A New view of the approximations in H v -groups

Let H is an H _v -group and the set of all finite products of elements of H. The relation β^* is the smallest equivalence relation on H such that the quotient H/ β^* is a group. The relation β^* is transitive closure of the relation β, where β is defined as follows: x β y if and only if for some . Based on the relation β, we define a neighborhood system for each element of H, and we presents a general framework for the study of approximations in H _v -groups. In construction approach, a pair of lower and upper approximation operators is defined. The connections between H _v -groups and approximation operators are examined.     

An estimate on the convergence of MKZ–Bézier operators

The pointwise approximation properties of the MKZ–Bézier operators M_n,α(f,x) for α≥1 have been studied in [X.M. Zeng, Rates of approximation of bounded variation functions by two generalized Meyer–König–Zeller type operators, Comput. Math. Appl. 39 (2000) 1–13]. The aim of this paper is to study the pointwise approximation of the operators M_n,α(f,x) for the other case 0<α<1. By means of some new estimate techniques and a result of Guo and Qi [S. Guo, Q. Qi, The moments for the Meyer–König and Zeller operators, Appl. Math. Lett. 20 (2007) 719–722], we establish an estimate formula on the rate of convergence of the operators M_n,α(f,x) for the case 0<α<1.     

Rate of convergence of nonlinear integral operators for functions of bounded variation

The aim of this paper is to study the behavior of the operators T _λ defined by

On the Decomposition of Interval-Valued Fuzzy Morphological Operators

Interval-valued fuzzy mathematical morphology is an extension of classical fuzzy mathematical morphology, which is in turn one of the extensions of binary morphology to greyscale morphology. The uncertainty that may exist concerning the grey value of a pixel due to technical limitations or bad recording circumstances, is taken into account by mapping the pixels in the image domain onto an interval to which the pixel’s grey value is expected to belong instead of one specific value. Such image representation corresponds to the representation of an interval-valued fuzzy set and thus techniques from interval-valued fuzzy set theory can be applied to extend greyscale mathematical morphology. In this paper, we study the decomposition of the interval-valued fuzzy morphological operators. We investigate in which cases the [α ₁,α ₂]-cuts of these operators can be written or approximated in terms of the corresponding binary operators. Such conversion into binary operators results in a reduction of the computation time and is further also theoretically interesting since it provides us a link between interval-valued fuzzy and binary morphology.     

Approximation of functions by means of a new generalized Bernstein operator

In this paper we first use a probabilistic method to construct a linear positive polynomial operatorL _{m, r} ^α,β Bernstein type, depending on a non-negative integer parameterr and on two real parameters α and β, such that 0≤α≤β. Then we investigate the approximation properties of this operator mapping into itself the Banach spaceC[0,1] of real-valued continuous functions on [0,1]. A special attention is accorded to the case of the operatorL _m,r=L _m,r ^0,0 . We prove that the remainder of the approximation formula of a functionfεC[0,1] byL _m,r f can be represented either by means of divided differences, or in an integral form, obtained by using a classical theorem of Peano. We give also an asymptotic estimate for this remainder. The operatorL _m,r enjoys the variation diminishing property—in the sense of I. J. Schoenberg [15]. By extending the known inequalities of T. Popoviciu [12] and G. G. Lorentz [7], we evaluate the orders of approximation in terms of the modulus of continuity of the functionf or of its derivative. In the last section of this paper we determine the point spectrum of the operatorL _m,r and , finally, we present a quadrature formula which can be constructed by means of this operator. Dedicated to Professor Aldo Ghizzetti on his 75th birthday     

Approximation of multilevel Toeplitz matrices via multilevel trigonometric matrix spaces and applications to preconditioning

In a previous paper [34] we discussed the approximation of multilevel Toeplitz matrices generated by multivariate rectangular matrix-valued continuous functions , with I=[−π,π], by means of multilevel trigonometric matrix spaces with unstructured s×t blocks and by a (multi) sequence of linear approximation operators . Here we prove some theorems about strong and weak clustering around the unity of the eigenvalues/singular values of and of other preconditioned matrices based on linear approximation operators. These results represent a very uniform tool for dealing with the preconditioning problem for large dimensions, for a variety of situations (e.g., control theory, Markov chain problems), both in the Hermitian and non-Hermitian/non-square case, by devising preconditioners in any multilevel trigonometric linear space of matrices. Received: January 1998 / Accepted: February 1999     

Optimal error estimates for linear parabolic problems under minimal regularity assumptions

We study the approximation of linear parabolic problems by means of Galerkin approximation in space and θ-method in time. The error is evaluated in norms of typeH _t ^δ (H ₁ ^x ) ⋂H _t ^δ+1/2 (L _x ² ) for |δ|≤1/2. We prove error estimates which are optimal with respect to the regularity assumptions on the right-hand side of the equation. Dedicated to Professor Aldo Ghizzetti on his 75th birthday Istituto di Analisi Numerica del C.N.R.     

