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What role does our knowledge about the ideal society play in guiding policymaking in the real world? One intuitive answer is to approximate. Namely, we have a duty to approximate the ideal within the relevant constraints of feasibility. However, political philosophers seem to have what might be called 'approximatophobia'. Many philosophers, including idealists such as David Estlund, warn against approximation. Their criticism is chiefly motivated by 'the problem of second best', which points out that your second-best option may not be closest to your best option. This paper aims to dispel 'approximatophobia'. The difficulty posed by the problem of second best is often overstated. More positively, I present a novel defence of approximation, arguing that approximation of an ideal can be a reasonably reliable default strategy of action guidance in the real world. Difficulties that may afflict the project of approximation can be mitigated by sophistication of the project of approximation. After showing that critics of approximation overstep their mark in issuing the strong or moderate warning against approximation, I propose an account of sophisticated approximation. It seeks a series of reforms that make existing social institutions closer to the ideal based on careful selection of frame of analysis. Three virtuous correlations are identified as indicators for successful project of approximation, linking descriptive similarity to desirability, feasibility and knowledge of the ideal. I also explain a two-staged strategy of sophisticating the way you approximate the ideal, with an expected positive feedback effect. [ABSTRACT FROM AUTHOR]

Singular integral operators play an important role in approximation theory and harmonic analysis. In this paper, we consider a weighted Lebesgue space L p , w , define a modified Gauss–Weierstrass singular integral on it, and study direct and inverse approximation properties of the operator followed by a Korovkin-type approximation theorem for a function f ∈ L p , w . We use the modulus of continuity of the functions to measure the rate of convergence. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with absolute, relative and Tate Tor modules. In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander—Buchweitz approximation theory, and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules. In the second part of the paper we consider a depth equality, called the depth formula, which has been initially introduced by Auslander and developed further by Huneke and Wiegand. As an application of our main result, we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension. [ABSTRACT FROM AUTHOR]

The approximation properties of the Aldaz–Kounchev–Render (AKR) operators where investigated in several papers. We improve some existing quantitative results concerning these approximation properties. Moreover, we describe classes of functions for which these operators approximate better than the classical Bernstein operators and classes of functions for which Bernstein operators approximate better than AKR operators. The new results, in particular involving monotonic convergence and Voronovskaja type formulas, are then extended to the bivariate case on the square [ 0 , 1 ] 2 and compared with other existing results. Several numerical examples, illustrating the relevance and supporting the theoretical findings, are presented. [ABSTRACT FROM AUTHOR]

This paper presents some efficient classes of estimators of population mean on current occasion in the presence of random non-response under a two-phase successive sampling set-up. The suggested classes of estimators are proposed for simple random sampling under various situations of non-response. The properties of proposed estimators have been discussed up to first order of approximation. The efficiency of the presented estimators has been contrasted with the estimators for the complete response scenarios. Two real and two artificially generated data sets are used. The efficacy of the proposed classes of estimators over the existing estimators is checked theoretically and empirically. The numerical comparison supports the proposed estimators. [ABSTRACT FROM AUTHOR]

The present paper deals with the problem of estimation of finite population mean of study variable using two auxiliary variables in two-phase sampling scheme using predictive approach in case of missing values of the study variable and unknown population mean of first auxiliary variable. Four classes of such estimators have been proposed using this predictive approach. The expressions of bias and mean square errors are derived up to first order of approximation. The optimal values of the constants involved in the proposed classes of estimators have been obtained and thus minimum mean square errors of the proposed classes are obtained in this study. The empirical and graphical comparisons with regression type estimators (under single phase and double phase sampling scheme) and also among themselves have been made for evaluating the performance of the proposed classes for different choices of non-responding units. Five real data sets and three simulated data sets following normal distribution have been used to evaluate the performance of the proposed classes. Numerical findings confirm the theoretical results obtained regarding superiority of proposed classes of estimators over the conventional regression type estimators in terms of percent relative efficiencies. [ABSTRACT FROM AUTHOR]

Extreme learning machine (ELM) is a training algorithm for single hidden layer feedforward neural network (SLFN), which converges much faster than traditional methods and yields promising performance. In this paper, we hope to present a comprehensive review on ELM. Firstly, we will focus on the theoretical analysis including universal approximation theory and generalization. Then, the various improvements are listed, which help ELM works better in terms of stability, efficiency, and accuracy. Because of its outstanding performance, ELM has been successfully applied in many real-time learning tasks for classification, clustering, and regression. Besides, we report the applications of ELM in medical imaging: MRI, CT, and mammogram. The controversies of ELM were also discussed in this paper. We aim to report these advances and find some future perspectives. [ABSTRACT FROM AUTHOR]

