\\end{document}.

Primitives of continuous functions via continuous piecewise linear functions

In classical calculus textbooks, the existence of primitive functions of continuous functions is proved by using Riemann integrals. Recently, Patrik Lundström gave a proof via polynomials, based on the Weierstrass approximation theorem. In this note, it is shown that the proof will be easy by using continuous piecewise linear functions.

On sofic approximations of non amenable groups

In this paper we exhibit for every non amenable group that is initially sub-amenable (sometimes also referred to as LEA), two sofic approximations that are not conjugate by any automorphism of the universal sofic group. This addresses a question of Pǎunescu and generalizes the Elek–Szabo uniqueness theorem for sofic approximations.

Data-Driven Initial Gap Identification of Piecewise-Linear Systems Using Sparse Regression and Universal Approximation Theorem

Abstract This paper proposes a method for identifying an initial gap in piecewise-linear (PWL) systems from data. Piecewise-linear systems appear in many engineered systems such as degraded mechanical systems and infrastructures and are known to show strong nonlinearities. To analyze the behavior of such piecewise-linear systems, it is necessary to identify the initial gap, at which the system behavior switches. The proposed method identifies the initial gap by discovering the governing equations using sparse regression and calculating the gap based on the universal approximation theorem. A key step to achieve this is to approximate a piecewise-linear function by a finite sum of piecewise-linear functions in sparse regression. The equivalent gap is then calculated from the coefficients of the multiple piecewise-linear functions and their respective switching points in the obtained equation. The proposed method is first applied to a numerical model to confirm its applicability to piecewise-linear systems. Experimental validation of the proposed method has then been conducted with a simple mass-spring-hopping system, where the method successfully identifies the initial gap in the system with high accuracy.

Parametric Bernstein operators based on contagion distribution

This paper proposes the extended form of the ‐Bernstein operators with the Pólya–Eggenberger distribution, also known as contagion distribution, using the integral operators for . We will refer to these sequence of operators as the “Parametric‐Bernstein operators based on contagion distribution” or simply the modified form of ‐Bernstein operators. These operators are defined with the help of two parameters, namely, and . We observe the significance of both these parameters. Subsequently, we calculate their moments and on the basis of Korovkin's approximation theorem we assert that our defined operators converge for a real valued continuous function . We give some basic properties, study the Voronovskaya type asymptotic result and discuss the ‐statistical approximation of our operators. We also use the modulus of continuity to study the rate of convergence. Alongside our primary work, we have also defined the King‐type modification that preserves the operators at and . Further, using modulus of continuity, we compare the properties and behavior of both the operators.

On application of Gaussian kernels and Laplace functions combined with Kolmogorov's theorem for approximation of functions of several variables

A special class of approximations of measurable functions of several variables on the unit coordinate cube is investigated. The class is constructed on the base of Kolmogorov's theorem (in version by Sprecher–Golubkov) stating that a continuous function $f$ of several variables can be represented as a finite superposition of continuous single-variable functions — so called outer functions (which depend of $f$) and inner one $\Psi$ (which is independent of $f$ and is monotone). In the case of continuous functions $f$ the class under study is obtained with outer functions approximated by linear combinations of quadratic exponentials (also known as Gaussian functions or Gaussian kernels) and with the inner function $\Psi$ approximated by Laplace functions. As is known, a measurable function $f$ can be approximated by a continuous one (up to a set of small measure) with the help of classical Luzin's theorem. The effectiveness of such approach is based on assertions that, firstly, the Mexican hat mother wavelet on any fixed bounded interval can be approximated as accurately as desired by a linear combination of two Gaussian functions, and, secondly, that a continuous monotone function on such an interval can be approximated as accurately as desired by a linear combination of translations and dilations of the Laplace integral (in other words, Laplace functions). It is proved that the class of approximations under study is dense everywhere in the class of continuous multivariable functions on the coordinate cube. For the case of continuous and piecewise continuous functions of two variables, numerical results are presented that confirm the effectiveness of approximations of the studied class.