Improved Approximation Algorithms for Maximum Resource Bin Packing and Lazy Bin Covering Problems

In this paper, we study two variants of the bin packing and covering problems called Maximum Resource Bin Packing (MRBP) and Lazy Bin Covering (LBC) problems, and present new approximation algorithms for them. For the offline MRBP problem, the previous best known approximation ratio is \frac65\frac{6}{5} (=1.2) achieved by the classical First-Fit-Increasing (FFI) algorithm (Boyar et al. in Theor. Comput. Sci. 362(1–3):127–139, 2006). In this paper, we give a new FFI-type algorithm with an approximation ratio of \frac8071\frac{80}{71} (≈1.12676). For the offline LBC problem, it has been shown in Lin et al. (COCOON, pp. 340–349, 2006) that the classical First-Fit-Decreasing (FFD) algorithm achieves an approximation ratio of \frac7160\frac{71}{60} (≈1.18333). In this paper, we present a new FFD-type algorithm with an approximation ratio of \frac1715\frac{17}{15} (≈1.13333). Our algorithms are based on a pattern-based technique and a number of other observations. They run in near linear time (i.e., O(nlog n)), and therefore are practical.     

Convergence speed estimates for the norms of the inverses of large truncated Toeplitz matrices

Given a continuous function a on the complex unit circle, let T(a) denote the infinite Toeplitz matrix generated by a and let T _n (a) stand for the (n+1)×(n+1) principal section of T(a). We think of T(a) and T _n (a) as operators on l ² spaces. A classical result by Gohberg and Feldman says that if T(a) is invertible, then so is T _n (a) for all sufficiently large n≥n ₀ and . Only in 1994 did we realize that in fact . In this paper, we provide estimates for the speed with which converges to . We prove that in the “generic case” the convergence speed can be estimated by the smoothness of a, whereas in some “exceptional cases” (e.g., if T(a) is Hermitian or triangular) it is not the smoothness of $a$ but the orders of certain zeros which determine the convergence speed. Some of the results are extended to operators on l ^p spaces. Received: September 1998 / Accepted: November 1998     

Numerical solution of parabolic problems by the generalized Crank-Nicolson scheme

A Galerkin procedure is used to obtain a semi-discretization for parabolic equations such as the heat equationu _t =t _xx . The time variable being left continuous, the higher order approximation thus obtained for the space variable is then matched by a higher order discretization of the system of ordinary differential equations that results. Specifically we choose the Padé (2,2), and show how complex factorization it can be practically used. Moreover we prove that the operation count is 0 (h ⁻²) as compared to 0(h ⁻³) with the classical Crank-Nicolson. Numerical calculations are available. This work was supported by the Office of Naval Research, and the Lebanese Council for Scientific Research.     

Lower estimation of approximation rate for neural networks

Let SFd and Πψ,n,d = { nj=1bjψ(ωj·x+θj) :bj,θj∈R,ωj∈Rd} be the set of periodic and Lebesgue’s square-integrable functions and the set of feedforward neural network (FNN) functions, respectively. Denote by dist (SF d, Πψ,n,d) the deviation of the set SF d from the set Πψ,n,d. A main purpose of this paper is to estimate the deviation. In particular, based on the Fourier transforms and the theory of approximation, a lower estimation for dist (SFd, Πψ,n,d) is proved. That is, dist(SF d, Πψ,n,d) (nlogC2n)1/2 . T...     

The δ-process for acceleration of outer iterations in reactor problems

A new δ-process method is proposed and justified for the acceleration of outer iterations in reactor problems of the eigenvalue (K _eff) calculation in a multigroup approximation. It is proved that the δ-process is asymptotically equivalent to Newton’s method. To investigate the efficiency of this method, the initial state of critical assembly BZD/1 in the ZEBRA experiments is computed in approximation of the discrete ordinates method in X-Y-Z geometry with acceleration for the different value of parameter δ in the interval (0, 1). The best acceleration factor of 3 was obtained in the S ₈ P ₃-approximation for the value δ = 0.8.     

Semigroup generation and stabilization by A-bounded feedback perturbations

Let A be a generator of a strongly continuous semigroup of operators, and assume that C and H are operators such that A + CH generates a strongly continuous semigroup S_H(t) on X. Let λ₀ be a real number in the resolvent set of A, and let ε [−1, 1]. Then there are some fairly unrestrictive conditions under which A+(λ₀ − A)CH(λ₀ − A)⁻ also generates a strongly continuous semigroup S_K(t) on X which has the same exponential growth rate as S_H(t). Given an input operator B, we can use this to identify a class of feedback perturbations K such that A + BK generates a strongly continuous semigroup. We can also use this result to identify classes of feedbacks which can and cannot uniformly stabilize a system. For example, we show that if the control on a cantilever beam in the state space H₀²[0, 1] × L²[0, 1] is a moment force on the free end, then we cannot stabilize the beam with an A^−1/2-bounded feedback, but we can find an A^−1/4-bounded feedback, for any > 0, which does stabilize the beam.     

