A Decision Procedure for Nonperiodic Sequents of the First-Order Linear Temporal Logic

A deduction-based decision procedure is presented for the nonperiodic D-sequents of the first-order linear temporal logic. The D-sequents are obtained from D ₂-sequents [7], [8] by removing the periodicity condition. The deductive procedure proposed consists of decidable deductive procedures that replace infinitary and finitary induction rules for the temporal operator ``always'. The soundness and completeness of the deduction-based decision procedure proposed is proved.     

On the Asymptotics of Trimmed Best k-Nets

Trimmed best k-nets were introduced in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413) as a robustified L_∞-based quantization procedure. This paper focuses on the asymptotics of this procedure. Also, some possible applications are briefly sketched to motivate the interest of this technique. Consistency and weak limit law are obtained in the multivariate setting. Consistency holds for absolutely continuous distributions without the (artificial) requirement of a trimming level varying with the sample size as in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413). The weak convergence will be stated toward a non-normal limit law at a O_P(n^−1/3) rate of convergence. An algorithm for computing trimmed best k-nets is proposed. Also a procedure is given in order to choose an appropriate number of centers, k, for a given data set.     

The modified bordering method to evaluate eigenvalues and eigenvectors of normal matrices

A bordering procedure is here proposed to evaluate the eigensystem of hermitian matrices, and more in general of normal matrices, when the spectral decomposition is known of then–1×n–1 principal minor. The procedure is also applicable to special real and nonsymmetric matrices here named quasi-symmetric. The computational cost to write the characteristic polynomial isO(n ²), using a new set of recursive formulas. A modified Brent algorithm is used to find the roots of the polynomial. The eigenvectors are evaluated in a direct way with a computational cost ofO(n ²) for each one. Some numerical considerations indicate where numerical difficulties may occur. Numerical results are given comparing this method with the Givens-Householder one.     

A new algorithm for digraph isomorphism

A new and efficient procedure for testing a pair of digraphs for isomorphism is developed. It is based on conducting a depth-first search on one of the digraphs followed by a systematic matching of edges using backtracking with very effective pruning. It is proved that for digraphs (ofn vertices) the expected time complexity of this procedure isO(n ^logn ). This theoretical result is verified empirically on more than 300 large random digraphs. This procedure is shown to be more efficient than any of the existing general isomorphism procedures.     

A globalization procedure for solving nonlinear systems of equations

A new globalization procedure for solving a nonlinear system of equationsF(x)=0 is proposed based on the idea of combining Newton step and the steepest descent step WITHIN each iteration. Starting with an arbitrary initial point, the procedure converges either to a solution of the system or to a local minimizer off(x)=1/2F(x) ^T F(x). Each iteration is chosen to be as close to a Newton step as possible and could be the Newton step itself. Asymptotically the Newton step will be taken in each iteration and thus the convergence is quadratic. Numerical experiments yield positive results. Further generalizations of this procedure are also discussed in this paper.     

An efficient method for calculating smoothing splines using orthogonal transformations

Summary A procedure for calculating the mean squared residual and the trace of the influence matrix associated with a polynomial smoothing spline of degree 2m–1 using an orthogonal factorization is presented. The procedure substantially overcomes the problem of ill-conditioning encountered by a recently developed method which employs a Cholesky factorization, but still requires only orderm ² n operations and ordermn storage.     

The structure of saturated critical blocks

A 2‐connected graph G is a critical block if G ? v is not 2‐connected for every vertex v ∈ V(G). A critical block G is a saturated critical block if G + e is not a critical block for any new edge e. The structure of all saturated critical blocks and a procedure for constructing every saturated critical block are determined. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 223–237, 2003     

The algebraic theory approach for ordinary differential equations: Highly accurate finite differences

This article reports further developments of Herrera's algebraic theory approach to the numerical treatment of differential equations. A new solution procedure for ordinary differential equations is presented. Finite difference algorithms of 0(h^r), for arbitrary “r” are developed. The method consists in constructing local approximate solutions and using them to extract information about the sought solution. Only nodal information is derived. The local approximate solutions are constructed by collocation, using polynomials of degree G. When “n” collocation points are used at each subinterval, G = n + 1and the order of accuracy is 0(h^2n?1). The procedure here presented is very easy to implement. A program in which n can be chosen arbitrarily, was constructed and applied to selected examples.     

Smoothing noisy data with spline functions

Summary A procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m–1 fitted ton distinct, not necessarily equally spaced or uniformly weighted, data points is presented. The procedure requires orderm ² n operations and therefore permits efficient orderm ² n calculation of statistics associated with a polynomial smoothing spline, including the generalized cross validation. The method is a significant improvement over an existing method which requires ordern ³ operations.     

On the uniform nonsingularity of matrices of search directions and the rate of convergence in minimization algorithms

A convergent minimization algorithm made up of repetitive line searches is considered in ⁿ . It is shown that the uniform nonsingularity of the matrices consisting ofn successive normalized search directions guarantees a speed of convergence which is at leastn-step Q-linear. Consequences are given for multistep methods, including Powell's 1964 procedure for function minimization without calculating derivatives as well as Zangwill's modifications of this procedure.The authors wish to thank the Namur Department of Mathematics, especially its optimization group, for many discussions and encouragement. They also thank the reviewers for many helpful suggestions.     