A Geometric Approach to Polynomial and Rational Approximation

Abstract We strengthen the classical approximation theorems of Weierstrass, Runge, and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$ on a compact set $K$, the critical points of our approximants may be taken to lie in any given domain containing $K$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$.

Learning in Deep Radial Basis Function Networks.

Learning in neural networks with locally-tuned neuron models such as radial Basis Function (RBF) networks is often seen as instable, in particular when multi-layered architectures are used. Furthermore, universal approximation theorems for single-layered RBF networks are very well established; therefore, deeper architectures are theoretically not required. Consequently, RBFs are mostly used in a single-layered manner. However, deep neural networks have proven their effectiveness on many different tasks. In this paper, we show that deeper RBF architectures with multiple radial basis function layers can be designed together with efficient learning schemes. We introduce an initialization scheme for deep RBF networks based on k-means clustering and covariance estimation. We further show how to make use of convolutions to speed up the calculation of the Mahalanobis distance in a partially connected way, which is similar to the convolutional neural networks (CNNs). Finally, we evaluate our approach on image classification as well as speech emotion recognition tasks. Our results show that deep RBF networks perform very well, with comparable results to other deep neural network types, such as CNNs.

Approximation by Riemann–Liouville type fractional α$$ \alpha $$‐Bernstein–Kantorovich operators

In this research paper, we construct a new sequence of Riemann–Liouville type fractional ‐Bernstein–Kantorovich operators. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first order modulus of continuity of these operators. Further, we study Voronovskaja type theorem, quantitative Voronovskaya type theorem, Chebyshev–Grüss inequality and Grüss–Voronovskaya type theorem.

Markov and Division Inequalities on Algebraic Sets

A compact set E⊂CN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E\\subset \\mathbb {C}^N$$\\end{document} satisfies the Markov inequality if the supremum norm on E of the gradient of a polynomial p can be estimated from above by the norm of p multiplied by a constant polynomially depending on the degree of p. This inequality is strictly related to the Bernstein approximation theorem, Schur-type estimates and the extension property of smooth functions. Additionally, the Markov inequality can be applied to the construction of polynomial grids (norming sets or admissible meshes) useful in numerical analysis. We expect such an inequality with similar consequences not only on polynomially determining compacts but also on some nowhere dense sets. The primary goal of the paper is to extend the above definition of Markov inequality to the case of compact subsets of algebraic varieties in CN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^N$$\\end{document}. Moreover, we characterize compact sets satisfying such a Markov inequality on algebraic hypersurfaces as well as on certain varieties defined by several algebraic equations. We also prove a division inequality (a Schur-type inequality) on these sets. This opens up the possibility of establishing polynomial grids on algebraic sets. We also provide examples that complete and ilustrate the results.

Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure

We prove that every open Riemann surface [Formula: see text] is the complex structure of a complete surface of constant mean curvature [Formula: see text] ([Formula: see text]) in the three-dimensional hyperbolic space [Formula: see text]. We go further and establish a jet interpolation theorem for complete conformal [Formula: see text] immersions [Formula: see text]. As a consequence, we show the existence of complete densely immersed [Formula: see text] surfaces in [Formula: see text] with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in [Formula: see text] which is also established in this paper.

Approximation of Periodic Functions by Wavelet Fourier Series

This paper aims to examine the expansion of periodic functions using wavelet bases. M. Skopina [8] obtained a Wavelet analog of the classical Jackson’s theorem for trigonometric approximation. Our result generalizes the result of M. Skopina [8] and V. Karanjgaokar et al. [15].

Approximate Noether theorem and its inverse for nonlinear dynamical systems with approximate nonstandard Lagrangian

The approximate Noether theorem and its inverse theorem for the nonlinear dynamical systems with approximate exponential Lagrangian and approximate power-law Lagrangian are investigated. For each case, the approximate differential equations of motion for the nonlinear dynamical systems with approximate nonstandard Lagrangian are established, the generalized Noether identities are given. The relationship between the approximate Noether symmetries and approximate conserved quantities for the system with approximate nonstandard Lagrangian are established, and the approximate Noether theorems and their inverse theorems are obtained. Two examples are given to illustrate the application of the results.