Tighter Approximation Bounds for Minimum CDS in Unit Disk Graphs

Connected dominating set (CDS) in unit disk graphs has a wide range of applications in wireless ad hoc networks. A number of approximation algorithms for constructing a small CDS in unit disk graphs have been proposed in the literature. The majority of these algorithms follow a general two-phased approach. The first phase constructs a dominating set, and the second phase selects additional nodes to interconnect the nodes in the dominating set. In the performance analyses of these two-phased algorithms, the relation between the independence number α and the connected domination number γ _c of a unit-disk graph plays the key role. The best-known relation between them is a ￡ 3\frac23g_c+1\alpha\leq3\frac{2}{3}\gamma_{c}+1. In this paper, we prove that α≤3.4306γ _c +4.8185. This relation leads to tighter upper bounds on the approximation ratios of two approximation algorithms proposed in the literature.     

The essential order of approximation for nearly exponential type neural networks

1 Introduction Artificial neural networks have been extensively applied in various fields of science and engineering. Why is so is mainly because the feedforward neural networks (FNNs) have the universal approximation capability[1-9]. A typical example of…     

Maps and isometries between indistinguishability operators

 In this paper, some geometric aspects of indistinguishability operators are studied by using the concept of morphism between them. Among all possible types of morphisms, the paper is focused on the following cases: Maps that transform a T-indistinguishability operator into another of such operators with respect to the same t-norm T and maps that transform a T-indistinguishability operator into another one of such operators with respect to a different t-norm T ^′. The group of isometries of a given T-indistinguishability operator is also studied and it is determined for the case of one-dimensional operators, in particular for the natural indistinguishability operators E _T on [0, 1]. Finally, the indistinguishability operators invariant under translations on the real line are characterized.     

A Note on the Spectral Collocation Approximation of Some Differential Equations with Singular Source Terms

The solution of differential equations with singular source terms contains the local jump discontinuity in general and its spectral approximation is oscillatory due to the Gibbs phenomenon. To minimize the Gibbs oscillations near the local jump discontinuity and improve convergence, the regularization of the approximation is needed. In this note, a simple derivative of the discrete Heaviside function H _c (x) on the collocation points is used for the approximation of singular source terms δ(x−c) or δ ⁽ⁿ⁾(x−c) without any regularization. The direct projection of H _c (x) yields highly oscillatory approximations of δ(x−c) and δ ⁽ⁿ⁾(x−c). In this note, however, it is shown that the direct projection approach can yield a non-oscillatory approximation of the solution and the error can also decay uniformly for certain types of differential equations. For some differential equations, spectral accuracy is also recovered. This method is limited to certain types of equations but can be applied when the given equation has some nice properties. Numerical examples for elliptic and hyperbolic equations are provided. The current address: Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, USA.     

Image Compression and Reconstruction Using pi<Subscript><Emphasis Type="Italic">t</Emphasis></Subscript>-Sigma Neural Networks

A high-order feedforward neural architecture, called pi _t -sigma (π _t σ) neural network, is proposed for lossy digital image compression and reconstruction problems. The π _t σ network architecture is composed of an input layer, a single hidden layer, and an output layer. The hidden layer is composed of classical additive neurons, whereas the output layer is composed of translated multiplicative neurons (π _t -neurons). A two-stage learning algorithm is proposed to adjust the parameters of the π _t σ network: first, a genetic algorithm (GA) is used to avoid premature convergence to poor local minima; in the second stage, a conjugate gradient method is used to fine-tune the solution found by GA. Experiments using the Standard Image Database and infrared satellite images show that the proposed π _t σ network performs better than classical multilayer perceptron, improving the reconstruction precision (measured by the mean squared error) in about 56%, on average.     

ℓ 2 2 Spreading Metrics for Vertex Ordering Problems

We design approximation algorithms for the vertex ordering problems Minimum Linear Arrangement, Minimum Containing Interval Graph, and Minimum Storage-Time Product, achieving approximation factors of $O(\sqrt{\log n}\log\log n)We design approximation algorithms for the vertex ordering problems Minimum Linear Arrangement, Minimum Containing Interval Graph, and Minimum Storage-Time Product, achieving approximation factors of O(?{logn}loglogn)O(\sqrt{\log n}\log\log n) , O(?{logn}loglogn)O(\sqrt{\log n}\log\log n) , and O(?{logT}loglogT)O(\sqrt{\log T}\log\log T) , respectively, the last running in time polynomial in T (T being the sum of execution times). The technical contribution of our paper is to introduce “ℓ ₂² spreading metrics” (that can be computed by semidefinite programming) as relaxations for both undirected and directed “permutation metrics,” which are induced by permutations of {1,2,…,n}. The techniques introduced in the recent work of Arora, Rao and Vazirani (Proc. of 36th STOC, pp. 222–231, 2004) can be adapted to exploit the geometry of such ℓ ₂² spreading metrics, giving a powerful tool for the design of divide-and-conquer algorithms. In addition to their applications to approximation algorithms, the study of such ℓ ₂² spreading metrics as relaxations of permutation metrics is interesting in its own right. We show how our results imply that, in a certain sense we make precise, ℓ ₂² spreading metrics approximate permutation metrics on n points to a factor of O(?{logn}loglogn)O(\sqrt{\log n}\log\log n) .