Let H be a separable Hilbert space and let { x n } be a sequence in H that does not contain any zero elements. We say that { x n } is a Bessel-normalizable or frame-normalizable sequence if the normalized sequence { x n ‖ x n ‖ } is a Bessel sequence or a frame for H, respectively. In this paper, several necessary and sufficient conditions for sequences to be frame-normalizable and not frame-normalizable are proved. Perturbation theorems for frame-normalizable sequences are also proved. As applications, we show that the Balazs–Stoeva conjecture holds for Bessel-normalizable sequences. Finally, we apply our results to partially answer the open question raised by Aldroubi et al. as to whether the iterative system { A n x ‖ A n x ‖ } n ≥ 0 , x ∈ S associated with a normal operator A : H → H and a countable subset S of H, is a frame for H. In particular, if S is finite, then we are able to show that { A n x ‖ A n x ‖ } n ≥ 0 , x ∈ S is not a frame for H whenever { A n x } n ≥ 0 , x ∈ S is a frame for H. [ABSTRACT FROM AUTHOR]

In this paper, we describe an algorithm for fitting an analytic and bandlimited closed or open curve to interpolate an arbitrary collection of points in R 2 . The main idea is to smooth the parametrization of the curve by iteratively filtering the Fourier or Chebyshev coefficients of both the derivative of the arc-length function and the tangential angle of the curve and applying smooth perturbations, after each filtering step, until the curve is represented by a reasonably small number of coefficients. The algorithm produces a curve passing through the set of points to an accuracy of machine precision, after a limited number of iterations. It costs O(N log N) operations at each iteration, provided that the number of discretization nodes is N. The resulting curves are smooth, affine invariant, and visually appealing and do not exhibit any ringing artifacts. The bandwidths of the constructed curves are much smaller than those of curves constructed by previous methods. We demonstrate the performance of our algorithm with several numerical experiments. [ABSTRACT FROM AUTHOR]

We present a scheme for finding all roots of an analytic function in a square domain in the complex plane. The scheme can be viewed as a generalization of the classical approach to finding roots of a function on the real line, by first approximating it by a polynomial in the Chebyshev basis, followed by diagonalizing the so-called “colleague matrices.” Our extension of the classical approach is based on several observations that enable the construction of polynomial bases in compact domains that satisfy three-term recurrences and are reasonably well-conditioned. This class of polynomial bases gives rise to “generalized colleague matrices,” whose eigenvalues are roots of functions expressed in these bases. In this paper, we also introduce a special-purpose QR algorithm for finding the eigenvalues of generalized colleague matrices, which is a straightforward extension of the recently introduced structured stable QR algorithm for the classical cases (see Serkh and Rokhlin 2021). The performance of the schemes is illustrated with several numerical examples. [ABSTRACT FROM AUTHOR]

This paper refutes the erroneous statements that arose in theoretical cosmology almost 90 years ago. The method accepted in the literature for deriving the local properties of the Friedmann cosmological model, using only Newton theory, without resorting to Einstein theory, is considered. We showed that the usual method of such inference is not sufficient to obtain the correct result, and leads to errors. Requirements are formulated that are sufficient for the Newtonian model to really be an approximation to the relativistic theory. [ABSTRACT FROM AUTHOR]

Facility location is a prominent optimization problem that has inspired a large quantity of both theoretical and practical studies in combinatorial optimization. Although the problem has been investigated under various settings reflecting typical structures within the optimization problems of practical interest, little is known on how the problem behaves in conjunction with parity constraints. This shortfall of understanding was rather discouraging when we consider the central role of parity in the field of combinatorics. In this paper, we present the first constant-factor approximation algorithm for the facility location problem with parity constraints. We are given as the input a metric on a set of facilities and clients, the opening cost of each facility, and the parity requirement– odd , even , or unconstrained –of every facility in this problem. The objective is to open a subset of facilities and assign every client to an open facility so as to minimize the sum of the total opening costs and the assignment distances, but subject to the condition that the number of clients assigned to each open facility must have the same parity as its requirement. Although the unconstrained facility location problem as a relaxation for this parity-constrained generalization has unbounded gap, we demonstrate that it yields a structured solution whose parity violation can be corrected at small cost. This correction is prescribed by a T-join on an auxiliary graph constructed by the algorithm. This auxiliary graph does not satisfy the triangle inequality, but we show that a carefully chosen set of shortcutting operations leads to a cheap and sparseT-join. Finally, we bound the correction cost by exhibiting a combinatorial multi-step construction of an upper bound. We also consider the parity-constrained k-center problem, the bottleneck optimization variant of parity-constrained facility location. We present the first constant-factor approximation algorithm also for this problem. [ABSTRACT FROM AUTHOR]

