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In this paper, we propose a novel model to restore an image corrupted by blur and Cauchy noise. The model is composed of a data fidelity term and two regularization terms including total variation and high-order total variation. Total variation provides well-preserved edge features, but suffers from staircase effects in smooth regions, whereas high-order total variation can alleviate staircase effects. Moreover, we introduce a strategy for adaptively selecting regularization parameters. We develop an efficient alternating minimization algorithm for solving the proposed model. Numerical examples suggest that the proposed method has the advantages of better preserving edges and reducing staircase effects.

A kind of robust control of electromechanical actuator (EMA) system for thrust vector control in a spacecraft was investigated. In the flight of a spacecraft, the EMA system must overcome the influence of load disturbance and working point alteration to improve the robust control performances. Addressing this problem and considering the large inertia and low stiffness load of the EMA system, this paper studied the robust control technology of the EMA system by H"~ control and @m synthesis theory based on the degree of freedom (DOF) mathematical model. The H"~ hybrid controller was designed and the EMA system experiments of three controllers including proportion-integral-differential controller, H"~ controller and H"~ hybrid controller were conducted. Furthermore, the @m synthesis controller was designed and the simulation was made on the @m synthesis controller and the comparison between the control effects of the H"~ controller and the @m synthesis controller was conducted. The comparison results show that the robust synthesis control can more effectively handle the parameters perturbation and load disturbance.

In this paper we study the Frechet differentiability of the objective functional for a quite general class of coefficient inverse problems. We present sufficient conditions for the existence of arbitrary-order differentials as well as formulae for the calculation of first and second order derivatives. The general case theorems are then applied to two real-world geophysical inverse problems.

We have for some time studied lattice solutions within the nonintegrable Aesthetic Field Theory. These lattices arise when we specify an integration path. In this paper, we study small deviations from the lattice structure for certain three space-time dimensional systems. We find that x locations within a family of minima differ from the lattice in a sinusoidal-looking way. The motion of the maxima (minima) was found to be a simple harmonic motion in both x and y . In addition, we found that the magnitude of the maxima (minima) within a family varies slightly but in a sinusoidal way. The magnitude for a representative maximum (minimum) varies slightly in time, again in a sinusoidal-looking way. Such systems are referred to as dynamical lattices. The conclusion we reach is that dynamical lattices exist within the nonintegrable Aesthetic Field Theory, although the dynamical lattice system discussed is not a general one (for example, the y separation between members of a family appear constant). Difficulties in obtaining a more general dynamical lattice are illustrated by an example. A study is made of how the integration scheme associated with nonintegrable systems discussed in the latter part of [1] affects the dynamical lattice; y locations for members of a family deviate from the ladder symmetry (obtained when we are dealing with simple lattices) and show a sinusoidal-looking dependence. Magnitudes for maxima (minima) within a family also exhibit a sinusoidal behavior. The integration scheme affects the motion of the maxima (minima) in a rather dramatic way. We find examples of maxima (minima) that cannot be followed in time for as long as we wish as they appear and disappear.

The uniform asymptotic expansion of the modified Bessel function of the first kind lv(vz), except for an auxiliary multiplier, is in the form ∑ k = 0 ∞ U k ( u ) / v k , u = ( 1 + z 2 ) − 1 / 2 ⋅ u − k U k ( u ) is a polynomial in u2 of degree k and Uk(u) is determined by use of a recurrence equation which is in the form of a difference-differential-integral equation. As a consequence these coefficients are not easily evaluated and the calculation becomes more cumbersome as k increases. In this paper, we show that the above series can be rearranged as ∑ k = 0 ∞ ( − ) k c k ( v ) u k where ck(v) satisfies a difference equation of order three, but there are only three terms as the coefficient of one of the terms is always zero. We also show how Kv(vz), the Hankel functions, etc. can be handled in a similar fashion.

