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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]

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]

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 survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms' theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian's results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, "Fractional derivatives and the Cauchy problem for differential equations of fractional order"), and were invited by the "FCAA" editors to publish its re-edited version in this same issue of the journal. [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]

The Padé approximation is a useful method for creating new inequalities and improving certain inequalities. In this paper we use the Padé approximant to give the refinements of some remarkable inequalities involving inverse trigonometric functions, it is shown that the new inequalities presented in this paper are more refined than that obtained in earlier papers. [ABSTRACT FROM AUTHOR]

We study the classical problem introduced by R. Isaacs and S. Gal of minimizing the time to find a hidden point H on a network Q moving from a known starting point. Rather than adopting the traditional continuous unit speed path paradigm, we use the dynamic "expanding search" paradigm recently introduced by the authors. Here the regions S(t) that have been searched by time t are increasing from the starting point and have total length t. Roughly speaking the search follows a sequence of arcs ai such that each one starts at some point of an earlier one. This type of search is often carried out by real life search teams in the hunt for missing persons, escaped convicts, terrorists or lost airplanes. The paper which introduced this type of search solved the adversarial problem (where H is hidden to maximize the time to be found) for the cases where Q is a tree or is 2-arc-connected. This paper's main contribution is to give two strategy classes which can be used on any network and have expected search times which are within a factor close to 1 of the value of the game (minimax search time). These strategies classes are respectively optimal for trees and 2-arc-connected networks. We also solve the game for circle-and-spike networks, which can be considered as the simplest class of networks for which a solution was previously unknown. [ABSTRACT FROM AUTHOR]

This paper presents an analysis of an efficient parallel implementation of the active-set Newton algorithm (ASNA), which is used to estimate the nonnegative weights of linear combinations of the atoms in a large-scale dictionary to approximate an observation vector by minimizing the Kullback–Leibler divergence between the observation vector and the approximation. The performance of ASNA has been proved in previous works against other state-of-the-art methods. The implementations analysed in this paper have been developed in C, using parallel programming techniques to obtain a better performance in multicore architectures than the original MATLAB implementation. Also a hardware analysis is performed to check the influence of CPU frequency and number of CPU cores in the different implementations proposed. The new implementations allow ASNA algorithm to tackle real-time problems due to the execution time reduction obtained. [ABSTRACT FROM AUTHOR]

The notion of general mixed modulus of smoothness of periodic functions of several variables in the spaces L is introduced. The proposed construction is, on the one hand, a natural generalization of the general modulus of smoothness in the one-dimensional case, which was introduced in a paper of the first author and in which the coefficients of the values of a given function at the nodes of a uniform lattice are the Fourier coefficients of a 2 π-periodic function called the generator of the modulus; while, on the other hand, this construction is a generalization of classical mixed moduli of smoothness and of mixed moduli of arbitrary positive order. For the modulus introduced in the paper, in the case 1 ≤ p ≤ +∞, the direct and inverse theorems on the approximation by the 'angle' of trigonometric polynomials are proved. The previous estimates of such type are obtained as direct consequences of general results, new mixed moduli are constructed, and a universal structural description of classes of functions whose best approximation by 'angle' have a certain order of convergence to zero is given. [ABSTRACT FROM AUTHOR]

Interpolation plays an important role in the areas of signal processing and applied mathematics. Among the various interpolation methods, those related to Chebyshev polynomial interpolation have received much interest recently. In this paper, we propose a new interpolation method using a type I discrete cosine transform (type I DCT) and the nonuniform roots of the second type of Chebyshev polynomials. In this method, the interpolation coefficients are derived using the type I DCT of the Chebyshev nonuniform sampling points. Simulations show the correctness of the proposed method, and a comparison of the proposed method with existing methods is also discussed in detail. [ABSTRACT FROM AUTHOR]