In this work, a Bayesian Regularization Network based Geometric Deviation Control (BRN-GDC) algorithm is developed to mitigate thermal distortion in 3D printing. Inspired by points registration in computer vision and function approximation theory, the Bayesian regularization network method is used to quantify thermal distortion in 3D printed products. Because of "shallow" regularization network architecture, the BRN-GDC method is training-free and does not require lots of data. Due to the lack of one-to-one correspondence between the design point data and the scan point data in 3D printing, conventional point set registration methods, e.g. Coherent Point Drift method, may fail in finding the global geometric deviation field, while the Bayesian regularization network approach works. In the two experiments presented in this paper, we showed that the BRN-GDC algorithm has the capability to control the thermal distortion in 3D printing that is parameter- and location-dependent. [ABSTRACT FROM AUTHOR]

This paper considers the relationship between electromagnetic waves and the geometric optics approximation in general relativity. The approximation is widely used in predictions of observations in astronomy and cosmology, for example in gravitational lensing and the cosmic microwave background. Its properties are thus of considerable significance. The paper's author was a Golden Oldies board member and a prominent author and editor for Springer. It complements other Golden Oldies by Ehlers showing the importance and influence of his attention to issues concerning the relation of exact physical theories and their approximations. [ABSTRACT FROM AUTHOR]

This paper presents a link between unbalanced non-linear urn model (a two-colored urn model) and stochastic approximation theory. Findings of our study reveal a successful establishment of limit laws for the urn composition, obtained under a drawing rule reinforced by an ℝ + -valued concave function and a non-balanced replacement matrix. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the (p, q)-Kirchhoff type problem - 1 + a ∫ R N | ∇ u | p d x Δ p u - 1 + b ∫ R N | ∇ u | q d x Δ q u + V ε (x) | u | p - 2 u + | u | q - 2 u = f (u) in R N , where ε > 0 is a small parameter, a , b > 0 , 1 < p < q < N < 2 q , Δ s u = div (| ∇ u | s - 2 ∇ u) , with s ∈ { p , q } , is the s-Laplacian, V : R N → R is a continuous potential, V ε (x) = V (ε x) and f : R → R is a continuous non-linearity without satisfying usual Ambrosetti–Rabinowitz condition. By using Nehari manifold techniques and perturbation methods, we establish the existence of positive ground state solution, and by those methods combined with Ljusternik–Schnirelmann theory, we investigate the multiplicity and concentration of solutions. In particular, our results extend and improve the works of He and Zou (J Differ Equ 252:1813–1834, 2012). [ABSTRACT FROM AUTHOR]

Approximate computing is gaining grip as a computing paradigm for computer vision, data analytics, and image/signal processing applications. In the era of real-time applications, approximate computing plays a significant role. In many computers including digital signal processors (DSP) and a microprocessor, adders are the main element for the implementation of signal processing applications and digital circuit design. The major problem for addition is the propagation delay in the carry chain. As the bit length of the input operand increases, the length of the carry chain increases. To address the carry propagation problem in digital systems, the most efficient adder architectures for VLSI implementation are classified as a parallel prefix adder (PPA) structure. In this paper, a novel methodology to implement and synthesize different adders (non-speculative and speculative) for any ASIC-based system is proposed. The proposed hybrid Han-Carlson and Kogge-stone speculative adders show improved performance (low power and delay) over the state-of-the-art approximate adders. If the approximation fails, then the proposed efficient error correction technique is activated. The proposed speculative H_C adder results in a 23.79% speed improvement over the proposed K_S adder, and 23.86% of energy is saved. The proposed architectures were synthesized for an operand bit length of 16 bits. Finally, the proposed adder is validated for an error-tolerant image processing application resulting in 41.2 dB PSNR. [ABSTRACT FROM AUTHOR]

It is well known that the a-priori, uniformly time-independent L∞ estimate of solutions is the key point to ensure the large time behavior and the relaxation time limit of entropy solutions for both the unipolar and the bipolar hydrodynamic model of semiconductors. With the technical help of the artificial damping coefficient functions ai(x) ≢ 1, in the previous papers, we introduced a technique to obtain the uniform L∞ estimates of the viscosity-flux approximation solutions, in which, the maximum principle is applied to the combination of the Riemann invariants with the integrals of the density and the concentration of a fixed background charge. In this paper, we remove the auxiliary condition on the functions ai(x), and obtain the uniformly, time-independent estimate of entropy solutions for the physical case ai(x) ≡ 1. [ABSTRACT FROM AUTHOR]

In the present paper we show that in Pólya's urn model, for an arbitrarily fixed initial distribution of the urn, the corresponding random variables satisfy a natural convex ordering with respect to the replacement parameter. As an application, we show that in the class of convex functions, the error of approximation for Bernstein–Stancu operators is a non-decreasing (strictly increasing under an additional hypothesis) function of the corresponding parameter. The proofs rely on two results of independent interest: an interlacing lemma of three sets and the monotonicity of the (partial) first moment of Pólya random variables with respect to the replacement parameter. [ABSTRACT FROM AUTHOR]

