Multivariable interpolation by weighted arithmetic means at arbitrary points

A method is considered for constructing weighted arithmetic means from IR ^m to IR, which possess the interpolation property. Necessary and sufficient conditions are then given in order to solve the convergence problem to a continuous real functionf(ξ), ξ∈IR ^m of such interpolating mean functions. Finally a few formulas are considered as an application of the proposed method. This work has been supported by the Italian Ministry of Scientific and Technological Research (M.U.R.S.T.) and the National Research Council (C.N.R.).     

Fuzzy <Emphasis Type="Italic">R</Emphasis>-subgroups with thresholds of near-rings and implication operators

Using the belongs to relation (∈) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (α, β)-fuzzy R-subgroup of a near-ring where α , β are any two of {∈, q, ∈∨q , ∈∧q} with α ≠ ∈∧q is introduced and related properties are investigated. We also introduce the notion of a fuzzy R-subgroup with thresholds which is a generalization of an ordinary fuzzy R-subgroup and an (∈, ∈∨q)-fuzzy R-subgroup. Finally, we give the definition of an implication-based fuzzy R-subgroup.     

Series expansion of weighted pseudoinverse matrices and iterative methods for calculating weighted pseudoinverse matrices and weighted normal pseudosolutions

Weighted pseudoinverse matrices are expanded into matrix power series with negative exponents and arbitrary positive parameters. Based on this expansion, iterative methods for evaluating weighted pseudoinverse matrices and weighted normal pseudosolutions are designed and analyzed. The iterative methods for weighted normal pseudosolutions are extended to solving constrained least-squares problems. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 32–62, January–February 2006.     

Methods of computing weighted pseudoinverse matrices with singular weights

Three iterative processes are constructed and investigated for computing weighted pseudoinverse matrices with singular weights and ML-weighted pseudoinverse matrices. Two of them are based on the decompositions of the weighted pseudoinverse matrix with singular weights into matrix power series, and the third is a generalization of the Schulz method to nonsingular square matrices. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 150–169, September–October, 1999.     

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. II. Singular weights

The paper reviews studies on the representations and expansions of weighted pseudoinverse matrices with positive semidefinite weights and on the construction of iterative methods and regularized problems for the calculation of weighted pseudoinverses and weighted normal pseudosolutions based on these representations and expansions. The use of these methods to solve constrained least squares problems is examined. Continued from Cybernetics and Systems Analysis, 44, No. 1, 36–55 (2008). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 75–102, May–June 2008.     

Expansion of weighted pseudoinverse matrices with positive definite weights into matrix power products. Iterative methods

Weighted pseudoinverse matrices with positive definite weights are expanded into matrix power products with negative exponents and arbitrary positive parameters. These expansions are used to develop and analyze iterative methods for evaluating weighted pseudoinverse matrices and weighted normal pseudosolutions and solving constrained least-squares problems. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 45–64, January–February 2007.     

Non-approximability of just-in-time scheduling

We consider the following single machine just-in-time scheduling problem with earliness and tardiness costs: Given n jobs with processing times, due dates and job weights, the task is to schedule these jobs without preemption on a single machine such that the total weighted discrepancy from the given due dates is minimum. NP-hardness of this problem is well established, but no approximation results are known. Using the gap-technique, we show in this paper that the weighted earliness–tardiness scheduling problem and several variants are extremely hard to approximate: If n denotes the number of jobs and b∈ℕ is any given constant, then no polynomial-time algorithm can achieve an approximation which is guaranteed to be at most a factor of O(b ⁿ ) worse than the optimal solution unless P = NP.     

On signal reconstruction in white noise using dictionaries

Assume that we observe a Gaussian vector Y = Xβ + σζ, where X is a known p × n matrix with p ≥ n, β ∈ ℝ ⁿ is an unknown vector, and ζ ∈ ℝ ⁿ is a standard Gaussian white noise. The problem is to reconstruct Xβ from observations Y, provided that β is a sparse vector.     

A fast cellular method of matrix multiplication

This paper proposes a cellular method of matrix multiplication. The method reduces the multiplicative and additive complexities of well-known matrix multiplication algorithms by 12.5%. The computational complexities of cellular analogs of such algorithms are estimated. A fast cellular analog is presented whose multiplicative and additive complexities are equal to ≈0.382n³ multiplications and ≈1.147n³ additions, respectively, where n is the order of the matrices being multiplied. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 55–59, May–June 2008.     

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. I. Positive definite weights

The paper reviews studies on the representations and expansions of weighted pseudoinverse matrices with positive definite weights and on iterative methods and regularized problems for calculation of weighted pseudoinverse matrices and weighted normal pseudosolutions. The use of these methods to solve constrained least-squares problems is examined. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 47–73, January–February 2008.     

Correlation between the gradients of the quadratic functional and its clipped prototype

Applicability of clipping of quadratic functional E = −0.5x ⁺ Tx + Bx in the minimization problem is considered (here x is the configurational vector and B ∈ R ^N is real valued vector). The probability that the gradient of this functional and the gradient of clipped functional ɛ = −0.5x ⁺ τx + bx are collinear is shown to be very high (the matrix τ is obtained by clipping of original matrix T: τ_ij = sgnT _ij ). It allows the conclusion that minimization of functional ɛ implies minimization of functional E. We can therefore replace the laborious process of minimizing functional E by the minimization of its clipped prototype ɛ. Use of the clipped functional allows sixteen-times reduction of the computation time and computer memory usage.     

Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

The paper addresses the problem of determining an outer interval solution of the parametric eigenvalue problem A(p)x = λx, A(p) ∈ ℝ^n×n for the general case where the matrix elements a_ij(p) are continuous nonlinear functions of the parameter vector p, p belonging to the interval vector p. A method for computing an interval enclosure of each eigenpair (λ_μ, x^(μ)), μ = 1, ..., n, is suggested for the case where λ_μ is a simple eigenvalue. It is based on the use of an affine interval approximation of a_ij(p) in p and reduces, essentially, to setting up and solving a real system of n or 2n incomplete quadratic equations for each real or complex eigenvalue, respectively.     

On Parabolic Boundary Layers for Convection–Diffusion Equations in a Channel: Analysis and Numerical Applications

In this article we discuss singularly perturbed convection–diffusion equations in a channel in cases producing parabolic boundary layers. It has been shown that one can improve the numerical resolution of singularly perturbed problems involving boundary layers, by incorporating the structure of the boundary layers into the finite element spaces, when this structure is available; see e.g. [Cheng, W. and Temam, R. (2002). Comput. Fluid. V.31, 453–466; Jung, C. (2005). Numer. Meth. Partial Differ. Eq. V.21, 623–648]. This approach is developed in this article for a convection–diffusion equation. Using an analytical approach, we first derive an approximate (simplified) form of the parabolic boundary layers (elements) for our problem; we then develop new numerical schemes using these boundary layer elements. The results are performed for the perturbation parameter ε in the range 10⁻¹–10⁻¹⁵ whereas the discretization mesh is in the range of order 1/10–1/100 in the x-direction and of order 1/10–1/30 in the y-direction. Indications on various extensions of this work are briefly described at the end of the Introduction.Dedicated to David Gottlieb on his 60th birthday.     

Solving Some Discrepancy Problems in NC

We show that several discrepancy-like problems can be solved in NC nearly achieving the discrepancies guaranteed by a probabilistic analysis and achievable sequentially. For example, we describe an NC algorithm that given a set system (X, S) , where X is a ground set and S⊆2 ^X , computes a set R⊆X so that for each S∈ S the discrepancy ||R S|-|R^-∩S|| is . Whereas previous NC algorithms could only achieve discrepancies with ɛ>0 , ours matches the probabilistic bound within a multiplicative factor 1+o(1) . Other problems whose NC solution we improve are lattice approximation, ɛ -approximations of range spaces with constant VC-exponent, sampling in geometric configuration spaces, approximation of integer linear programs, and edge coloring of graphs. Received June 26, 1998; revised February 18, 1999.     

Efficient MFS Algorithms for Inhomogeneous Polyharmonic Problems

In this work we develop an efficient algorithm for the application of the method of fundamental solutions to inhomogeneous polyharmonic problems, that is problems governed by equations of the form Δ ^ℓ u=f, ℓ∈ℕ, in circular geometries. Following the ideas of Alves and Chen (Adv. Comput. Math. 23:125–142, 2005), the right hand side of the equation in question is approximated by a linear combination of fundamental solutions of the Helmholtz equation. A particular solution of the inhomogeneous equation is then easily obtained from this approximation and the resulting homogeneous problem in the method of particular solutions is subsequently solved using the method of fundamental solutions. The fact that both the problem of approximating the right hand side and the homogeneous boundary value problem are performed in a circular geometry, makes it possible to develop efficient matrix decomposition algorithms with fast Fourier transforms for their solution. The efficacy of the method is demonstrated on several test problems.     

A Partial Inverse Linear-Quadratic Optimization Problem

A partial linear-quadratic problem is considered. It is formulated as the determination of the perturbation of a matrix that belongs to a control block under the constraint that the perturbation is such that an extreme value of a quadratic functional has a preassigned form in the “perturbed system” obtained. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 183–188, May–June 2005.     

Singular Linear Behaviors and Their AR-Representations

A linear behavior is defined essentially as a family of finite but sufficiently long trajectories that is invariant with respect to restriction operators. This may be singular in the sense that it may contain trajectories that cannot be continued to the right. An AR-model is defined essentially as a pair (D,R), where D is a nonsingular rational matrix and R is a full row rank polynomial matrix such that D ⁻¹ R is proper. A canonical one-to-one correspondence is established between linear behaviors and equivalence classes of AR-models. The result is a natural generalization of a well-known result due to Willems. Date received: April 19, 1999. Date revised: April 18, 2000.     

Block Toeplitz matrices and preconditioning

We study the asymptotic behaviour of the eigenvalues of Hermitiann×n block Topelitz matricesT _n , withk×k blocks, asn tends to infinity. No hypothesis is made concerning the structure of the blocks. Such matrices{T _n } are generated by the Fourier coefficients of a Hermitian matrix valued functionf∈L ², and we study the distribution of their eigenvalues for largen, relating their behaviour to some properties of the functionf. We also study the eigenvalues of the preconditioned matrices{P _n ⁻¹ T_n}, where the sequence{P _n } is generated by a positive definite matrix valued functionp. We show that the spectrum of anyP _n ⁻¹ T _n is contained in the interval [r, R], wherer is the smallest andR the largest eigenvalue ofp ⁻¹ f. We also prove that the firstm eigenvalues ofP _n ⁻¹ T_n tend tor and the lastm tend toR, for anym fixed. Finally, exact limit values for both the condition number and the conjugate gradient convergence factor for the preconditioned matricesP _n ⁻¹ T_n are computed.     

Linear Time Algorithms for Generalized Edge Dominating Set Problems

We prove that a generalization of the edge dominating set problem, in which each edge e needs to be covered b _e times for all e∈E, admits a linear time 2-approximation for general unweighted graphs and that it can be solved optimally for weighted trees. We show how to solve it optimally in linear time for unweighted trees and for weighted trees in which b _e ∈{0,1} for all e∈E. Moreover, we show that it solves generalizations of weighted matching, vertex cover, and edge cover in trees.