Order Determination for Multivariate Autoregressive Processes Using Resampling Methods

LetX₁, …, X_nbe observations from a multivariate AR(p) model with unknown orderp. A resampling procedure is proposed for estimating the orderp. The classical criteria, such as AIC and BIC, estimate the orderpas the minimizer of the function[formula]wherenis the sample size,kis the order of the fitted model, Σ²_kis an estimate of the white noise covariance matrix, andC_nis a sequence of specified constants (for AIC,C_n=2m²/n, for Hannan and Quinn's modification of BIC,C_n=2m²(ln ln n)/n, wheremis the dimension of the data vector). A resampling scheme is proposed to estimate an improved penalty factorC_n. Conditional on the data, this procedure produces a consistent estimate ofp. Simulation results support the effectiveness of this procedure when compared with some of the traditional order selection criteria. Comments are also made on the use of Yule–Walker as opposed to conditional least squares estimations for order selection.     

Estimation of Weak Lensing Parameters by Stochastic Integration

Abstract

A procedure for computing distance–redshift statistics in inhomogeneous universes is described. We interpret the generalized Dyer–Roeder equation [Dyer, C. C.; Roeder, R. C. The distance-redshift relation for universes with no intergalactic medium. Astrophys. J. 1972, 174, L115–L117; Dyer, C. C.; Roeder, R. C. Distance-redshift relations for universes with some intergalactic medium. Astrophys. J. 1972, 180, L31–L34] as a stochastic differential equation which treats the shearing forces as Brownian movement and yields a Monte-Carlo procedure. Shear distributions taken from N-body simulations produce a redshift dependent diffusion coefficient.     

Minimax approximation by rational fractions of the inverse polynomial type

Rational fractions of the formR(x)/(c ₁ +c ₂ x +c ₃ x ² + ...) ^r are useful for approximating decay type functions over infinite and semi-infinite domains. A procedure is given which produces the optimal coefficients with no more effort than for linear approximations. No initial guess is needed for the values of the coefficients nor for the maximum error of approximation.     

Factorization of a class of matrix‐functions with stable partial indices

A new effective method for factorization of a class of nonrational n × n matrix‐functions with stable partial indices is proposed. The method is a generalization of one recently proposed by the authors, which was valid for the canonical factorization only. The class of matrices being considered is motivated by their applicability to various problems. The properties and steps of the asymptotic procedure are discussed in detail. The efficiency of the procedure is highlighted by numerical results. Copyright © 2016 John Wiley & Sons, Ltd.     

The simultaneous computation of a few of the algebraically largest and smallest eigenvalues of a large,sparse, symmetric matrix

A heuristic argument and supporting numerical results are given to demonstrate that a block Lanczos procedure can be used to compute simultaneously a few of the algebraically largest and smallest eigenvalues and a corresponding eigenspace of a large, sparse, symmetric matrixA. This block procedure can be used, for example, to compute appropriate parameters for iterative schemes used in solving the equationAx=b. Moreover, if there exists an efficient method for repeatedly solving the equation (A–I)X=B, this procedure can be used to determine the interior eigenvalues (and corresponding eigenvectors) ofA closest to .     

The trigonometric triangle

A procedure is presented by which the integration of a function such as cos ⁿ (x) or sin ⁿ (x) with n = 1, 2, 3, 4 … is facilitated.     

Estimation of the arrival times of seismic waves by multivariate time series model

A computationally efficient procedure was developed for the fitting of many multivariate locally stationary autoregressive models. The details of the Householder method for fitting multivariate autoregressive model and multivariate locally stationary autoregressive model (MLSAR model) are shown. The proposed procedure is quite efficient in both accuracy and computation. The amount of computation is bounded by a multiple of Nm ² with N being the data length and m the highest model order, and does not depend on the number of models checked. This facilitates the precise estimation of the change point of the AR model. Based on the AICs' of the fitted MLSAR models and Akaike's definition of the likelihood of the models, a method of evaluating the posterior distribution of the change point of the AR model is also presented. The proposed procedure is, in particular, useful for the estimation of the arrival time of the S wave of a microearthquake. To illustrate the usefulness of the proposed procedure, the seismograms of the foreshocks of the 1982 Urakawa-Oki Earthquake were analyzed. These data sets have been registered to AISM Data Library and the readers of this Journal can access to them by the method described in this issue.A part of this research was carried out under the ISM Cooperative Research Program (89-ISM.CRP-57).Also with the Faculty of Economics, the University of Tokyo. The author was supported in part by the Japanese Ministry of Education, Science and Culture under Grant-in-Aid for Developmental Scientific Research 63830002.     

Explicit estimators under <Emphasis Type="Italic">m</Emphasis>-dependence for a multivariate normal distribution

The problem of estimating parameters of a multivariate normal p-dimensional random vector is considered for a banded covariance structure reflecting m-dependence. A simple non-iterative estimation procedure is suggested which gives an explicit, unbiased and consistent estimator of the mean and an explicit and consistent estimator of the covariance matrix for arbitrary p and m.     

A Mixed Symmetric Problem of Electroelasticity Theory for a Layer Weakened by Through-the-Thickness Tunnel Holes

A new solution procedure for spatial mixed symmetric problems of electroelasticity theory for a layer weakened by through-the-thickness holes is proposed. The boundary-value problem is reduced to a system of one-dimensional singular integral equations consisting of 4k (k = 1, 2, ...) equations. Calculation results for characteristic stresses are presented.     

An algorithm proposed for busy-period subcomponent analysis of bulk queues

This paper presents an algorithmic procedure for a busy-period subcomponent analysis of bulk queues. A component of interest for many server queues is the periodt _k to reduce congestion from a levelk to levelk-1. For anM ^(x)/M/c system with the possibility of total or partial rejection of batches, it is demonstrated that the expected length of busy periods, the proportion of delayed batch and the steady state queue length probabilities can be easily obtained. The procedure is based on the nested partial sums and monotonic properties of expected lengths of the busy periods.Formerly University of Ife.