In this paper, we study the metric theory of dyadic approximation in the middle-third Cantor set. This theory complements earlier work of Levesley et al. (Math Ann 338(1):97–118, 2007), who investigated the problem of approximation in the Cantor set by triadic rationals. We find that the behaviour when we consider dyadic approximation in the Cantor set is substantially different to considering triadic approximation in the Cantor set. In some sense, this difference in behaviour is a manifestation of Furstenberg's times 2 times 3 phenomenon from dynamical systems, which asserts that the base 2 and base 3 expansions of a number are not both structured. [ABSTRACT FROM AUTHOR]

This paper studies the consensus issue of nonlinear singular multi-agent systems under undirected graphs based on fuzzy logic systems. The main objective is to develop a novel collaborative control algorithm to achieve consensus and impulse free for each agent described by nonlinear singular systems while only knowing local information, namely the state information of the agent itself and its neighbors. To overcome the analysis limitation of nonlinear singular multi-agent systems, the fuzzy logic system method is applied to deal with nonlinear items. Specifically, the consensus issue of singular multi-agent systems with fuzzy logic systems is equivalently simplified to an asymptotic stability analysis process for error systems. To achieve this objective, a novel distributed adaptive collaborative controller is designed, where the adaptive rate adjusts the coupling weight between each agent. In the research of the singular linear multi-agent system, a tolerance region is introduced, and stability of the closed-loop system is established according to Lyapunov theory. Then, fuzzy approximation theory is used to transform unknown nonlinear functions into fuzzy functions satisfying Lipschitz conditions, and the stability and impulse-free property are addressed by constructing LMI conditions. Finally, the effectiveness of the presented approach is demonstrated through two numerical examples. [ABSTRACT FROM AUTHOR]

This paper concerns late 1920 s attempts to construct unitary theories of gravity and electromagnetism. A first attempt using a non-standard connection—with torsion and zero-curvature—was carried out by Albert Einstein in a number of publications that appeared between 1928 and 1931. In 1929, Tullio Levi-Civita discussed Einstein's geometric structure and deduced a new system of differential equations in a Riemannian manifold endowed with what is nowadays known as Levi-Civita connection. He attained an important result: Maxwell's electromagnetic equations and the gravitational equations were obtained exactly, while Einstein had deduced them only as a first order approximation. A main feature of Levi-Civita's theory is the essential use of the Ricci's rotation coefficients, introduced by Gregorio Ricci Curbastro many years before. We trace the history of Ricci's coefficients that are still used today, and highlight their geometric and mechanical meaning. [ABSTRACT FROM AUTHOR]

In this paper, the authors introduce the index of subgeneral position for closed subschemes and obtain a second main theorems based on this notion. They also give the corresponding Schmidt's subspace type theorem via the analogue between Nevanlinna theory and Diophantine approximation. [ABSTRACT FROM AUTHOR]

In this paper, we present a detailed investigation of the Generalized Bessel-Laguerre-Gaussian (GBLG) beam propagating in a maritime turbulent atmosphere. Based on the born approximation and the Rytov theory with the use of the Von Karman spectrum, the scintillation index of the GBLG beam is evaluated and examined numerically. The obtained results indicate that the comportment of the GBLG beam is influenced by the turbulent strength, the inner scale of the maritime turbulent atmosphere, and the input beam parameters. The scintillation indexes of Laguerre-Bessel-Gaussian, Laguerre-Gaussian, Bessel-Gaussian and Gaussian beams through the considered medium are deduced as particular cases from our study. [ABSTRACT FROM AUTHOR]

In a recent contribution, Kumar et al. (J Mech Phys Solids 142:104027, 2020) have introduced a comprehensive macroscopic phase-field theory for the nucleation and propagation of fracture in linear elastic brittle materials under arbitrary quasistatic loading conditions. The theory can be viewed as a natural generalization of the phase-field approximation of the variational theory of brittle fracture of Francfort and Marigo (J Mech Phys Solids 46:1319–1342, 1998) to account for the material strength at large. This is accomplished by the addition of an external driving force—which physically represents the macroscopic manifestation of the presence of inherent microscopic defects in the material—in the equation governing the evolution of the phase field. The main purpose of this paper is to continue providing validation results for the theory by confronting its predictions with direct measurements from three representative types of experimentally common yet technically challenging problems: (i) the indentation of glass plates with flat-ended cylindrical indenters and the three-point bending of (ii) U-notched and (iii) V-notched PMMA beams. [ABSTRACT FROM AUTHOR]

