Search

This Forum Note concerns the question of necessary optimality conditions in optimal control problems subject to state constraints. Some critical remarks about a recently published paper are made. [ABSTRACT FROM AUTHOR]

In this note, we take care of a gap in Remark 2.1 of Mishra and Laha ('On approximately star-shaped functions and approximate vector variational inequalities', Mishra and Laha in J. Optim. Theory Appl. 156:278-293, ) and modify the corresponding results. [ABSTRACT FROM AUTHOR]

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

In this paper we study the well-posedness of different models of population of leaky integrate-and-fire neurons with a population density approach. The synaptic interaction between neurons is modeled by a potential jump at the reception of a spike. We study populations that are self excitatory or self inhibitory. We distinguish the cases where this interaction is instantaneous from the one where there is a repartition of conduction delays. In the case of a bounded density of delays both excitatory and inhibitory population models are shown to be well-posed. But without conduction delay the solution of the model of self excitatory neurons may blow up. We analyze the different behaviours of the model with jumps compared to its diffusion approximation. [ABSTRACT FROM AUTHOR]

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

In this paper, to introduce approximate efficiency notions in variable ordering structures, some coradiant-valued maps are dealt with. Approximate nondominated and minimal elements are defined and some of their properties are studied. Corresponding to these concepts, necessary and sufficient conditions are provided. To obtain such conditions, some scalarization methods are investigated. The paper also investigates possible relationships between the Pascoletti-Serafini radial scalarization, approximate efficiency, approximate nondominance, and minimality utilizing some coradiant-valued maps. [ABSTRACT FROM AUTHOR]

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

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

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

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

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

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

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

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

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

In cases where multiple prices change, this paper develops methods for calculating compensating and equivalent variations from the prices, the consumption bundle, and the wealth elasticities of demand for goods. The methods are a natural extension and generalization of the method in Willig (Am Econ Rev 66(4):589-597, 1976), and have the ability to provide second-order approximations of compensating and equivalent variations. In addition, this paper considers two types of price paths: indifference price paths, along which utility levels are kept constant, and iso-price-ratio paths, along which prices change proportionally. Along these two paths, changes in consumer surplus, compensating variation, and equivalent variation are easily calculated, and that there are interesting and precise relationships among them. Furthermore, the methods and relationships above correspond to those of the Laspeyres-Konüs and Paasche-Konüs true cost of living price indices and the Divisia indices. [ABSTRACT FROM AUTHOR]

In actual applications, aggregation is an important operation to return statistical characterizations of subset of the data set. On massive data, approximate aggregation often is preferable for its better timeliness and responsiveness. This paper focuses on deterministic approximate aggregation to return running aggregate within progressive deterministic error interval. The existing methods either return approximate results with fixed accuracy, or return online approximate aggregate with probabilistic confidence interval, or incur a high I/O cost on massive data. This paper proposes LDA algorithm to compute deterministic approximate aggregate on massive data efficiently. LDA utilizes selection attribute lattice of hierarchical structure to distribute tuples and obtain a horizontal partitioning of the table. In each partition, each selection attribute is kept in column file and each ranking attribute is transposed to bit-slices. Given the selection condition, only relevant partitions are involved to compute the running aggregate. The compact storage scheme based on Z-order space filling curve is proposed to reduce the management cost of the partitions. An error reduction method is devised to reduce the error interval when computing running aggregate. The extensive experimental results on synthetic and real data sets show that LDA has a significant performance advantage over the existing algorithms. [ABSTRACT FROM AUTHOR]

This paper investigates fixed-time synchronization of coupled neural networks with discontinuous activation functions and nonidentical perturbations under the framework of Filippov solution. In order to overcome uncertainties of the Filippov solution and the effects of nonidentical perturbations, 1-norm based techniques are developed. By designing new state feedback controllers, constructing new Lyapunov functional, and utilizing differential inclusion theory, several sufficient conditions are obtained to ensure that the coupled discontinuous neural networks (CDNNs) to be synchronized in a fixed settling time. Results of this paper improve corresponding ones which only finite-time synchronization can be achieved for CDNNs. Finally, numerical simulations are offered to verify the effectiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

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

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

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

This 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]

This paper deals with variational inclusions of the form 0∈K−f(x) where f:Rn→Rm is a semismooth function and K is a nonempty closed convex cone in Rm. We show that the previous problem can be solved by a Newton-type method using the Clarke generalized Jacobian of f. The results obtained in this paper extend those obtained by Robinson in the famous paper (Robinson in Numer. Math. 19:341-347, 1972). We provide a semilocal method with a superlinear convergence that is new in the context of semismooth functions. Finally, numerical results are also given to illustrate the convergence. [ABSTRACT FROM AUTHOR]

For a given stiffness tensor (tensor of elastic moduli) of a generally anisotropic medium, we can estimate the extent to which the medium is transversely isotropic, and determine the direction of its reference symmetry axis. In this paper, we rotate the given stiffness tensor about this reference symmetry axis, and determine the reference transversely isotropic (uniaxial) stiffness tensor as the average of the rotated stiffness tensor over all angles of rotation. The obtained reference transversely isotropic (uniaxial) stiffness tensor represents an analytically differentiable approximation of the given generally anisotropic stiffness tensor. The proposed analytic method is compared with a previous numerical method in two numerical examples. [ABSTRACT FROM AUTHOR]

