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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]

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]

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]

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]

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]

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]

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 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]

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]

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]

Confidence interval for the difference of two proportions has been studied for decades. Many methods were developed to improve the approximation of the limiting distribution of test statistics, such as the profile likelihood method, the score method, and the Wilson method. For the Wilson interval developed by Beal (Biometrics 43:941, 1987), the approximation of the Z test statistic to the standard normal distribution may be further improved by utilizing the continuity correction, in the observation of anti-conservative intervals from the Wilson interval. We theoretically prove that the Wilson interval is nested in the continuity corrected Wilson interval under mild conditions. We compare the continuity corrected Wilson interval with the commonly used methods with regards to coverage probability, interval width, and mean squared error of coverage probability. The proposed interval has good performance in many configurations. An example from a Phase II cancer trial is used to illustrate the application of these methods. [ABSTRACT FROM AUTHOR]

We study polynomials given by the discrete Rodrigues formula, which generalizes a similar formula for Meixner polynomials. Such polynomials are associated with the theory of Diophantine approximations. The saddle point method is used to find the limit distribution of zeros of scaled polynomials. An answer is received in terms of a meromorphic function on a compact Riemann surface and is interpreted using the vector equilibrium problem of the logarithmic potential theory. [ABSTRACT FROM AUTHOR]

In a recent work (Koc et al., SIAM J. Numer. Anal. 59(4), 2163–2196, 2021), the authors showed that including difference quotients (DQs) is necessary in order to prove optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order models of the heat equation. In this work, we introduce a new approach to including DQs in the POD procedure. Instead of computing the POD modes using all of the snapshot data and DQs, we only use the first snapshot along with all of the DQs and special POD weights. We show that this approach retains all of the numerical analysis benefits of the standard POD DQ approach, while using a POD data set that has approximately half the number of snapshots as the standard POD DQ approach, i.e., the new approach requires less computational effort. We illustrate our theoretical results with numerical experiments. [ABSTRACT FROM AUTHOR]

For more than three decades, nearly free-electron elemental metals have been a topic of debate because the computed bandwidths are significantly wider in the local density approximation to density-functional theory (DFT) than indicated by angle-resolved photoemission (ARPES) experiments. Here, we systematically investigate this using first principles calculations for alkali and alkaline-earth metals using DFT and various beyond-DFT methods such as meta-GGA, G0W0, hybrid functionals (YS-PBE0, B3LYP), and LDA + eDMFT. We find that the static non-local exchange, as partly included in the hybrid functionals, significantly increase the bandwidths even compared to LDA, while the G0W0 bands are only slightly narrower than in LDA. The agreement with the ARPES is best when the local approximation to the self-energy is used in the LDA + eDMFT method. We infer that even moderately correlated systems with partially occupied s orbitals, which were assumed to approximate the uniform electron gas, are very well described in terms of short-range dynamical correlations that are only local to an atom. [ABSTRACT FROM AUTHOR]

We establish a number of theorems of geometric approximation theory in asymmetrically normed spaces. Sets with continuous selection of the near-best approximation operator are studied and properties of such sets are discussed in terms of -solar points and the distance function. A result on the coincidence of the classes of - and -suns in asymmetric spaces is given. An asymmetric analogue of the Kolmogorov criterion for an element of best approximation for suns, strict suns, and -suns is put forward. [ABSTRACT FROM AUTHOR]

An analytical solution is developed to study the free vibration of a thin functionally graded (FG) spherical shell under initial internal static pressure based on Love's first approximation theory. A coupled vibro-acoustic analytical model is presented for spherical shells filled with compressible nonviscous fluid. The non-homogenous material properties are assumed to be graded according to a power-law distribution of the constituents through the shell thickness. By introducing a stress function, the reformulated coupled equations of motion of FG spherical shells under the influence of initial stresses are obtained. The wave equation is used to model the internal acoustic domain. The boundary conditions of continuity of fluid and shell velocities, as well as the normal pressure acting on the internal surface of the shell from the fluid are imposed. The frequency equation of the coupled system is obtained utilizing modal expansion along with the orthogonality properties of the mode shapes. Exact solutions for the free vibration of pressurized empty and fluid-filled shells are obtained in terms of products of trigonometric and Legendre functions in a spherical coordinate system. Numerical results are validated with the results of simple cases available in the literature as well as finite element modeling. Effects of different parameters including material constants, geometry, initial pressure and vibro-acoustic coupling on natural frequencies are studied. The presented analytical solution is an attempt to describe the vibrational behavior of FG pressurized fluid-filled spherical shells. [ABSTRACT FROM AUTHOR]

A signature result in compressed sensing is that Gaussian random sampling achieves stable and robust recovery of sparse vectors under optimal conditions on the number of measurements. However, in the context of image reconstruction, it has been extensively documented that sampling strategies based on Fourier measurements outperform this purportedly optimal approach. Motivated by this seeming paradox, we investigate the problem of optimal sampling for compressed sensing. Rigorously combining the theories of wavelet approximation and infinite-dimensional compressed sensing, our analysis leads to new error bounds in terms of the total number of measurements m for the approximation of piecewise α -Hölder functions. Our theoretical findings suggest that Fourier sampling outperforms random Gaussian sampling when the Hölder exponent α is large enough. Moreover, we establish a provably optimal sampling strategy. This work is an important first step towards the resolution of the claimed paradox and provides a clear theoretical justification for the practical success of compressed sensing techniques in imaging problems. [ABSTRACT FROM AUTHOR]

We introduce three developments within the stochastic many-body perturbation theory: efficient evaluation of off-diagonal self-energy terms, construction of Dyson orbitals, and stochastic constrained random phase approximation. The stochastic approaches readily handle systems with thousands of atoms. We use them to explore the electronic states of twisted bilayer graphene (tBLG) characterized by giant unit cells and correlated electronic states. We document the formation of electron localization under compression; weakly correlated states are merely shifted in energy. We demonstrate how to efficiently downfold the correlated subspace on a model Hamiltonian with a screened frequency-dependent two-body interaction. For the 6° tBLG system, the onsite interactions are between 200 and 300 meV under compression. The Dyson orbitals exhibit spatial distribution similar to the mean-field single-particle states. Under pressure, the electron-electron interactions increase in the localized states; however, the dynamical screening does not fully balance the dominant bare Coulomb interaction. [ABSTRACT FROM AUTHOR]

When a photon collides with a laser pulse, an electron-positron pair can be produced via the nonlinear Breit–Wheeler process. A simulation framework has been developed to calculate this process, which is based on a ponderomotive approach that includes strong-field quantum electrodynamical effects via the locally monochromatic approximation (LMA). Here we compare simulation predictions for a variety of observables, in different physical regimes, with numerical evaluation of exact analytical results from theory. For the case of a focussed laser background, we also compare simulation with a high-energy theory approximation. These comparisons are used to quantify the accuracy of the simulation approach in calculating harmonic structure, which appears in the lightfront momentum and angular spectra of outgoing particles, and the transition from multi-photon to all-order pair creation. Calculation of the total yield of pairs over a range of intensity parameters is also used to assess the accuracy of the locally constant field approximation (LCFA). [ABSTRACT FROM AUTHOR]