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of constructioning such approximations using polynomials in the Bernstein basis. We consider a family of inequality-constrained quadratic programs. In the univariate case, a quadratic cone constraint allows us to search over all nonnegative polynomials of a given degree. In both the univariate and multivariate cases, we consider approximate problems with linear inequality constraints. Additionally, our method can be modified slightly to include equality constraints such as mass preservation. [ABSTRACT FROM AUTHOR]

In this paper, weighted differential evolution (WDE)-based evolutionary heuristic computing is presented for parameter estimation of nonlinear dynamics of the electrically stimulated muscle system based on a Hammerstein model structure as an element of rehabilitation program for the stroke patient. The approximation theory is defined on the basis of mean square error of actual and estimated responses for the system's fitness function. Optimization technique based on WDE is employed to find the optimal parameters of ESM model with sigmoidal, polynomial and spline as input nonlinear functions. Statistical analysis verified the worth of WDE as computational heuristic technique for identification of electrically stimulated muscle system. [ABSTRACT FROM AUTHOR]

This paper proposes a multidimensional Hilbert transform approach for pricing discretely monitored multi-asset barrier options and computing joint survival probability in multivariate exponential Lévy asset price models. We generalize the univariate Hilbert transform method of Feng and Linetsky (Math Financ 18(3), 337–384, 2008) for single-asset barrier options and the well-known Sinc approximation theory of Stenger (Numerical methods based on sinc and analytic functions. Springer, New York, 1993) for computing the one-dimensional Hilbert transform to any dimension. We prove that, for Lévy processes with joint characteristic functions having an exponentially decaying tail, the error of our method decays exponentially in some power of the number of terms used in the expansion for each dimension. Numerical experiments demonstrate the efficiency of our method in the two-dimensional and three-dimensional problems for some popular multivariate Lévy models. [ABSTRACT FROM AUTHOR]

Hadamard type fractional calculus involves logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenge in numerical treatment. In this paper we present a spectral collocation method with mapped Jacobi log orthogonal functions (MJLOFs) as basis functions and obtain an efficient algorithm to solve Hadamard type fractional differential equations. We develop basic approximation theory for the MJLOFs and derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e, the fractional Helmholtz equation, and the fractional Burgers equation. [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]

We study extremal problems related to the best joint polynomial approximation of functions analytic in the unit disk and belonging to the Bergman space . The problem of joint approximation of periodic functions and their derivatives by trigonometric polynomials was considered by Garkavy [1] in 1960. Then, in the same year, Timan [2] considered this problem for classes of entire functions defined on the entire axis. The problem of joint approximation of functions and their derivatives is considered in more detail in Malozemov's monograph [3], where some classical theorems of the theory of approximation of functions are presented and generalized. In the present paper, a number of exact theorems are obtained and sharp upper bounds for the best joint approximations of a function and its successive derivatives by polynomials and their respective derivatives on some classes of complex functions belonging to the Bergman space are calculated. [ABSTRACT FROM AUTHOR]

This paper introduces a novel technique to approximate a given continuous function f defined on a real compact interval by a new class of zipper α -fractal functions which contain a scaling vector and a binary vector or signature. For specific choices of scaling and signature vectors, the corresponding zipper fractal functions simultaneously interpolate and approximate f. When signature is zero, the proposed zipper fractal functions coincide with existing α -fractal functions. Hence, the zipper approximation proposed in this manuscript generalizes the existing fractal and classical approximations. Zipper fractal analogue of some elementary results in the classical approximation theory are obtained. Using convex optimization technique, we investigate the existence of optimal zipper fractal function for a given continuous function. The parameter identification problems for zipper α -fractal approximants are investigated. We derive sufficient conditions on the parameters of zipper α -fractal functions so that these functions preserve the positivity, monotonicity and convexity of the original function f. Also, we have studied the constructions of k-times continuously differentiable zipper α -fractal functions and one sided zipper fractal approximants for f. Numerical illustrations are provided to support the proposed theoretical results on zipper α -fractal functions. [ABSTRACT FROM AUTHOR]

In recent years, the information technology industry around the world has grown strong. At the same time, we also face a new challenge with the explosion in the amount of information. Although there is a huge amount of data, the information that we actually have is lacking, and the implications behind the data have not been fully exploited. Scientists have researched new ways to fully exploit the information contained in the database. Since the late 1980s, the concept of knowledge discovery in databases was first mentioned. This is the process of detecting latent, unknown, and useful knowledge in large databases, while overcoming the limitations of traditional database models with only data query tools that cannot find new information, and is information hidden in the database. Knowledge mining in a database is the process of discovering new, useful, and information hidden in a database. Since the early 1980s, Z. Pawlak has proposed the Rough Set theory with a very solid mathematical basis. This theory is practiced by many research groups working in the field of general information technology and exploring knowledge in the database and applied in research. Rough Set theory is more widely applied in the field of knowledge discovery, while being useful in solving problems of data classification and association rules through discovery, and especially useful in problems dealing with ambiguous and uncertain data. Specifically, in theory, the raw set of data is displayed using information systems or tables. With large data tables having imperfect data, redundant data, or continuous data or represented in the form of symbols, the Rough Set theory allows knowledge exploration in databases like this to detect hidden knowledge from these "raw" blocks of data. The found knowledge is expressed in the form of rules and patterns. After finding the most general rules for data representation, one can calculate the strength and dependence between attributes in the information system. In this paper, the authors research the recommendation system, rough set theory, theory of approximation, and fuzzy rough set theory, thereby building a partial model. Software enables users to exploit association rules of their database, thereby facilitating appropriate purchase or import decisions. The system can support user design options of database features, load data from the SQL Server by Apache Spark, and export the statistics to website to be reported. [ABSTRACT FROM AUTHOR]