In this work an improved numerical solution of the singular boundary integral equation of the 2D compressible fluid flow around obstacles is obtained by a boundary element method based on modified shape functions and cubic boundary elements. The singular boundary integral equation with sources distribution is considered in this paper, and for its discretization cubic boundary elements are used. The integrals of singular kernels are evaluated using modified shape functions which are deduced by using series expansions for the basis functions we choose for the local approximation models. A computer code is made using Mathcad programming language and, based on it, some particular cases are solved. In order to validate the proposed method, comparisons between numerical solutions and exact ones are performed for the considered test problems. The advantage of using modified shape functions for evaluating the singularities is pointed out through a comparison study between the numerical solution obtained by the method proposed in this paper and the one obtained by using a truncation method for evaluating the singularities. [ABSTRACT FROM AUTHOR]

The AMSAA-BISE model is derived from another approach. This certainly shows the correctness of the AMSAA-BISE model, and indicates the incorrectness of the approximate model given in this paper. The engineering example illustrating these conclusions is given. Merits and demerits of AMSAA and AMSAA-BISE model are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we examine the applicability of the approximation, fg‾≈f‾g‾, within Backus (J. Geophys. Res. 67(11):4427-4440, 1962) averaging. This approximation is a crucial step in the method proposed by Backus (J. Geophys. Res. 67(11):4427-4440, 1962), which is widely used in studying wave propagation in layered Hookean solids. According to this approximation, the average of the product of a rapidly varying function and a slowly varying function is approximately equal to the product of the averages of those two functions.Considering that the rapidly varying function represents the mechanical properties of layers, we express it as a step function. The slowly varying function is continuous, since it represents the components of the stress or strain tensors. In this paper, beyond the upper bound of the error for that approximation, which is formulated by Bos et al. (J. Elast. 127:179-196, 2017), we provide a statistical analysis of the approximation by allowing the function values to be sampled from general distributions.Even though, according to the upper bound, Backus (J. Geophys. Res. 67(11):4427-4440, 1962) averaging might not appear as a viable approach, we show that—for cases representative of physical scenarios modelled by such an averaging—the approximation is typically quite good. We identify the cases for which there can be a deterioration in its efficacy.In particular, we examine a special case for which the approximation results in spurious values. However, such a case—though physically realizable—is not likely to appear in seismology, where Backus (J. Geophys. Res. 67(11):4427-4440, 1962) averaging is commonly used. Yet, such values might occur in material sciences, in general, for which Backus (J. Geophys. Res. 67(11):4427-4440, 1962) averaging is also considered. [ABSTRACT FROM AUTHOR]

Distributionally robust optimization (DRO) has become a popular research topic since it can solve stochastic programs with ambiguous distribution information. In this paper, as the background of economic dispatch (ED) in renewable integration systems, we present a new DRO-based ED optimization framework (DRED). The new DRED is addressed with a coupled format of distribution uncertainty for objective and chance constraints, which is different from most existing DRO frameworks. Some approximation strategies are adopted to handle the complicated DRED: the data-driven approach, the approximation of chance constraints by conditional value-at-risk, and the discrete scheme. The approximate reformulations are solvable nonconvex nonlinear programming problems. The approximation error analysis and convergence analysis are also established. Numerical results using an IEEE-30 buses system are presented to demonstrate the approach proposed in this paper. [ABSTRACT FROM AUTHOR]

In order to resolve the co-channel inter-cell interference problem in heterogeneous networks (HetNet), the feature of almost blank subframes (ABS) in the time domain of the enhanced inter-cell interference coordination (eICIC) is utilized. In this paper, an ABS configuration design is developed on downlink in HetNet and the associated resource allocation problem for maximizing the system performance with fairness among user equipments (UEs) is considered. Compared to conventional problems, the resource assignment problems include the configuration of ABS pattern and the resource allocation for macro UEs and pico UEs, which aims to maximize the downlink throughput and balance the traffic offloading in intra-frequency HetNet deployments. First, this paper introduces an ABS pattern design by using the channel condition, which is developed in terms of the time domain resource. Subframes are categorized as protected or normal subframes for reducing interference impact to pico UEs. Based on the configuration of the ABS pattern, we develop a grouping strategy to determine which pico UEs use either protected or normal subframes. Besides, the assignment of resource blocks with respect to the resource in the frequency domain is developed along with the fairness among UEs. The proposed joint allocation scheme takes the system throughput and the fairness into account, and has better performance than the existing schemes. Simulation results also reveal that the performance of the proposed joint allocation scheme approximates the optimal solution with the full search scheme. [ABSTRACT FROM AUTHOR]

