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A unified study of necessary and sufficient conditions for the existence of intersection points and other important points in mathematical analysis is presented. The results are established in topologically based settings and shown to have equivalent formulations for various kinds of important points. For set-valued maps from a set to a topological space, instead of a convexity structure on this set, we use two general structures. The first structure is based on continuous single-valued maps from simpleces to the aforementioned set in order to extend the ideas of Knaster–Kuratowski–Mazurkiewicz and Fan for convex hulls. The second one is based on single-valued maps from the unit interval of the real line to the considered set and extends the classical connectedness properties. The general existence theorems of the above types are crucial tools in studies of the solution existence in optimization and other areas of applied mathematics. Here we choose minimax problems for applications of our general theorems. Our results are new, or improve or include as special cases many known results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sparse portfolio selection models with a regularized l p -norm term (0 < p ≤ 1) and negatively bounded shorting constraints. We obtain some basic properties of several linear l p -sparse minimax portfolio models in terms of the regularization parameter. In particular, we introduce an l 1 -sparse minimax Sharpe ratio model by guaranteeing a positive denominator with a pre-selected parameter and design a parametric algorithm for finding its global solution. We carry out numerical experiments of linear l p -sparse minimax portfolio models with 1200 stocks from Hang Seng Index, Shanghai Securities Composite Index, and NASDAQ Index and compare their performance with l p -sparse mean-variance models. We test the effect of the regularization parameter and the negatively bounded shorting parameter on the level of sparsity, risk, and rate of return respectively and find that portfolios including fewer stocks of the linear l p -sparse minimax models tend to have lower risks and lower rates of return. However, for the l p -sparse mean-variance models, the corresponding changes are not so significant. [ABSTRACT FROM AUTHOR]

We propose a minimax design method for finite impulse response (FIR) graph filters in this brief. We first convert the graph filter design problem into a numerically stable equivalent problem using a shifted monomial basis. Then, we convert the minimax graph filter design problem into a semidefinite programming problem. Furthermore, we propose a computationally-efficient structure to implement the FIR graph filters designed using the proposed method. Design examples confirm that the proposed method can be used to design FIR graph filters having reasonably high orders, and the proposed implementation structure substantially saves additions required to process a graph signal compared to graph filters designed using a previously proposed Chebyshev polynomial approximation method. [ABSTRACT FROM AUTHOR]

We consider the nonparametric estimation of an S‐shaped regression function. The least squares estimator provides a very natural, tuning‐free approach, but results in a non‐convex optimization problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal‐dual bases algorithm for its efficient, sequential computation. After developing a projection framework that demonstrates the consistency and robustness to misspecification of the estimator, our main theoretical results provide sharp oracle inequalities that yield worst‐case and adaptive risk bounds for the estimation of the regression function, as well as a rate of convergence for the estimation of the inflection point. These results reveal not only that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost‐parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations and a real data application to air pollution modelling also confirm the desirable finite‐sample properties of the estimator, and our algorithm is implemented in the R package Sshaped. [ABSTRACT FROM AUTHOR]

An efficient hybrid method based on subdomain adaptive integral method (SAIM) and Chebyshev approximation technology (CAT) is proposed for analyzing the wideband radar cross section (RCS) of complex multiscale objects in this letter. The SAIM cannot only be used to effectively deal with the difficulty of slow convergence caused by the mutiscale problems, but also greatly reducing the computational memory. By introducing the CAT into SAIM, we can use the currents at a few frequency sampling points to predict the currents at any frequency point over the entire band. In addition, the application of the Maehly approximation can obtain a better accuracy. Numerical examples show that the proposed SAIM-CAT can significantly improve the calculation efficiency without loss of accuracy. [ABSTRACT FROM AUTHOR]

The Fuzzy transform is applied mainly to 1-D signals and 2-D data organized as a regular grid (e.g., 2-D images), thus, limiting its potential application to arbitrary data in terms of dimensionality and structure. This article defines and analyzes the properties of the data-driven F-transform, with a focus on the construction of the class of data-driven membership functions, which are multiscale, local, linearly independent, intrinsic, and robust to data discretization. Data-driven membership functions are defined by applying a 1-D filter to the Laplace–Beltrami operator, which encodes the geometric and topological properties of the input data. Then, we address the efficient computation of the data-driven F-transform through a polynomial or a rational polynomial approximation of the input filter. In this way, the computation of the data-driven F-transform is independent of the evaluation of the membership functions at any point of the input domain and reduces to the solution of a small set of sparse and symmetric linear systems. Finally, the data-driven F-transform is efficiently evaluated on large and arbitrary data, in terms of dimensionality, structure, and size. [ABSTRACT FROM AUTHOR]