The following problem was raised by H.-J. Stoss in connection with certain questions related to the complexity of Boolean functions. An acyclic directed graph G is said to have property P(m,n) if for any set X of m vertices of G, there is a directed path of length n in G which does not intersect X. Let f(m,n) denote the minimum number of edges a graph with porperty P(m,n) can have. The problem is to estimate f(m,n). For the remainder of the paper, we shall restrict ourselves to the case m = n. We shall prove (1) $c_1$n log n/log log n > f(n,n) > $c_2$n log n (where $c_1$,$c_2$,..., will hereafter denote suitable positive constaints). In fact, the graph we construct in order to establish the upper bound on f(n,n) in (1) will have just $c_3$n vertices. In this case the upper bound in (1) is essentially best possible since it will also be shown that for $c_4$ sufficiently large, every graph on $c_4$n vertices having property P(n,n) must have at least $c_5$n log n edges.

can be determinedusing interpolating polynomials. Although integral quadratures with a variety of interpolatingpolynomials are fully developed, the differential quadratures are still in an early stage ofdevelopment. Differentialquadrature hasobviousapplicationsinthe numericalsolutionof partialdifferential equations. ordinary differential equations withboundary conditions and identificationproblems. In this paper, we deal mostly with the former case and refer the reader to [1-3Jfor thediscussions of other applications.Ingeneral, the interpolationformulas are expressed intwoways,either intermsof differencesof the function or in terms of values of the function. The derivatives of a function can also beexpressed in the same mannerin both cases. The firstcase, i.e. the approximationof derivativeswith differences, provides the basis of many methods of solving differential equations (thedifference methods). This approximation can also be used for the estimation of errors innumerical differentiation. However, the application of the second case (i.e. approximatingthederivatives in terms of the values of the function) for the purpose of numerical differentiationwas hastily dropped, mainly due to the hazards of the roundoff errors in calculation, especiallywhen dealing with low accuracy empirical data. A better approach used for numericaldifferentiation was to first "smooth" the data and then to differentiate[4J. In the differentialquadrature method we use the second approximationfor solutionof partialdifferentialequations,and effectively smooth the "data" by employing spline interpolation for determination ofcoeffi cient

This paper advances a procedure to compute the a-posteriori error bounds for the solution of a wide class of ordinary differential equations with given initial conditions. The method is based on an iterative scheme which yields upper and lower bounds which approach each other. The convergence of these iterations is proved analytically. Application of the proposed method is demonstrated by several examples. The method is independent of the integration scheme used. Also, separate bounds for each component of the solution are specifically available as a function of time compared to the bound on the norm yielded by conventional methods.

A class of cubic spline estimators of probability density functions over a finite interval are considered in this paper. The precise asymptotic behavior of the bias and covariance of such estimators is obtained in the interior of the interval. The estimators are shown to be asymptotically normally distributed. The properties of these estimators are compared with those of kernel estimators. The kernel and spline estimators are compared in some Monte Carlo simulations as well as in the analysis of some data obtained in turbulent wind flow.

Let 0 b ≤ a c be integers. Two players play alternately with a pile of stones. Each player at his turn selects one move from the following two: (i) Remove k stones from the pile subject to 1 ≤ k ≤ a or c + 1 ≤ k ≤ c + a . (ii) If the number m of stones in the pile satisfies m ≡ 2 b (mod 2 a ), add a stones to the pile. The player making the last move wins. If there is no last move, the game is a (dynamic) tie. The Generalized Sprague—Grundy function G is determined, thus giving the strategy of play for the game and its disjunctive compound. An algorithm requiring O( a 2 ) steps for computing G is given. It turns out that G = G (a, b, c) is of a rather complicated form. The main interest of the paper is in presenting a complete strategy for a class of games with dynamic ties.

This paper deals with the maximization of linearly constrained positive polynomials ( posynomials ). While posynomials are not necessarily convex, they can be transformed to convex functions and thus the numerical maximization of such functions may result in the location of a local solution. We present a characterization of the global maximum for linearly constrained posynomials in terms of an associated nonlinear convex program. Also, we develop a hybrid algorithm based on a cutting plane and a modified gradient projection algorithm which guarantees the global maximum solution to this class of problems.