In order to solve the problem of low measurement efficiency of volumetric errors and predict the volumetric errors of CNC machine tools online, a new volumetric error modeling method based on information fusion technology is proposed in this paper, which was used for the volumetric error prediction of a two turntable five-axis machine tool. In order to fulfill the volumetric error modeling, the displacement variables, temperature variables, and cutting force variables were determined firstly, then the determined variables and the volumetric errors were measured and analyzed, and a volumetric error model was presented based on the determined variables. In order to optimize the parameters of the presented model, grey correlation analysis method is used for the second time prediction of the volumetric errors. Finally, the performance of the model was tested by an experiment. The result shows that the approximation ability of the model is very well, and the residual error is smaller than 2 μm. Comparing with conditional modeling methods, this method can be used in different types of machine tools with different operating conditions by regulating the weight matrix, and the robustness of the fusion model is improved. [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]

In this paper a meshfree weak-strong (MWS) form method is considered to solve the coupled equations in velocity and magnetic field for the unsteady magnetohydrodynamic flow throFor this modified estimaFor this modified estimaFor this modified estimaugh a pipe of rectangular and circular sections having arbitrary conducting walls. Computations have been performed for various Hartman numbers and wall conductivity at different time levels. The MWS method is based on applying a meshfree collocation method in strong form for interior nodes and nodes on the essential boundaries and a meshless local Petrov–Galerkin method in weak form for nodes on the natural boundary of the domain. In this paper, we employ the moving least square reproducing kernel particle approximation to construct the shape functions. The numerical results for sample problems compare very well with steady state solution and other numerical methods. [ABSTRACT FROM AUTHOR]

Some controversial results that arise in the problem of the scattering of gravitational objects are analyzed in this paper. First, the approach leading to possible use of the approximation of pairwise (binary) collisions in the theory of scattering in plasmas and in gravitational systems is examined. It is stressed that this possibility is not a consequence of neglecting long-range (distant) collisions. Next, results obtained for the number of scattering events of a test particle in an infinite gravitational medium with a fixed velocity change are discussed. The conclusion that the function describing the number of particle collisions is divergent is shown to be incorrect. This is because the finite time duration of collisions with large impact parameters is neglected. In concluding, we show that the previously proposed variant of a modified Holtsmark distribution is erroneous. [ABSTRACT FROM AUTHOR]

This paper describes the block element method for spatial integral equations with a difference kernel in the boundary-value problems of continuum mechanics and mathematical physics. The basis of the proposed method is the Wiener - Hopf method, whose generalization for a spatial case is called integral factorization method. The block element method is applied to solve problems in domains with piecewise smooth boundaries containing corner points. The developed method was used to solve the contact problem for a wedge-shaped stamp occupying the first quadrant. This paper describes in detail the methods of obtaining various characteristics of the solution constructed by reversing the system of one-dimensional linear integral equations typical for dynamics and static contact problems for stamps in the form of a strip. [ABSTRACT FROM AUTHOR]

We provide a new proof of the nonemptiness of approximate cores of games with many players of a finite number of types. Earlier papers in the literature proceed by showing that, for games with many players, equal-treatment cores of their 'balanced cover games,' which are nonempty, can be approximated by equal-treatment $$\varepsilon $$ -cores of the games themselves. Our proof is novel in that we develop a limiting payoff possibilities set and rely on a fixed point theorem. [ABSTRACT FROM AUTHOR]