In this work, we investigate the poroelastic waves by solving the time‐domain Biot‐JKD equation with an efficient numerical method. The viscous dissipation occurring in the pores depends on the square root of the frequency and is described by the Johnson‐Koplik‐Dashen (JKD) dynamic tortuosity/permeability model. The temporal convolutions of order 1/2 shifted fractional derivatives are involved in the time‐domain Biot‐JKD model, causing the problem to be stiff and challenging to be implemented numerically. Based on the best relative Chebyshev approximation of the square‐root function, we design an efficient algorithm to approximate and localize the convolution kernel by introducing a finite number of auxiliary variables that satisfy a local system of ordinary differential equations. The imperfect hydraulic contact condition is used to describe the interface boundary conditions and the Runge‐Kutta discontinuous Galerkin (RKDG) method together with the splitting method is applied to compute the numerical solutions. Several numerical examples are presented to show the accuracy and efficiency of our approach. [ABSTRACT FROM AUTHOR]

We study the problem of off-policy evaluation in the multi-armed bandit model with bounded rewards, and develop minimax rate-optimal procedures under three settings. First, when the behavior policy is known, we show that the Switch estimator, a method that alternates between the plug-in and importance sampling estimators, is minimax rate-optimal for all sample sizes. Second, when the behavior policy is unknown, we analyze performance in terms of the competitive ratio, thereby revealing a fundamental gap between the settings of known and unknown behavior policies. When the behavior policy is unknown, any estimator must have mean-squared error larger—relative to the oracle estimator equipped with the knowledge of the behavior policy— by a multiplicative factor proportional to the support size of the target policy. Moreover, we demonstrate that the plug-in approach achieves this worst-case competitive ratio up to a logarithmic factor. Third, we initiate the study of the partial knowledge setting in which it is assumed that the minimum probability taken by the behavior policy is known. We show that the plug-in estimator is optimal for relatively large values of the minimum probability, but is sub-optimal when the minimum probability is low. In order to remedy this gap, we propose a new estimator based on approximation by Chebyshev polynomials that provably achieves the optimal estimation error. Numerical experiments on both simulated and real data corroborate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a linear Chebyshev pseudospectral method (LCPM) is proposed to solve the nonlinear optimal control problems (OCPs) with hard terminal constraints and unspecified final time, which uses Chebyshev collocation scheme and quasi-linearization. First, Taylor expansion around the nonlinear differential equations of the system is used to obtain a set of linear perturbation equations. Second, the first-order necessary conditions for OCPs with these linear equations and unspecified terminal time are derived, which provide the successive correction formulas of control and terminal time. Traditionally, these formulas are linear time varying and cannot be solved in an analytical manner. Third, Lagrange interpolation, whose supporting points are orthogonal Chebyshev–Gauss–Lobatto (CGL), is employed to discretize the resulting problem. Therefore, a series of analytical correction formulas are successfully derived in approximating polynomial space. It should be noted that Chebyshev approximation is close to the best polynomial approximation, and CGL points can be solved in closed form. Finally, LCPM is applied to the air-to-ground missile guidance problem. The simulation results show that it has high computational efficiency and convergence rate. A comparison with the other typical OCP solvers is provided to verify the optimality of the proposed algorithm. In addition, the results of Monte Carlo simulations are presented, which show that the proposed algorithm has strong robustness and stability. Therefore, the proposed method has potential to be onboard application. [ABSTRACT FROM AUTHOR]

A wideband bandpass filter (BPF) with in-band Chebyshev equal-ripple responses and extended upper stopband is proposed, which is composed of composite microstrip, slotline, and coplanar waveguide (CPW) structure. According to the odd-mode and even-mode equivalent circuits, there are four in-band transmission poles constructing the ultrabroad bandwidth can be directly deduced. In addition, there are two sets of lowpass filter (LPF) embedding in the CPW connecting line to greatly widen the upper stopband without increasing circuit size. In order not to affect original passband Chebyshev equal-ripple responses, the LPFs are synthesized using the low-pass prototype network. Furthermore, there are two finite frequency transmission zeros (TZs) which generated by cross-coupling effects located at lower and upper side of passband to effectively improve the shirt selectivity. Finally, for experimental verification, the proposed BPF prototype is manufactured and tested. Experimental results are consistent with predicted theoretical responses, which verify the validity and feasibility of proposed approach. [ABSTRACT FROM AUTHOR]

A technique for trading off the main lobe width against sidelobe magnitude for any arbitrary window was reported in Lim et al. and subsequently, a fast convergence method for its implementation was proposed by the same authors. These methods require the computation of derivatives involving the evaluation of trigonometric and hyperbolic functions. In this paper, we show that the derivatives can be computed without evaluating trigonometric and hyperbolic functions if the window function is a sum of exponentials such as a Fourier series. [ABSTRACT FROM AUTHOR]

In this paper, we generalize He's frequency approach for solving the damped Duffing equation by introducing a time varying amplitude. We also solve this equation by means of the homotopy method and the Lindstedt-Poincaré method. High accurate formulas for approximating the Jacobi elliptic function cn are formally derived using Chebyshev and Pade approximation techniques. [ABSTRACT FROM AUTHOR]