This paper is a continuation of the study of classification learning algorithms generated by regularization schemes associated with Gaussian kernels and general convex loss functions. In previous papers Xiang and Zhou (2009) [5], Xiang (2010) [7], it is assumed that the convex loss @f has a zero. This excludes some useful loss functions without zero such as the logistic loss @?(t)=log(1+exp(-t)). The main purpose of this paper is to conduct error analysis for the classification learning algorithms associated with such loss functions. The learning rates are derived by a novel application of projection operators to overcome the technical difficulty.

Classifying large quantities of multidimensional data (e.g. remotely sensed agricultural data)[1] requires efficient and effective classification techniques and the construction of certain transformations of a dimension-reducing, information-preserving nature. This paper will deal with the construction of transformations that minimally degrade information (i.e. class separability). We will only consider the construction of linear dimension-reducing transformations for multivariate normal populations and information content will be measured by divergence[2].

In this paper we develop a number of probabilistic results related to a combinatorial representation of the real number system. This representation employs an algorithmic definition of the arithmetic operations analogous to that used by a computer. A carry function for each place is defined and the distribution of these functions is characterized in terms of classical combinatorial polynomials.

This paper presents an efficient computational technique for the construction of the exact-state capacity levels attainable from a set of generators operating in parallel. This method used both the recursive model found in most of the literature on the loss-of-load and frequency-and-duration methods of reliability analysis, and also a method that uses the binomial formula. Here, the binomial method is used for construction of those exact-capacity states due to identical generators, or batches of identical generators. These states may then be joined with those constructed by applying the recursive approach to the distinct units, which yields the final set of capacity states

Interpolation methods fit a model to a given objective function by evaluating the objective function at, say, M points of a grid. If the model has, say, N independent coefficients which have to be determined, they are found by solving a set of M linear simultaneous equations in N unknowns.In this paper the effect on these methods is tested of enlarging the size of the grid (M) to include more than N points. Numerical results show that the optimal data grid size tends to occur when M = N.

This paper describes an exact method for computer generation of random variables with a gamma distribution. The method is based on the Wilson-Hilferty transformation and an improvement on the rejection technique. The idea is to “squeeze” a target density between two functions, the top one easy to sample from, the bottom one easy to evaluate.

In this paper an implicit enumeration algorithm is presented for solving the general integer programming problem. The algorithm is a generalization of algorithms of Balas and Geoffrion for solving 0-1 problems. In order that each variable may be treated directly, a special data structure has been designed. Based on this data structure, a procedure is developed which efficiently keeps track of the status of the algorithm and of the potential actions to be taken at any stage in the procedure. The concept of a ‘best’ surrogate constraint is extended and incorporated into the algorithm. The algorithm can be modified for solving mixed integer programming problems.

In a paper by Melcher[6] the stabilization of a vibrating string has been studied from a theoretical point of view. The purposes of this paper are to study the same problem from both mathematical and numerical points of view and, furthermore, to look at stabilization and control problems. The main mathematical and numerical interests of this problem lie in the presence of a Dirac measure in the wave equation.

Let ψ be a continuous function from Mnk (the class of all rank k k × n matrices) into R1 that is invariant under multiplication on the left by k × k invertible matrices. Then there exists a matrix H1 from the class Hn of Householder transformations such that ψ([Ik∣Z]H1=l.u.b.H∈Hn{ψ([Ik∣Z]H)}. Now for each positive integer i, let the element Hi ∈Hn be chosen such that ψ([Ik∣Z∣Hi⋅Hi−1⋯H1)=l.u.b.H∈Hn{ψ([Ik∣Z]H⋅Hi−1⋯H1The question of whether or not the sequence of matrices {[Ik|Z]Hi…H1} converges to an absolute ψ-extremum (rank k maximal statistic) appeared in [1]. In this paper, we show that there exists a function ψ as above for which the above sequence does not converge to an absolute ψ-extremum.