A flattenable mesh surface is a polygonal mesh surface that can be unfolded into a planar patch without stretching any polygon. This paper presents a new method for computing a slightly stretched flattenable mesh surface M from a piecewise-linear surface patch P∈ R , where the shape approximation error between M and P is minimized and the strain of stretching on M is controlled. Prior approaches result in either a flattenable surface that could be quite different from the input shape or a (discrete) developable surface has relative simple shape. The techniques investigated in this paper overcome these difficulties. First, we introduce a new surface modeling method to conduct a sequence of nearly isometric deformations to morph a flattenable mesh surface to a new shape which has a better approximation of the input surface. Second, in order to get better initial surfaces for fitting and overcome topological obstacles, a shape perturbation scheme is investigated to obtain the optimal surface fitting result. Last, to improve the scalability of our optimal surface fitting algorithm, a coarse-to-fine fitting framework is exploited so that very dense flattenable mesh surfaces can be modeled and boundaries of the input surfaces can be interpolated. [ABSTRACT FROM AUTHOR]

This paper shows that two or more switched linear systems can generate chaotic dynamical behaviors by an appropriate switching rule as they at least consist of a controllable system and an unstable system with the expanding property. According to the results in the reference (Xie, L. L., Zhou, Y., and Zhao, Y. Criterion of chaos for switched linear systems with antrollers. International Journal of Bifurcation and Chaos, 20(12), 4105–4109 (2010)), a nonlinear feedback gain is needed to generate chaotic dynamics. A linear feedback control is usually used to approximate the nonlinear one for simulation. In order to obtain the exact control, as a main result of this paper, the controller is constructed by Russell’s result, and a block diagram is included to interpret the realization of the controller. Numerical simulations are given to illustrate the generated chaotic dynamical behavior of the switched linear systems with some parameters and show the effects of the constructed controller. [ABSTRACT FROM AUTHOR]

This paper deals with reduction of computational complexities in dynamic systems. This paper develops a novel method of reducing complexities with use of control moments of the system. Though the proposed method is validated through channel estimation in this paper, the same can be equally applied to any other dynamic systems. Encouraging results given in this paper prove that the computational complexities can be reduced up to 10 with a marginal affordable loss of performance. [ABSTRACT FROM AUTHOR]

In the econometric literature of high frequency data, it is often assumed that one can carry out inference conditionally on the underlying volatility processes. In other words, conditionally Gaussian systems are considered. This is often referred to as the assumption of 'no leverage effect'. This is often a reasonable thing to do, as general estimators and results can often be conjectured from considering the conditionally Gaussian case. The purpose of this paper is to try to give some more structure to the things one can do with the Gaussian assumption. We shall argue in the following that there is a whole treasure chest of tools that can be brought to bear on high frequency data problems in this case. We shall in particular consider approximations involving locally constant volatility processes, and develop a general theory for this approximation. As applications of the theory, we develop an ANOVA for processes with multiple regressors, and give an estimator for error bars on the Hayashi-Yoshida estimator of quadratic covariation. Other applications are considered in other papers. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to give a systematic account on privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold, it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies down the background for its sequel (Choi and Ponge 2017) by clarifying a few points on privileged coordinates and the nilpotent approximation of Carnot manifolds. In particular, we give a description of all the systems of privileged coordinates at a given point. We also give an algebraic characterization of all nilpotent groups that appear as the nilpotent approximation at a given point. In fact, given a nilpotent group G satisfying this algebraic characterization, we exhibit all the changes of variables that transform a given system of privileged coordinates into another system of privileged coordinates in which the nilpotent approximation is given by G. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical approximate solutions of singular fourth order four-point boundary value problems using reproducing kernel method (RKM). The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the RKM can not be used directly to solve singular fourth order four-point boundary value problems (BVPs), since there is no method of obtaining reproducing kernel (RK) satisfying four-point boundary conditions. The aim of this paper is to fill this gap. A method for obtaining RK satisfying four-point boundary conditions is proposed so that RKM can be used to solve singular fourth order four-point BVPs. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular fourth order four-point BVPs. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new technique for the asymmetric approximation of discrete functions arising in seasonal customer demand extrapolation. We adapt the technique for two different settings, the so-called pull and push models. Our main goal here is to find effectively extrapolations minimizing the loss. For bothmodels, we discuss several features related to sampling, approximation, and extrapolation. [ABSTRACT FROM AUTHOR]