Popular graph neural networks implement convolution operations on graphs based on polynomial spectral filters. In this paper, we propose a novel graph convolutional layer inspired by the auto-regressive moving average (ARMA) filter that, compared to polynomial ones, provides a more flexible frequency response, is more robust to noise, and better captures the global graph structure. We propose a graph neural network implementation of the ARMA filter with a recursive and distributed formulation, obtaining a convolutional layer that is efficient to train, localized in the node space, and can be transferred to new graphs at test time. We perform a spectral analysis to study the filtering effect of the proposed ARMA layer and report experiments on four downstream tasks: semi-supervised node classification, graph signal classification, graph classification, and graph regression. Results show that the proposed ARMA layer brings significant improvements over graph neural networks based on polynomial filters. [ABSTRACT FROM AUTHOR]

There has been a long history of using ordinary differential equations (ODEs) to understand the dynamics of discrete-time algorithms (DTAs). Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how to obtain a suitable ODE from a given DTA, and (ii) it is unclear the connection between the convergence of a DTA and its corresponding ODEs. In this paper, we propose a new machinery—an O (s r) -resolution ODE framework—for analyzing the behavior of a generic DTA, which (partially) answers the above two questions. The framework contains three steps: 1. To obtain a suitable ODE from a given DTA, we define a hierarchy of O (s r) -resolution ODEs of a DTA parameterized by the degree r, where s is the step-size of the DTA. We present a principal approach to construct the unique O (s r) -resolution ODEs from a DTA; 2. To analyze the resulting ODE, we propose the O (s r) -linear-convergence condition of a DTA with respect to an energy function, under which the O (s r) -resolution ODE converges linearly to an optimal solution; 3. To bridge the convergence properties of a DTA and its corresponding ODEs, we define the properness of an energy function and show that the linear convergence of the O (s r) -resolution ODE with respect to a proper energy function can automatically guarantee the linear convergence of the DTA. To better illustrate this machinery, we utilize it to study three classic algorithms—gradient descent ascent (GDA), proximal point method (PPM) and extra-gradient method (EGM)—for solving the unconstrained minimax problem min x ∈ R n max y ∈ R m L (x , y) . Their O(s)-resolution ODEs explain the puzzling convergent/divergent behaviors of GDA, PPM and EGM when L(x, y) is a bilinear function, and showcase that the interaction terms help the convergence of PPM/EGM but hurts the convergence of GDA. Furthermore, their O(s)-linear-convergence conditions not only unify the known scenarios when PPM and EGM have linear convergence, but also showcase that these two algorithms exhibit linear convergence in much broader contexts, including when solving a class of nonconvex-nonconcave minimax problems. Finally, we show how this ODE framework can help design new optimization algorithms for minimax problems, by studying the difference between the O(s)-resolution ODE of GDA and that of PPM/EGM. [ABSTRACT FROM AUTHOR]

In this paper, we deal with new applications of two minimax theorems of B. Ricceri ([5],[9]). Here is a particular case of one of the results that we obtain: Let (T,F,μ) be a non-atomic measure space, with μ(T) < +∞, (E,∥·∥) a real Banach space, I ⊆ E an unbounded set whose closure does not contain 0. Moreover, let p,q, r, s be four numbers such that 0 < s < q ≤ p, p ≥ 1, r > 1. Set X := { f ∈ Lp(T,E): f (T) ⊆ I}. Then, one has ... [ABSTRACT FROM AUTHOR]

To maintain the vibrations of the gyroscope proof mass, a trajectory tracking control system using a neural network estimator is proposed. The proposed control system incorporates a fractional controller based on the terminal sliding-mode and a recurrent Chebyshev fuzzy neural network using a self-evolving mechanism. The fractional-order terminal sliding-mode control can guarantee the tracking error exponential stable, and a self-evolving recurrent Chebyshev fuzzy neural network (SERCFNN) is introduced to relax the requirement of nonlinear functional certainty. In addition, the SERCFNN develops the advantages of the self-evolving fuzzy neural network (SEFNN), recurrent fuzzy neural network (RFNN), and Chebyshev function network (CFN). The SEFNN can adaptively update the dynamic structure through generating and adjusting fuzzy logic rules. The RFNN improves the performance for coping with a temporal problem with the capability to store prior information. The CFN is capable to enlarge the dimensionality of the input variables. Moreover, the asymptotic stability of the proposed control system can be proved by the Lyapunov stability theory. The effectiveness and superiority of the performance are exhibited with simulation studies and comprehensive comparisons. [ABSTRACT FROM AUTHOR]