In this paper a new parallel algorithm based on interlocking matrix quadrant factors similar to the Rutishauser LR method is presented. After a series of similarity transformations the eigenvalues are obtained from simple (2 × 2) linear systems derived from the principal and secondary diagonals of the limit matrix.

In this paper there is shown (for a model problem) that the consistence error of a finite element method—which is based on noncompatible trial functions—disappears, if the underlying variational principle is extended to the noncompatible trial function set.

The square-root iteration as presented by Ostrowski ([1], p. 110) for the real case can be extended to determine simultaneously all simple roots of a polynomial in the complex plane. The use of circular arithmetic[2] allows to establish the corresponding algorithm in terms of disks with automatic evaluation of error bounds for each approximation. As shown in [3], the convergence of the successive iterates to a zero is of order four. This paper analyses the algorithm given in [3] from the standpoint of the numerical stability.

This paper describes the implementation of rational approximation algorithms for evaluation of kth roots in short wordlength machines. The emphasis is on maintaining full machine precision in computers that use fixed point, truncated binary arithmetic with at most 16 bits of wordlength. Included is a table of coefficients for evaluation of kth roots on a 16 bit machine with 3 ≤ k ≤ 11.

Informal mathematical reasoning has a strong metamathematical component, which is used to expand the rules of proof once theorems which (informally) justify such expansion have been proved. For use of mechanised proof verifier systems to remain comfortable over a wide range of applications, they will have to include corresponding mechanisms. This paper formalizes these mechanisms, and also shows how verifier systems can be expanded by the progressive compilation of additional internal routines but without loss of logical soundness.

This paper concerns the construction of a quadrilateral finite element whose interpolation space admits of rational fractions for basis functions of “Wachspress type” [1, 2]. The construction of this finite element, which is in a way the “rational” equivalent of the ADINI finite element[3, 4], is founded on a method analogous to the one used for Serendip degree-two finite element construction in[2]. The study of interpolation error is dealt with in a paper by Apprato, Arcangeli and Gout in this journal “Rational interpolation of Wachspress error estimates”.

Multhopp's subsonic lifting surface theory includes the calculation of a singular double integral of the induced velocity, using Mangler's second order principal value. In order to make the method efficient, it is necessary to use some approximations to describe the integrand in the neighborhood of the singularity. The present paper presents a higher order approximation, relative to previous investigations. The development of the new approximation includes a discussion of certain mathematical problems that emerge during the derivation. The model can deal with lifting surfaces having arbitrary planform, twist and camber. The advantages of using the higher order approximation are illustrated by a few examples.

This paper describes a class of variable penalty methods for solving the general nonlinear programming problems. The algorithm poses a sequence of unconstrained optimization problems with mechanisms to control the quality of the approximation for the Hessian matrix. The Hessian matrix is proposed in terms of the constraint functions and their first derivatives. The unconstrained problems are solved using a modified Newton's algorithm. The convergence of the method is accelerated by choosing variable penalty function parameters which in a given constraint environment, during an unconstrained minimization process, best control the error in the approximation of the Hessian matrix. Several possibilities for obtaining such parameters are discussed. The numerical effectiveness of this algorithm is demonstrated on a relatively large set of test problems.

In the present paper an asymptotic solution of initial problem for disturbances of laminar flow obtained in [8] is employed for analysis of autooscillations.Numerical calculation has shown that there are two autooscillating regimes at values of the wave number α with the range 0.9649 ⩽ α ⩽ 1.0765 in subcritical region. The first regime is a well-known unstable limit cycle, and the second one is a stable autooscillating regime coming from the beyond-critical region. Cycles interflow at some critical value of the supercriticity parameter δ1. The nonlinear critical value of Reynolds number RH(α) corresponds to this value.All disturbances are quenching at values of α, R lying to the left of nonlinear neutral curve. There exist critical amplitudes A1(α, R) in the region between nonlinear and linear neutral curves. Increasing disturbances are stabilized at amplitude values equal to A2(α, R). Within the linear neutral curve all disturbances increase up to the amplitude A2(α, R).