Query reformulation, including query recommendation and query auto-completion, is a popular add-on feature of search engines, which provide related and helpful reformulations of a keyword query. Due to the dropping prices of smartphones and the increasing coverage and bandwidth of mobile networks, a large percentage of search engine queries are issued from mobile devices. This makes it possible to improve the quality of query recommendation and auto-completion by considering the physical locations of the query issuers. However, limited research has been done on location-aware query reformulation for search engines. In this paper, we propose an effective spatial proximity measure between a query issuer and a query with a location distribution obtained from its clicked URLs in the query history. Based on this, we extend popular query recommendation and auto-completion approaches to our location-aware setting, which suggest query reformulations that are semantically relevant to the original query and give results that are spatially close to the query issuer. In addition, we extend the bookmark coloring algorithm for graph proximity search to support our proposed query recommendation approaches online, and we adapt an A* search algorithm to support our query auto-completion approach. We also propose a spatial partitioning based approximation that accelerates the computation of our proposed spatial proximity. We conduct experiments using a real query log, which show that our proposed approaches significantly outperform previous work in terms of quality, and they can be efficiently applied online. [ABSTRACT FROM AUTHOR]

The increasing popularity of location-based applications creates new opportunities for users to travel together. In this paper, we study a novel spatio-social optimization problem, i.e., Optimal Group Route, for multi-user itinerary planning. With our problem formulation, users can individually specify sources and destinations, preferences on the Point-of-interest (POI) categories, as well as the distance constraints. The goal is to find a itinerary that can be traversed by all the users while maximizing the group’s preference of POI categories in the itinerary. Our work advances existing group trip planning studies by maximizing the group’s social experience. To this end, individual preferences of POI categories are aggregated by considering the agreement and disagreement among group members. Furthermore, planning a multi-user itinerary on large road networks is computationally challenging. We propose two efficient greedy algorithms with bounded approximation ratio, one exact solution which computes the optimal itinerary by exploring a limited number of paths in the road network, and a scaled approximation algorithm to speed up the dynamic programming employed by the exact solution. We conduct extensive empirical evaluations on two real-world road network/POI datasets and our results confirm the effectiveness and efficiency of our solutions. [ABSTRACT FROM AUTHOR]

The present paper is a continuation of the previous papers of the author dealing with this subject. We study the cascade search of a given subset A, i.e., the construction on the metric space X of a multicascade with given limit subset A in X. A multicascade is a multivalued dynamical system with translation semigroup equal to the additive semigroup of nonnegative integers. We propose a finer (than in the author’s previous papers) version of cascade search for cases in which (1) A is the complete preimage of a closed subspace under the multivalued mapping of metric spaces; (2) A is the set of coincidence points n, n > 1, of the multivalued mappings. An estimate of the distance from the initial to any corresponding limit point is given. In particular, in case (2), a new generalization of a recent theorem due to Arutyunov is obtained for n = 2. [ABSTRACT FROM AUTHOR]

In this paper we describe the implementation of the charged current decays of the type t→bl+νl(γ) in the framework of the SANC system. All calculations are done taking into account the one-loop electroweak correction in the standard model. The emphasis of this paper is on the presentation of numerical results. Various distributions are produced by means of a Monte Carlo integrator and event generator. Comparison with the results of the CompHEP and PYTHIA packages are presented for the Born and hard photon contributions. The validity of the cascade approximation at one-loop level is also studied. [ABSTRACT FROM AUTHOR]