This article investigates the $\eta$ -exponential stabilization and positivity of the controlled discrete-time polynomial fuzzy model (DPFM) system with time delay. $\eta$ -exponential polynomial copositive Lyapunov candidate is proposed to detect the system stability under the convergence (decay) rate. Static output feedback fuzzy controller is then synthesized to assure the DPFM closed-loop system with time delay. We propose Chebyshev membership functions (CMFs) to approach the primary membership function and attenuate the conservativeness of $\eta$ -exponential stability formulation and positivity analysis. Chebyshev norm approximation error is utilized to introduce the error between CMFs and primary membership functions using slack matrices. A numerical example is presented to validate the proposed methods. [ABSTRACT FROM AUTHOR]

This brief proposes a wideband interdigital bandpass filter (IBPF), exploiting the through-silicon via (TSV)-based 3-D integrated circuit (3-D IC) technology. IBPF is a structure composed of multiple groups of parallel-coupled line resonators, which can increase the bandwidth and out-band suppression (OBS) characteristics of a bandpass filter. Through the transformation of the Chebyshev low-pass circuit model, a seventh-order IBPF centered at 85 GHz is obtained. By combining TSV technology with the coupling coefficient method, the design process from the Chebyshev low-pass circuit model to the final optimization of IBPF is given. The fractional bandwidth (FBW) of IBPF based on TSV is 63%, the out-of-band suppression is greater than 80 dB, the insertion loss is 1.3 dB, the return loss is 20 dB, and the size of the compact IBPF is only $0.50\times0.34$ mm2 ($0.48\times 0.33\,\,\lambda \text{g}^{2}$). [ABSTRACT FROM AUTHOR]

This paper deals with Chebyshev wavelets. We analyze their properties computing their Fourier transform. Moreover, we discuss the differential properties of Chebyshev wavelets due to the connection coefficients. Uniform convergence of Chebyshev wavelets and their approximation error allow us to provide rigorous proofs. In particular, we expand the mother wavelet in Taylor series with an application both in fractional calculus and fractal geometry. Finally, we give two examples concerning the main properties proved. [ABSTRACT FROM AUTHOR]

The deceptive jamming technique can threaten a synthetic aperture radar (SAR) imaging system by repeatedly transmitting copied radar signals back to the radar. A conventional phased array can generate only an angle-dependent beam and cannot distinguish the jammer from other targets. Instead, a frequency diverse array (FDA) can offer a range-angle-dependent beampattern by introducing small frequency offsets to array elements, which can be adopted to distinguish signals from different range cells. Thus, considering that the position of the jammer is unknown, it is necessary to suppress the range peak sidelobe as much as possible. To lower this value, this communication proposes a weighted FDA to achieve the optimal solution at the expense of raising other range sidelobes. [ABSTRACT FROM AUTHOR]

Assume that we observe i.i.d. points lying close to some unknown d-dimensional C k submanifold M in a possibly high-dimensional space. We study the problem of reconstructing the probability distribution generating the sample. After remarking that this problem is degenerate for a large class of standard losses ( L p , Hellinger, total variation, etc.), we focus on the Wasserstein loss, for which we build an estimator, based on kernel density estimation, whose rate of convergence depends on d and the regularity s ≤ k - 1 of the underlying density, but not on the ambient dimension. In particular, we show that the estimator is minimax and matches previous rates in the literature in the case where the manifold M is a d-dimensional cube. The related problem of the estimation of the volume measure of M for the Wasserstein loss is also considered, for which a minimax estimator is exhibited. [ABSTRACT FROM AUTHOR]

This paper aims to build a supervised classifier for dealing with imbalanced datasets, uncertain class proportions, dependencies between features, the presence of both numeric and categorical features, and arbitrary loss functions. The Bayes classifier suffers when prior probability shifts occur between the training and testing sets. A solution is to look for an equalizer decision rule whose class-conditional risks are equal. Such a classifier corresponds to a minimax classifier when it maximizes the Bayes risk. We develop a novel box-constrained minimax classifier which takes into account some constraints on the priors to control the risk maximization. We analyze the empirical Bayes risk with respect to the box-constrained priors for discrete inputs. We show that this risk is a concave non-differentiable multivariate piecewise affine function. A projected subgradient algorithm is derived to maximize this empirical Bayes risk over the box-constrained simplex. Its convergence is established and its speed is bounded. The optimization algorithm is scalable when the number of classes is large. The robustness of our classifier is studied on diverse databases. Our classifier, jointly applied with a clustering algorithm to process mixed attributes, tends to equalize the class-conditional risks while being not too pessimistic. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an efficient quadrature method adapted to a large class of functions. The idea is to replace the polynomial approximation by a function in Chebyshev space. We describe an algorithm to select the best quadrature method for a given class of functions. This method is determined by solving a minimization of the approximation error. Numerical results are given to show the efficiency of our method. [ABSTRACT FROM AUTHOR]