Some recent papers relate the criticality of complex systems to their maximal capacity of information processing. In the present paper, we consider high dimensional point processes, known as age-dependent Hawkes processes, which have been used to model spiking neural networks. Using mean-field approximation, the response of the network to a stimulus is computed and we provide a notion of stimulus sensitivity. It appears that the maximal sensitivity is achieved in the sub-critical regime, yet almost critical for a range of biologically relevant parameters. [ABSTRACT FROM AUTHOR]

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

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

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

We consider the problem of approximating a smooth function from finitely many pointwise samples using $$\ell ^1$$ minimization techniques. In the first part of this paper, we introduce an infinite-dimensional approach to this problem. Three advantages of this approach are as follows. First, it provides interpolatory approximations in the absence of noise. Second, it does not require a priori bounds on the expansion tail in order to be implemented. In particular, the truncation strategy we introduce as part of this framework is independent of the function being approximated, provided the function has sufficient regularity. Third, it allows one to explain the key role weights play in the minimization, namely, that of regularizing the problem and removing aliasing phenomena. In the second part of this paper, we present a worst-case error analysis for this approach. We provide a general recipe for analyzing this technique for arbitrary deterministic sets of points. Finally, we use this tool to show that weighted $$\ell ^1$$ minimization with Jacobi polynomials leads to an optimal method for approximating smooth, one-dimensional functions from scattered data. [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]

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

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

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

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]

A kind of adaptive sliding mode control scheme for tracking control of robot manipulator which has structured uncertainties and unstructured uncertainties is proposed in this paper. Multi-input Multi-output fuzzy logical system (FLS) is used as a compensator in the control law to compensate all the uncertainties. To reduce the number of the fuzzy rules and the burden of the computation, we design FLS based on second order approximation theorem which can approximate the uncertain function with less fuzzy rules at arbitrary precision than traditional FLS. Besides, to further reduce the number of the fuzzy rules and the amount of calculation, a new decomposed fuzzy logical system based on the decomposition of membership function is proposed. From the simulation results we can see that the control scheme and the fuzzy compensator proposed in this paper can perform fairly. [ABSTRACT FROM AUTHOR]

In this paper, we build, for the first time in the literature, a new four-stages symmetric six-step finite difference pair with optimized properties. The method:is a symmetric non-linear six-step method,is of four stagesis of fourteenth algebraic order,has eliminated the phase-lag,has eliminated the first and second derivatives of the phase-lag. An analysis of the new proposed method is given in details in this paper. More specifically, we present:the building of the new four-stages symmetric six-step method,the computation of the local truncation error of the new proposed method,the comparative local truncation error analysis of the new proposed method with other finite difference pairs of the same family.the stability and the interval of periodicity analysis andfinally, the investigation and evaluation of the computational efficiency of the new proposed scheme for the approximate solution of the Schrödinger equation. The theoretical, computational and numerical results for the new proposed method show its effectiveness compared with other known or recently obtained finite difference pairs in the literature. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of adaptive finite-time synchronization for neutral type dynamical network model with double derivative coupling. Based on the Lyapunov functional theory, inequality theorem, and adaptive control technique, some synchronization criteria of neutral type dynamical network are established by using adaptive finite-time control, which is different from the existence of finite-time synchronization of neutral type network using LMI and matrix equality constraint methods. Moreover, when the dynamics nodes of the neutral type network contain delay or no delay, the control gain of the response network are also provided. Finally, the effectiveness of the synchronization criteria proposed in this paper is verified by numerical simulations. [ABSTRACT FROM AUTHOR]

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

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

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

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

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]

The article presents an approximate method of solving direct and inverse problems described by Bernoulli-Euler inhomogeneous equation of vibrations of a beam. A semianalytical solution is approximated by a linear combination of the Trefftz functions (T-functions, solving functions), which satisfies identically the homogenous equation describing the vibrations of a beam. In the paper, the properties of the solving functions have been investigated, theorems concerning their linear independence have been formulated and proved. A method of obtaining the particular solution of the inhomogeneous equation has been shown. To get this solution, recurrent formulas enabling us to determine the inverse operator for monomials have been derived. The paper discusses two kinds of inverse problems. The first one is a boundary inverse problem, in which the boundary conditions are to be determined, based on known displacements within the area. In the second one, the load on the beam needs to be found (identification of the source). The solving functions can be used as a finite element method base functions. This approach is tested for solving inverse problems. The paper includes examples which illustrate the usefulness of the method. [ABSTRACT FROM AUTHOR]

This paper considers a large class of linear operator equations, including linear boundary value problems for partial differential equations, and treats them as linear recovery problems for functions from their data. Well-posedness of the problem means that this recovery is continuous. Discretization recovers restricted trial functions from restricted test data, and it is well-posed or stable, if this restricted recovery is continuous. After defining a general framework for these notions, this paper proves that all well-posed linear problems have stable and refinable computational discretizations with a stability that is determined by the well-posedness of the problem and independent of the computational discretization, provided that sufficiently many test data are used. The solutions of discretized problems converge when enlarging the trial spaces, and the convergence rate is determined by how well the data of the function solving the analytic problem can be approximated by the data of the trial functions. This allows new and very simple proofs of convergence rates for generalized finite elements, symmetric and unsymmetric Kansa-type collocation, and other meshfree methods like Meshless Local Petrov-Galerkin techniques. It is also shown that for a fixed trial space, weak formulations have a slightly better convergence rate than strong formulations, but at the expense of numerical integration. Since convergence rates are reduced to those coming from Approximation Theory, and since trial spaces are arbitrary, this also covers various spectral and pseudospectral methods. All of this is illustrated by examples. [ABSTRACT FROM AUTHOR]