It is our contention that the recently developed Remes-difcor algorithm is the best general purpose algorithm available for computing best uniform rational approximations, and should be widely used as a library routine. The purpose of this paper is to support this contention by theoretical arguments and by a numerical comparison of the Remes-difcor algorithm, the differential correction algorithm, and the widely-used Remes algorithm. The Remes-difcor algorithm is shown to be more robust than the Remes algorithm and faster than the differential correction algorithm. The three algorithms are briefly described and discussed, and the experimental results for 70 examples are presented in six tables.

A new fourth order method using quintic polynomials is designed in this paper for the smooth approximation of the two point boundary value problems involving second order differential equations lacking the first derivative. The present method enables us to approximate the unknown function as well as its derivative at every point of the range of integration and thus it has obvious advantages over other discrete numerical methods. Our present method outperforms the well-known fourth order Noumerov's finite difference scheme. The convergence of the method is briefly outlined using matrix algebra and two numerical illustrations are provided to demonstrate the practical suitability of our approach.

In this paper we obtain the best possible constants in three inequalities between two vectors. The inequalities depend on a parameter p . These inequalities were used to bound the error in the finite element approximation of a nonlinear diffusion problem.

Frequently the need arises for the computer generation of variates that are exactly distributed as Z = max(X1, …, Xn) where X1, …, Xn form a sequence of independent identically distributed random variables. For large n the individual generation of the Xi's is unfeasible, and the inversion-of-a-beta-variate is potentially inaccurate. In this paper, we discuss and compare the corrected inversion method, the log(n)/n-tail method and the record time method. The latter two methods have an average complexity 0(log(n)), are very accurate and do not require the inversion of a distribution function. The normal, exponential and gamma densities are treated in detail. The existence of fast and accurate inversion methods for the error function makes the corrected inversion method faster than the other ones for n large enough when the Xi's are normal random variables.

In this paper, we first present an impulsive version of the Filippov–Wazewski theorem and a continuous version of the Filippov theorem for fractional differential inclusions of the form D∗αy(t)∈F(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c, where J=[0,b],D∗α denotes the Caputo fractional derivative, and F is a set-valued map. The functions Ik,I¯k characterize the jump of the solutions at impulse points tk(k=1,…,m). Additional existence results are obtained under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on the nonlinear alternative of Leray–Schauder type, a Bressan–Colombo selection theorem, and Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of the solution set is also investigated. Finally, some geometric properties of solution sets, Rδ sets, acyclicity and contractibility, corresponding to Aronszajn–Browder–Gupta type results, are obtained. We also consider the impulsive fractional differential equations D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=Ik(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c, and D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(0,1],y(tk+)=Ik(y(tk−)),k=1,…,m,y(0)=a, where f:J×R→R is a single map. Finally, we extend the existence result for impulsive fractional differential inclusions with periodic conditions, D∗αy(t)∈φ(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=y(b),y′(0)=y′(b), where φ:J×R→P(R) is a multivalued map. The study of the above problems use an approach based on the topological degree combined with a Poincare operator.

WOS: 000278887600028, In this paper, we investigate the asymptotic behavior and periodic nature of positive solutions of the difference equation x(n) = max {A/x(n-1), 1/x(n-2)(alpha)}, n >= 0, where A >= 0 and 0

Methods of maximal monotone operators are used in order to study, from a general point of view, duality numerical algorithms for solving variational inequalities. With classical algorithms, such as Uzawa's method for the standard and augmented Lagrangian, this paper presents some new algorithms, which appear to have very good numerical performances.

In this paper, the application of preconditioning to improve the convergence rates of iterative methods for solving large sparse linear systems of finite difference equations arising from the discretisation of elliptic partial differential equations of self adjoint form on a rectangular grid is extended to the alternating direction implicit (A. D. I) methods. Theoretical results are presented for the model problem.