Signal and power integrity design in the time domain requires equivalent circuit models for interconnects and packages, whose descriptions may only be available as tabulated impedance or admittance parameters. Accurate models for these components should maintain their physical properties including causality, stability, and passivity. Even though a heuristic approach, pole flipping (i.e., changing the sign of the real part of an unstable pole) has proven to be sufficiently accurate for many applications, resulting in models with ensured causality and stability. In this article, we address the problem of generating passive scalar models, such as driving point impedances or admittances, based on an existing causal, stable, but nonpassive model. Our approach is based on using sum-of-squares (SOS) polynomials and results in a convex optimization problem, hence a global optimum is obtained. We obtain approximations through two distinct SOS algorithms: a coefficient and a sampling-based method. We also present a simple passivity test for such functions based on calculating the roots of numerator and denominator polynomials, which provides an intuitive link among existing approaches based on solving an eigenvalue problem. [ABSTRACT FROM AUTHOR]

A wideband planar multilayer filtering magic-T is proposed and developed. The proposed magic-T uses coupled microstrip slotlines (CMSs) and microstrip-slot transitions. With the orthogonality of even and odd modes of the CMS, the Chebyshev filtering responses of both the in-phase divider part (H-T) and out-of-phase divider part (E-T) are independent of each other. The microstrip-slot transitions act as a part of filtering structures to realize vertical transmission of energy. With equivalent impedance transformer models, the synthesis processes of a four-pole Chebyshev filtering H-T and a three-pole Chebyshev filtering E-T are derived and analyzed. To verify the proposed concept, a prototype with an operating center frequency of 3 GHz, a 100% equal ripple fractional bandwidth, and a different H-T and E-T in-band ripple are synthesized and implemented on a three-layer printed circuit board. The synthesized, simulated, and measured results show good agreements among them, verifying the feasibility of the proposed planar magic-T. [ABSTRACT FROM AUTHOR]

Principal component analysis (PCA) is a commonly used pattern analysis method that maps high-dimensional data into a lower-dimensional space maximizing the data variance, that results in the promotion of separability of data. Inspired by the principle of PCA, a novel type of learning discriminative matrix information geometry (MIG) detectors in the unsupervised scenario are developed, and applied to signal detection in nonhomogeneous environments. Hermitian positive-definite (HPD) matrices can be used to model the sample data, while the clutter covariance matrix is estimated by the geometric mean of a set of secondary HPD matrices. We define a projection that maps the HPD matrices in a high-dimensional manifold to a low-dimensional and more discriminative one to increase the degree of separation of HPD matrices by maximizing the data variance. Learning a mapping can be formulated as a two-step mini-max optimization problem in Riemannian manifolds, which can be solved by the Riemannian gradient descent algorithm. Three discriminative MIG detectors are illustrated with respect to different geometric measures, i.e., the Log-Euclidean metric, the Jensen–Bregman LogDet divergence and the symmetrized Kullback–Leibler divergence. Simulation results show that performance improvements of the novel MIG detectors can be achieved compared with the conventional detectors and their state-of-the-art counterparts within nonhomogeneous environments. [ABSTRACT FROM AUTHOR]

Approximations based on Chebyshev polynomials have several astrodynamic applications. The performance of these approximations can be improved by parallel implementations exploiting parallel architectures, such as OpenMP and CUDA. In this paper, we introduce the parallel implementation to two astrodynamic applications. The first is the gravitational finite element model (FEM): a piecewise Chebyshev approximation that replaces high degree and order gravitational spherical harmonic models (SHMs). Thus, much lower degree, locally valid functions can efficiently model and compute local gravity perturbations in parallel structure for efficient performance. For this model, the total gravity acceleration is split into a reference and disturbance term. The reference includes two-body plus J 2 , which are relatively cheap to compute. The FEM approximates the higher-order gravity terms. It is developed from a 2D mesh grid covering a sphere of a specified radius, and a family of spherical shells is sampled using a cosine distribution in the radial direction. To reduce the required memory when seeking a specific accuracy, an adaptive version of the gravitational FEM is introduced. In addition, a parallel implementation of the FEM using OpenMP is preseneted. We show the runtime comparison for the 200 degree × 200 order EGM2008 SHM and the serial and parallel equivalent FEM algorithms. The other application is the Chebyshev-Picard method (CPM): a numerical integrator that solves an ordinary differential equation by approximating the integrand using a Chebyshev approximant and iterates over the trajectory via Picard iteration. A parallel CUDA implementation of the CPM method in conjunction with the EGM2008 SHM and the FEM is introduced. We present numerical examples for propagating four Earth-orbiting satellites considering both the 200 × 200 EGM2008 SHM and the equivalent FEM representation to test the algorithm's performance via parallel and serial computation (i.e., a single CPU thread). [ABSTRACT FROM AUTHOR]