Abstract??We study convergence and rate of convergence of expansions of elements in a Banach spaceXinto series with regard to a given dictionary$$f\sim\sum_{j=1}^{\infty}c_{j}(f)g_{j}(f),\quad g_{j}(f)\in\mathcal{d},\ c_{j}(f)>0,\ j=1,2,\dots.$$ In building such a representation we should construct two sequences: {gj(f)}j=1?and {cj(f)}j=1?. In this paper the construction of {gj(f)}j=1?will be based on ideas used in greedy-type nonlinear approximation. This explains the use of the termgreedy expansion. We use a norming functionalof a residualfm?1obtained afterm?1 steps of an expansion procedure to select themth elementfrom the dictionary. This approach has been used in previous papers on greedy approximation. The greedy expansions in Hilbert spaces are well studied. The corresponding convergence theorems and estimates for the rate of convergence are known. Much less is known about greedy expansions in Banach spaces. The first substantial result on greedy expansions in Banach spaces has been obtained recently by Ganichev and Kalton. They proved a convergence result for theLp, 1m(f) that provides convergence of the corresponding greedy expansions in any uniformly smooth Banach space. Moreover, we obtain estimates for the rate of convergence of such greedy expansions for? the closure (inX) of the convex hull of. [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]

In this paper, we consider a perturbed compound Poisson risk model with stochastic premiums and constant interest force. We obtain the upper bound and Lundberg-Cramér approximation for the infinite-time ruin probability, and consider the asymptotic formula for the finite-time ruin probability when the claim size is heavy-tailed. We show that the model in our paper has similar results to the classical risk process and some existing generalized models. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of an approximately controllable system perturbed by a certain Lipschitz-continuous nonlinearity. By assuming that the intensity of the nonlinear influence is 'weak', we prove that the perturbed system is still approximately controllable. Using this perturbation theory, we prove that a third-order semilinear dispersion equation on a finite interval is approximately controllable whenever the nonlinearity effect is 'weak'. The result in the paper is a complement of the results obtained in (Zhou in SIAM J. Control Optim. 21:551-565, 1983), in which the control operator (resp. infinitesimal generator of the dynamics) of the unperturbed linear system is assumed to be bounded (resp. generated a differentiable semigroup on the state space). [ABSTRACT FROM AUTHOR]

Designers of MMOs such as Diablo 3 face economic problems much like policy makers in the real world, e.g. inflation and distributional issues. Solving economic problems through regular updates (patches) became as important to those games as traditional gameplay issues. In this paper we provide an agent framework inspired by the economic features of Diablo 3 and analyze the effect of monetary policy in the game. Our model reproduces a number of features known from the Diablo 3 economy such as a heterogeneous price development, driven almost exclusively by goods of high quality, a highly unequal wealth distribution and strongly decreasing economic mobility. The basic framework presented in this paper is meant as a stepping stone to further research, where our evidence is used to deepen our understanding of the real-world counterparts of such problems. The advantage of our model is that it combines simplicity that is inherent to model economies with a similarly simple observable counterpart (namely the game environment where real agents interact). By matching the dynamics of the game economy we can thus easily verify that our behavioral assumptions are good approximations to reality. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the investigation of a parabolic equation with moving boundaries arising in ductal carcinoma in situ (DCIS) model. Approximation solution of this problem is implemented by Ritz-Galerkin, which is a first attempt at tackling such problem. In process of dealing with this moving boundary condition, we use a trick of introducing two transformations to convert moving boundary to nonclassical boundary that can be handled with Ritz-Galerkin method. Also, existence and uniqueness are proved. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. [ABSTRACT FROM AUTHOR]

The paper presents new approximability results for single machine scheduling problems with jobs requiring some non-renewable resources (like raw materials, energy, or money) beside the machine. Each resource has an initial stock and additional supplies over time. A feasible schedule specifies a starting time for each job such that no two jobs overlap in time, and when a job is started, enough resources are available to cover its requirements. The goal is to find a feasible schedule of minimum makespan. This problem is strongly NP-hard. Recently, the authors of this paper have proposed a PTAS for the special case with a single non-renewable resource and with a constant number of supply dates, as well as an FPTAS for the special case with two supply dates and one resource only. In this paper we prove APX-hardness of the problem when the number of resources is part of the input, and new polynomial time approximation schemes are devised for some variants, including (1) job release dates, and more than one, but constant number of resources and resource supply dates, and (2) only one resource, arbitrary number of supply dates and job release dates, but with resource requirements proportional to job processing times. [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]