The goal of this paper is threefold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work in proving lower bounds for recently developed new type of discrepancy—the smooth discrepancy. Third, we consider numerical integration in classes, for which we do not impose any smoothness assumptions. We illustrate how nonlinear approximation, in particular greedy approximation, allows us to guarantee some rate of decay of errors of numerical integration even in such a general setting with no smoothness assumptions. [ABSTRACT FROM AUTHOR]

Existing depth separation results for constant-depth networks essentially show that certain radial functions in R d , which can be easily approximated with depth 3 networks, cannot be approximated by depth 2 networks, even up to constant accuracy, unless their size is exponential in d. However, the functions used to demonstrate this are rapidly oscillating, with a Lipschitz parameter scaling polynomially with the dimension d (or equivalently, by scaling the function, the hardness result applies to O (1) -Lipschitz functions only when the target accuracy ϵ is at most poly (1 / d) ). In this paper, we study whether such depth separations might still hold in the natural setting of O (1) -Lipschitz radial functions, when ϵ does not scale with d. Perhaps surprisingly, we show that the answer is negative: In contrast with the intuition suggested by previous work, it is possible to approximate O (1) -Lipschitz radial functions with depth 2, size poly (d) networks, for every constant ϵ . We complement it by showing that approximating such functions is also possible with depth 2, size poly (1 / ϵ) networks, for every constant d. Finally, we show that it is not possible to have polynomial dependence in both d , 1 / ϵ simultaneously. Overall, our results indicate that in order to show depth separations for expressing O (1) -Lipschitz functions with constant accuracy—if at all possible—one would need fundamentally different techniques than existing ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, I propose and defend a pair of necessary conditions on evidence-based knowledge which bear resemblance to the troubled sensitivity principles defended in the philosophical literature (notably by Fred Dretske). We can think of the traditional principles as simple but inaccurate approximations of the new proposals. Insofar as the old principles are intuitive and used in scientific and philosophical contexts, but are plausibly false, there's a real need to develop precise and correct formulations. These new renditions turned out to be more cautious, so they won't be able to do everything the old principled promised they could. For example, they respect closure for knowledge. But these sober formulations, or something like them, might be the best that we can do with respect to sensitivity. And there's value in understanding the limits to these types of principles. [ABSTRACT FROM AUTHOR]

Conventional reliability analysis requires the information from an existing structure, such as a mechanical model and random distributed inputs. In many engineering problems, a state monitoring system commonly provides experimental or monitoring data, which can be used to update the initial estimation for structural reliability to reduce prediction uncertainty. A critical issue in this process is the manner in which the existing information and new data can be reasonably integrated into a reliability estimation. In this paper, Bayesian updating approach is applied to incorporate the additional data. Firstly, a theoretical model is established to predict the prior distribution of the limit state function (LSF) with the first-order reliability method. Then, the Bayesian inference theory is applied to update the probability distribution parameters of LSF using the acquired experimental or monitoring data. The analytical form of the LSF's posterior distribution is derived under the assumption that the experimental test error follows a normal distribution. To improve accuracy, a second-order reliability method is proposed based on the theory of saddlepoint approximation. Markov chain Monte Carlo simulation is used to derive a general method for updating the LSF's distribution using the experimental or monitoring data. Finally, three numerical examples are provided to illustrate the proposed framework's validity. [ABSTRACT FROM AUTHOR]

Theories of truth approximation in terms of truthlikeness (or verisimilitude) almost always deal with (non-probabilistically) approaching deterministic truths, either actual or nomic. This paper deals first with approaching a probabilistic nomic truth, viz. a true probability distribution. It assumes a multinomial probabilistic context, hence with a lawlike true, but usually unknown, probability distribution. We will first show that this true multinomial distribution can be approached by Carnapian inductive probabilities. Next we will deal with the corresponding deterministic nomic truth, that is, the set of conceptually possible outcomes with a positive true probability. We will introduce Hintikkian inductive probabilities, based on a prior distribution over the relevant deterministic nomic theories and on conditional Carnapian inductive probabilities, and first show that they enable again probabilistic approximation of the true distribution. Finally, we will show, in terms of a kind of success theorem, based on Niiniluoto's estimated distance from the truth, in what sense Hintikkian inductive probabilities enable the probabilistic approximation of the relevant deterministic nomic truth. In sum, the (realist) truth approximation perspective on Carnapian and Hintikkian inductive probabilities leads to the unification of the inductive probability field and the field of truth approximation. [ABSTRACT FROM AUTHOR]

The theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fractional calculus has been found to be the best for modeling physical and engineering processes. More precisely, the proportional fractional operators are one of the recent important notions of fractional calculus. Our aim in this research paper is developing some novel ways of fractional integral Hermite–Hadamard inequalities in the frame of a proportional fractional integral with respect to another strictly increasing continuous function. The considered fractional integral is applied to establish some new fractional integral Hermite–Hadamard-type inequalities. Moreover, we present some special cases throughout discussing this work. [ABSTRACT FROM AUTHOR]

The "quantum complexity" of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. While the notion of quantum complexity was first introduced as a quantum generalization of the classical computational complexity, it has since been argued to hold a fundamental significance in its own right, as a physical quantity analogous to the thermodynamic entropy. In this paper, we present a unified perspective on various notions of quantum complexity, viewed as functions on the space of unitary operators. One striking feature of these functions is that they can exhibit non-smooth and even fractal behaviour. We use ideas from Diophantine approximation theory and sub-Riemannian geometry to rigorously quantify this lack of smoothness. Implications for the physical meaning of quantum complexity are discussed. [ABSTRACT FROM AUTHOR]

We study the binary q-voter model with generalized anticonformity on random Erdős–Rényi graphs. In such a model, two types of social responses, conformity and anticonformity, occur with complementary probabilities and the size of the source of influence q c in case of conformity is independent from the size of the source of influence q a in case of anticonformity. For q c = q a = q the model reduces to the original q-voter model with anticonformity. Previously, such a generalized model was studied only on the complete graph, which corresponds to the mean-field approach. It was shown that it can display discontinuous phase transitions for q c ≥ q a + Δ q , where Δ q = 4 for q a ≤ 3 and Δ q = 3 for q a > 3 . In this paper, we pose the question if discontinuous phase transitions survive on random graphs with an average node degree ⟨ k ⟩ ≤ 150 observed empirically in social networks. Using the pair approximation, as well as Monte Carlo simulations, we show that discontinuous phase transitions indeed can survive, even for relatively small values of ⟨ k ⟩ . Moreover, we show that for q a < q c - 1 pair approximation results overlap the Monte Carlo ones. On the other hand, for q a ≥ q c - 1 pair approximation gives qualitatively wrong results indicating discontinuous phase transitions neither observed in the simulations nor within the mean-field approach. Finally, we report an intriguing result showing that the difference between the spinodals obtained within the pair approximation and the mean-field approach follows a power law with respect to ⟨ k ⟩ , as long as the pair approximation indicates correctly the type of the phase transition. [ABSTRACT FROM AUTHOR]

The iterates of an operator and their limit are studied in ergodic theory, approximation theory and other related fields. In this paper we continue the study of iterates of certain Markov operators acting on C[0,1]. We provide new sufficient conditions under which a Markov operator L is uniquely ergodic, i.e., it admits a unique invariant probability measure ν . The determination of ν is reduced to solving an algebraic system of linear equations. Then ν is used in order to express the limit of the iterates of L. Another useful tool in this study is the eigenstructure of L. The general results are applied to several families of classical Markov operators. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new type of a neural network which is inspired by Gabor systems from harmonic analysis. In this regard, we construct a class of sparsely connected neural networks utilizing the concept of time–frequency shifts, and we show that their approximation error rates can be tied to the number of modulations in the corresponding Gabor frame and to the smoothness of the input function. Furthermore, we show that such networks are easily implementable and we illustrate their performance with some numerical examples. [ABSTRACT FROM AUTHOR]

Today's field programmable gate arrays (FPGAs) offer a significant computational power and are commonly used in modern commercial digital designs. However, they generally suffer from a large power consumption, which makes them unfit for battery-operated handheld devices. This paper addresses this problem by bringing the notion of approximate computing into the realm of reconfigurable devices such as LUT-based FPGAs. The proposed approximation is done by altering LUT contents in an exact design. The impact of this kind of approximation on output accuracy as well as design power consumption will be discussed. Once the theoretical foundation is established, we propose a method, called FERMAT (FPGA Energy Reduction Method by Approximation Theory), which takes an exact FPGA design and converts it into an approximated equivalent with a considerably reduced power consumption given a maximum error constraint. The effectiveness of FERMAT is shown by measuring the actual power consumption of an FPGA device performing an image processing application. Experimental results show about 8.5% power saving with an imperceptible loss in image quality. [ABSTRACT FROM AUTHOR]