In this letter, the modified elliptic low-pass filtering (LPF) matching network (MN) is proposed for harmonic suppression of a wideband gallium nitride (GaN) microwave monolithic integrated circuit (MMIC) power amplifier (PA) at millimeter-wave (mm-wave). To investigate the effectiveness of the method at mm-wave, a three-stage 25–31-GHz GaN MMIC PA is designed using a commercial 150-nm GaN HEMT foundry process, with a mask area of $4.5\times2.6$ mm2. Simulated results show that the second harmonic of the PA can be effectively suppressed by using the elliptic LPF MN. Experimental results show that the realized MMIC PA achieves a saturated power-added efficiency (PAE) higher than 26.1% from 25 to 31 GHz, with a peak PAE of 31.5% at 29 GHz. The saturated gain and output power are more than 20 and 34 dBm within the band, respectively. [ABSTRACT FROM AUTHOR]

In this letter, a behavior model, namely CP- LSTM, composed of Chebyshev polynomials (CP) and a long short-term memory (LSTM) network is built to linearize the wideband millimeter-wave power amplifier (mmW PA) in the fifth-generation (5G) mobile communication system. In order to verify the linearization performance of the CP- LSTM predistorter, a 100-MHz bandwidth 5G new radio (5G NR) signal is employed to test the 28-GHz mmW PA under the text. Experimental results show that the adjacent channel power ratio (ACPR) of the mmW PA with the CP- LSTM can be improved by 20 dB which is 5-dB better than with the LSTM and 3-dB better than with the generalized memory polynomial (GMP). Therefore, the proposed CP- LSTM model is very effective to linearize 5G mmW PAs. [ABSTRACT FROM AUTHOR]

In the contribution, 'Sharp estimates for approximate and exact solutions to quasi-optimization problems', M. Ait Mansour, M. A Bahraoui and A. El Bekkali consider an implicit set-valued map representing solutions to a parametric quasi-optimization problem (QOpt). It is our great pleasure to devote this special issue to Equilibrium Problems and Minimax Inequalities which includes seven papers selected among those presented at the autumn school ASEM19, which was held during 25-26 September 2019, in El Jadida Morocco. A special issue dedicated to the Autumn school on equilibrium problems and minimax inequalities, ASEM19, 25-26 September 2019, El Jadida, Morocco. [Extracted from the article]

In this paper, we give an overview of some recent applications of a minimax theorem. These applications deal with: uniquely remotal sets in normed spaces; multiple global minima for the integral functional of the Calculus of Variations; multiple periodic solutions for Lagrangian systems of relativistic oscillators; variational inequalities in balls of Hilbert spaces. [ABSTRACT FROM AUTHOR]

This paper considers optimal design problems for the Weibull distribution, which can be used to model symmetrical or asymmetrical data, in the presence of progressive interval censoring in life-testing experiments. Two robust approaches, Bayesian and minimax, are proposed to deal with the dependence of the D-optimality and c-optimality on the unknown model parameters. Meanwhile, the compound design method is applied to ensure a compromise between the precision of estimation of the model parameters and the precision of estimation of the quantiles. Furthermore, to make the design become more practical, the cost constraints are taken into account in constructing the optimal designs. Two algorithms are provided for finding the robust optimal solutions. A simulated example and a real life example are given to illustrate the proposed methods. The sensitivity analysis is also studied. These new design methods can help the engineers to obtain robust optimal designs for the censored life-testing experiments. [ABSTRACT FROM AUTHOR]

This study discusses how to find the best linear approximation p(x) = mx + n to a fundamental function y = √x on the interval [0, b], especially using the minimax error in Numerical Analysis. For this aim we employ two mathematical techniques: a) using the MATLAB code, positioning m and n values of the smallest maximum error on a broad range of m, and n value matrix in a rough scale and then repeatedly refining the regions in the smaller scales and b) finding three-point fitting line to a set of non-colinear three points. We see that both results are successfully obtained and identical each other neglecting the error tolerance. [ABSTRACT FROM AUTHOR]

The problem of approximation by low-rank matrices is found everywhere in computational mathematics. Traditionally, this problem is solved in the spectral or Frobenius norm, where the approximation efficiency is associated with the rate of decrease of the matrix singular values. However, recent results show that this requirement is not necessary in other norms. In this paper, a method for solving the problem of approximating by low-rank matrices in Chebyshev's norm is proposed. It makes it possible to construct effective approximations of matrices for which singular values do not decrease in an acceptable amount time. [ABSTRACT FROM AUTHOR]

An important aspect of multi-agent systems concerns the formation of coalitions that are stable or optimal in some well-defined way. The notion of popularity has recently received a lot of attention in this context. A partition is popular if there is no other partition in which more agents are better off than worse off. In this paper, we study popularity, strong popularity, and mixed popularity (which is particularly attractive because existence is guaranteed by the Minimax Theorem) in a variety of coalition formation settings. Extending previous work on marriage games, we show that mixed popular partitions in roommate games can be found efficiently via linear programming and a separation oracle. This approach is quite universal, leading to efficient algorithms for verifying whether a given partition is popular and for finding strongly popular partitions (resolving an open problem). By contrast, we prove that both problems become computationally intractable when moving from coalitions of size 2 to coalitions of size 3, even when preferences are strict and globally ranked. Moreover, we show that finding popular, strongly popular, and mixed popular partitions in symmetric additively separable hedonic games and symmetric fractional hedonic games is NP-hard. Together, these results indicate strong boundaries to the tractability of popularity in both ordinal and cardinal models of hedonic games. [ABSTRACT FROM AUTHOR]