Lévy-Khintchin Theorem for best simultaneous Diophantine approximations

Lévy-Khintchin Theorem for best simultaneous Diophantine approximations

Convolutional Neural Networks Approximation in Quasi-Orlicz Spaces on Sphers

It is necessary to study the theoretical bases of an approximation deep convolutional neural networks, because of its interesting developments in vital domains. The approximation abilities of deep-convolution neural networks produced by downsampling operators in quasi- Orlicz spaces have been studied, since this space is wider and more important than other spaces. In this paper, we define quasi-Orlicz norm on spherical spaces. In addition, modulus of smoothness is also studied in terms of quasi-Orlicz norm. Finally, Function approximation theorems are studied by using convolution neural networks with

Learning nonparametric ordinary differential equations from noisy data

Learning nonparametric systems of Ordinary Differential Equations (ODEs) x˙=f(t,x) from noisy data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for f for which the solution of the ODE exists and is unique. Learning f consists of solving a constrained optimization problem in an RKHS. We propose a penalty method that iteratively uses the Representer theorem and Euler approximations to provide a numerical solution. We prove a generalization bound for the L2 distance between x and its estimator. Experiments are provided for the FitzHugh–Nagumo oscillator, the Lorenz system, and for predicting the Amyloid level in the cortex of aging subjects. In all cases, we show competitive results compared with the state-of-the-art.

Bicomplex Runges approximation theorem

Bicomplex Runges approximation theorem

Geometric Quantization Results for Semi-positive Line Bundles on a Riemann Surface

In earlier work (Marinescu and Savale in Math Ann. https://doi.org/10.1007/s00208-023-02750-3, 2023) the authors proved the Bergman kernel expansion for semi-positive line bundles over a Riemann surface whose curvature vanishes to at most finite order at each point. Here we explore the related results and consequences of the expansion in the semi-positive case including: Tian’s approximation theorem for induced Fubini-Study metrics, leading-order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections, and the asymptotics of holomorphic torsion.

Tractable skew-normal approximations via matching

Many approximate Bayesian inference methods assume a particular parametric form for approximating the posterior distribution. A Gaussian distribution provides a convenient density for such approaches; examples include the Laplace, penalized quasi-likelihood, Gaussian variational, and expectation propagation methods. Unfortunately, these all ignore potential posterior skewness. The recent work of Durante et al. [Skewed Bernstein-von Mises theorem and skew-modal approximations; 2023. ArXiv preprint arXiv:2301.03038.] addresses this using skew-modal (SM) approximations, and is theoretically justified by a skewed Bernstein-von Mises theorem. However, the SM approximation can be impractical to work with in terms of tractability and storage costs, and uses only local posterior information. We introduce a variety of matching-based approximation schemes using the standard skew-normal distribution to resolve these issues. Experiments were conducted to compare the performance of this skew-normal matching method (both as a standalone approximation and as a post-hoc skewness adjustment) with the SM and existing Gaussian approximations. We show that for small and moderate dimensions, skew-normal matching can be much more accurate than these other approaches. For post-hoc skewness adjustments, this comes at very little cost in additional computational time.

Adaptive finite‐time control for a class of stochastic nonlinear systems with input saturation constraints: A new approach based on multidimensional Taylor network

AbstractThis article proposes a novel multidimensional Taylor network (MTN)‐based adaptive finite‐time control approach for stochastic nonlinear systems with input saturation constraints. First, by introducing the hyperbolic tangent function, a piecewise smooth function is proposed to approximate the input saturation constraints, which can solve the problem caused by input saturation. Then, with the aid of MTN approximation theorem, a novel adaptive finite‐time control approach is proposed, which can guarantee that all signals of the closed‐loop system are semi‐globally finite‐time stability in probability (SGFTSP), the system output follows the given reference signal, meanwhile, the tracking error converges to a small neighborhood of the origin in finite‐time. Compared with the existing results, the proposed approach has the characteristics of simple structure and low calculation, which can achieve the satisfactory control performance with a reduced computational burden. In the end, the effectiveness of the proposed approach is validated by Lyapunov stability theory and two simulation experiments.