Artificial Neural Networks (ANNs) are a tool in approximation theory widely used to solve interpolation problems. In fact, ANNs can be assimilated to functions since they take an input and return an output. The structure of the specifically adopted network determines the underlying approximation space, while the form of the function is selected by fixing the parameters of the network. In the present paper, we consider one-hidden layer ANNs with a feedforward architecture, also referred to as shallow or two-layer networks, so that the structure is determined by the number and types of neurons. The determination of the parameters that define the function, called training, is done via the resolution of the approximation problem, so by imposing the interpolation through a set of specific nodes. We present the case where the parameters are trained using a procedure that is referred to as Extreme Learning Machine (ELM) that leads to a linear interpolation problem. In such hypotheses, the existence of an ANN interpolating function is guaranteed. Given that the ANN is interpolating, the error incurred occurs outside the sampling interpolation nodes provided by the user. In this study, various choices of nodes are analyzed: equispaced, Chebychev, and randomly selected ones. Then, the focus is on regular target functions, for which it is known that interpolation can lead to spurious oscillations, a phenomenon that in the ANN literature is referred to as overfitting. We obtain good accuracy of the ANN interpolating function in all tested cases using these different types of interpolating nodes and different types of neurons. The following study is conducted starting from the well-known bell-shaped Runge example, which makes it clear that the construction of a global interpolating polynomial is accurate only if trained on suitably chosen nodes, ad example the Chebychev ones. In order to evaluate the behavior when the number of interpolation nodes increases, we increase the number of neurons in our network and compare it with the interpolating polynomial. We test using Runge’s function and other well-known examples with different regularities. As expected, the accuracy of the approximation with a global polynomial increases only if the Chebychev nodes are considered. Instead, the error for the ANN interpolating function always decays, and in most cases we observe that the convergence follows what is observed in the polynomial case on Chebychev nodes, despite the set of nodes used for training. Then we can conclude that the use of such an ANN defeats the Runge phenomenon. Our results show the power of ANNs to achieve excellent approximations when interpolating regular functions also starting from uniform and random nodes, particularly for Runge’s function. [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]

Aiming at the difficult problem of solving the conformation statistics of complex polymers, this study presents a novel and concise conformation statistics theoretical approach based on Monte Carlo and Neural Network method. This method offers a new research idea for investigating the conformation statistics of complex polymers, characterized by its simplicity and practicality. It can be applied to more complex topological structure, more higher degree of freedom polymer systems with higher dimensions, theory research on dynamic self-consistent field theory and polymer field theory, as well as the analysis of scattering experimental data. The conformation statistics of complex polymers determine the structure and response properties of the system. Using the new method proposed in this study, taking the semiflexible ring diblock copolymer as an example, Monte Carlo simulation is used to sample this ring conformation to construct the dataset of polymer. The structure factor describing conformation statistics are expressed as continuous functions of structure parameters by neural network supervised learning. This is the innovation of this work. As an application, the structure factors represented by neural networks were introduced into the random phase approximation theory to study the microphase separation of semiflexible ring diblock copolymers. The influence of the ring's topological properties on the phase transition behavior was pointed out. [ABSTRACT FROM AUTHOR]

Purpose: A novel piecewise symmetric two-dimensional tri-stable stochastic resonance system (NPSTDTSR) is one of two-dimensional systems. It is proposed with a greater signal amplification capability. Methods: First, the steady-state probability density function (SPD) and the mean first passage time (MFPT) of particles, as well as the output signal-to-noise ratio (SNR), are derived under the condition of adiabatic approximation theory, and the effects of system parameters on them are analyzed. Then NPSTDTSR system and the one-dimensional three-stable (ODTSR) system are numerically analyzed and simulated respectively, and the results show that the mean signal-to-noise ratio gain (MSNRG) of NPSTDTSR system is greater than that of ODTSR system. Finally, the parameters are optimized by a genetic algorithm and applied to the bearing fault detection. Results: In LDK UER204 ring fault detection, MSNRG in NPSTDTSR system is 0.8988 dB and 0.519 dB higher than ODTSR and two-dimensional tri-stable stochastic resonance (TDTSR) system. In the outer ring experiment on the experimental bench at CWRU, MSNRG in NPSTDTSR system is 0.1126 dB and 0.3301 dB higher than the other two systems, while in the inner ring experiment, MSNRG is 1.5139 dB and 5.9973 dB higher than the other two systems respectively. Conclusions: The experimental results show that the NPSTDTSR system has the most advantages, which provide good theoretical support and application feasibilities for the system in subsequent theoretical research and practical engineering applications. [ABSTRACT FROM AUTHOR]