Many statistical methods have been proposed for variable selection in the past century, but few balance inference and prediction tasks well. Here, we report on a novel variable selection approach called penalized regression with second-generation p-values (ProSGPV). It captures the true model at the best rate achieved by current standards, is easy to implement in practice, and often yields the smallest parameter estimation error. The idea is to use an l 0 penalization scheme with second-generation p-values (SGPV), instead of traditional ones, to determine which variables remain in a model. The approach yields tangible advantages for balancing support recovery, parameter estimation, and prediction tasks. The ProSGPV algorithm can maintain its good performance even when there is strong collinearity among features or when a high-dimensional feature space with p > n is considered. We present extensive simulations and a real-world application comparing the ProSGPV approach with smoothly clipped absolute deviation (SCAD), adaptive lasso (AL), and minimax concave penalty with penalized linear unbiased selection (MC+). While the last three algorithms are among the current standards for variable selection, ProSGPV has superior inference performance and comparable prediction performance in certain scenarios. [ABSTRACT FROM AUTHOR]

Non‐Gaussian distributions are commonly observed in collisionless space plasmas. Generating samples from non‐Gaussian distributions is critical for the initialization of particle‐in‐cell simulations that investigate their driven and undriven dynamics. To this end, we report a computationally efficient, robust tool, Chebsampling, to sample general distribution functions in one and two dimensions. This tool is based on inverse transform sampling with function approximation by Chebyshev polynomials. We demonstrate practical uses of Chebsampling through sampling typical distribution functions in space plasmas. Key Points: New computational tool for fast sampling of arbitrary particle distribution functions is presentedChebyshev polynomial interpolation allows to approximate grid‐based distributions and accelerates the solution of inversion problemWe illustrate the use of Chebsampling via sampling non‐Harris current sheets and non‐Maxwellian velocity distributions [ABSTRACT FROM AUTHOR]

In this paper, by virtue of the Kakutani–Fan–Glicksberg fixed point theorem and a nonlinear scalarization function, we first obtain an existence theorem of general n-person noncooperative games problems with set payoffs. Then, we also obtain existence theorems of minimax regret equilibria and ϵ-minimax regret equilibria with scalar set payoffs. Some examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

Time-equispaced inertial measurements are practically used as inputs for motion determination. Polynomial interpolation is a common technique of recovering the gyroscope signal but is subject to a fundamentally numerical stability problem due to the Runge effect on equispaced samples. This article reviews the theoretical results of Runge phenomenon in related areas and proposes a straightforward borrowing-and-cutting (BAC) strategy to depress it. It employs the neighboring samples for higher order polynomial interpolation but only uses the middle polynomial segment in the actual time interval. The BAC strategy has been incorporated into attitude computation by functional iteration, leading to accuracy benefit of several orders of magnitude under the classical coning motion. It would potentially bring significant benefits to the inertial navigation computation under sustained dynamic motions. [ABSTRACT FROM AUTHOR]

Alexander Rubinov was born in Leningrad, now S.-Petersburg, Russia, and graduated from Leningrad State University. This special issue is dedicated to the 80th birthday of Professor Alexander Rubinov (1940-2006) and celebrating his achievements and the impact of his work. [Extracted from the article]

In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalized rational approximation. In the second case, the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimization problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood. In this paper we demonstrate that in the case of multivariate generalized rational approximation the corresponding optimization problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimization. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalized rational approximation with the same number of decision variables. [ABSTRACT FROM AUTHOR]

By applying the perturbation function approach, we propose the Lagrangian and conjugate duals for minimization problems of the sum of two, generally nonconvex, functions. The main tools are the Φ-convexity theory and minimax theorems for Φ-convex functions. We provide conditions ensuring zero duality gap and introduce Φ-Karush–Kuhn–Tucker conditions that characterize solutions to primal and dual problems. We also discuss the relationship between the dual problems introduced in the present investigation and some conjugate-type duals existing in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate that a well-known linear inequality method developed for rational Chebyshev approximation is equivalent to the application of the bisection method used in quasiconvex optimization. Although this correspondence is not surprising, it naturally connects rational and generalized rational Chebyshev approximation problems with modern developments in the area of quasiconvex functions and, therefore, offers more theoretical and computational tools for solving this problem. The second important contribution of this paper is the extension of the linear inequality method to a broader class of Chebyshev approximation problems, where the corresponding objective functions remain quasiconvex. In this broader class of functions, the inequalities are no longer required to be linear: it is enough for each inequality to define a convex set and the computational challenge is in solving the corresponding convex feasibility problems. Therefore, we propose a more systematic and general approach for treating Chebyshev approximation problems. In particular, we are looking at the problems where the approximations are quasilinear functions with respect to their parameters that are also the decision variables in the corresponding optimization problems. [ABSTRACT FROM AUTHOR]

This paper deals with minimax fractional programs whose objective functions are the maximum of finite ratios of convex functions, with arbitrary convex constraints set. For such problems, Dinkelbach-type algorithms fail to work since the parametric subproblems may be nonconvex, whereas the latter need a global optimal solution of these subproblems. We give necessary optimality conditions for such problems, by means of convex analysis tools. We then propose a method, based on solving approximately a sequence of parametric convex problems, which acts as dc (difference of convex functions) algorithm, if the parameter is positive and as Dinkelbach algorithm if not. We show that every cluster point of the sequence of optimal solutions of these subproblems satisfies necessary optimality conditions of KKT criticality type, that are also of Clarke stationarity type. Finally we end with some numerical tests to illustrate the behaviour of the algorithm. [ABSTRACT FROM AUTHOR]

One of the major considerations for a short- or intermediate-distance wireless power transfer (WPT) system is its robustness against variations of the distance, displacement, misalignment between the transmitting and receiving coils, and variations of the loads. Designing such a robust system has not been easy since most methods require some forms of trial-and-errors or optimization that present different degrees of uncertainty. In this article, we propose a design method to overcome the above issue: we make the power transfer efficiency (PTE) function of the WPT equivalent to that of a modified filter. Then, we apply a well-established filter theory to develop a systematic WPT design method; it can keep the PTE stable within the specified ranges of the distance changes, displacements, misalignments, and load variations. We take the Chebyshev filter as an example to verify the proposed method and build and test the prototype. Experiments show that when distance and misalignments occur, the prototype can maintain the efficiency of around 90% with less than 5% in variations as predicted by the theoretical design. In comparison, the conventional magnetically coupled resonant WPT system can only maintain the efficiency of 82% with more than 20% in variations. Therefore, the proposed design method presents a new efficient and optimized way to develop a robust WPT system for practical applications. [ABSTRACT FROM AUTHOR]

This article presents a randomized matrix-free method for approximating the trace of f (A) , where A is a large symmetric matrix and f is a function analytic in a closed interval containing the eigenvalues of A. Our method uses a combination of stochastic trace estimation (i.e., Hutchinson's method), Chebyshev approximation, and multilevel Monte Carlo techniques. We establish general bounds on the approximation error of this method by extending an existing error bound for Hutchinson's method to multilevel trace estimators. Numerical experiments are conducted for common applications such as estimating the log-determinant, nuclear norm, and Estrada index. We find that using multilevel techniques can substantially reduce the variance of existing single-level estimators. [ABSTRACT FROM AUTHOR]

Over the last decades, there has been an increasing interest in dedicated preclinical imaging modalities for research in biomedicine. Especially in the case of positron emission tomography (PET), reconstructed images provide useful information of the morphology and function of an internal organ. PET data, stored as sinograms, involve the Radon transform of the image under investigation. The analytical approach to PET image reconstruction incorporates the derivative of the Hilbert transform of the sinogram. In this direction, in the present work we present a novel numerical algorithm for the inversion of the Radon transform based on Chebyshev polynomials of the first kind. By employing these polynomials, the computation of the derivative of the Hilbert transform of the sinogram is significantly simplified. Extending the mathematical setting of previous research based on Chebyshev polynomials, we are able to efficiently apply our new Chebyshev inversion scheme for the case of analytic preclinical PET image reconstruction. We evaluated our reconstruction algorithm on projection data from a small-animal image quality (IQ) simulated phantom study, in accordance with the NEMA NU 4-2008 standards protocol. In particular, we quantified our reconstructions via the image quality metrics of percentage standard deviation, recovery coefficient, and spill-over ratio. The projection data employed were acquired for three different Poisson noise levels: 100% (NL1), 50% (NL2), and 20% (NL3) of the total counts, respectively. In the uniform region of the IQ phantom, Chebyshev reconstructions were consistently improved over filtered backprojection (FBP), in terms of percentage standard deviation (up to 29% lower, depending on the noise level). For all rods, we measured the contrast-to-noise-ratio, indicating an improvement of up to 68% depending on the noise level. In order to compare our reconstruction method with FBP, at equal noise levels, plots of recovery coefficient and spill-over ratio as functions of the percentage standard deviation were generated, after smoothing the NL3 reconstructions with three different Gaussian filters. When post-smoothing was applied, Chebyshev demonstrated recovery coefficient values up to 14% and 42% higher, for rods 1–3 mm and 4–5 mm, respectively, compared to FBP, depending on the smoothing sigma values. Our results indicate that our Chebyshev-based analytic reconstruction method may provide PET reconstructions that are comparable to FBP, thus yielding a good alternative to standard analytic preclinical PET reconstruction methods. [ABSTRACT FROM AUTHOR]