In this paper, parallel algorithms are proposed for solving both systems of nonlinear algebraic equations and unconstrained optimization problems. Theoretical results for their convergence and orders of convergence are also presented. The proposed algorithms are analytically compared in terms of their time efficiencies and speed-up ratios.

Vincent's theorem of 1836, which was only recently discovered by the author of this article, is of extreme importance because it constitutes the basis of the fastest method existing for the isolation of the real roots of a polynomial equation (using exact integer arithmetic). In this paper this forgotten theorem is presented both in its original form and in an extended version, and is followed by a general discussion of its application.

In this paper the stock-out probabilities are computed at each depot in a parallel multi-installation inventory system operating under a periodic review joint ordering policy in which demand follows an Erlangian distribution. A recurrence relation is established for deriving expressions for the Laplace transform of certain “weighting factors” which appear in the expression for the stock-out probabilities. In addition a recursive scheme is developed for expanding into partial fractions rational functions having in their denominator linear factors with several multiple zeros. These two schemes form the basis for a com- putational efficient numerical procedure for calculating the required probabilities. Some of the computer output results are exhibited in graphical form.

Consider n independent uniform (0, 1) random variables, and let N1, …, Nn be the cardinalities of the intervals [ (i - 1) n) , (i n) ], 1 ≤ i ≤ n . Then E( max N 1 )∼ ( log n log log 4 ) as n → ∞ . This result (proved in the paper) and related results about the asymptotical behavior of E(g( max i N i )) for increasing functions g allow us to draw some conclusions about the average complexity of some bucketing algorithms in computational geometry. We illustrate this point by showing that Shamos' unpublished bucketing algorithm for finding the convex hull of n independent identically distributed random vectors X1 … Xn in R2 has an average complexity 0(n) whenever the Xi's have a bounded density with compact support.

Classically the solution of many functional equations is sought in the form of an infinite series of some kind. In practice, to get numerical results, the series must be truncated. In recent times, alternative finite series approximations have been sought as solutions of a functional equation slightly perturbed from the given one. This concept was first introduced by Lanczos who named it the τ-method. In previous studies the author has applied the τ-method to get polynomial and rational approximations for differential equations mostly of hypergeometric type. In the present paper, we apply the τ-method concept to get like approximations for solutions of difference equations. In particular we study the second order difference equation satisfied by the Bessel function of the first kind. Except for some recent work on the gamma function by the author, Wimp and Ting, the literature is void on use of the τ-method idea for solutions of difference equations.

In this paper, the authors prove several coefficient bounds, distortion inequalities of the class Sgn(p,λ,b,β) of p-valent analytic functions of complex order. By making use of the familiar concept of neighborhoods of p-valent analytic functions, they obtained several inclusion relations associated with the Np(n,δ)-neighborhood of this class and its certain derivatives, which is defined by means of a certain non-homogeneous Cauchy–Euler differential equations.

The convex hull of X1,…,Xn, a sample of independent identically distributed Rd-valued random vectors with density f is called a random convex hull with parameters f and n. In this paper, we give an algorithm for the computer generation of random convex hulls when f is radial, i.e. when f(x)=g(‖x‖) for some function g. Then we look at the average time E(T) of the algorithm under a convenient computational mode. We consider only d=2.We show that for any f, our algorithm takes average time ω(log n). This lower bound is achieved for all radial densities with a polynomially decreasing tail. For the radial densities with an exponentially decreasing tail, we show that E(T)=0(log3/2n). Finally, for the uniform density on the unit circle, we have E(T)=0(n13log23n). This rate is also shown to be optimal for this density.

In the paper the relationship between two formulations of the Signorini's Problem is investigated. The first one is based on the minimum-potential energy theorem, and is expressed in terms of displacements. In the second one, the so called “reciprocal formulation”, the unknown function appears to be a contact pressure. Based on Ekeland-Temam's results concerning the duality theory the equivalance of both formulations is shown. For the sake of some numerical experience the simple example of a beam supported on elastic supports is considered. Numerical results fully confirm the theoretical investigations.

The paper deals with certain boundedness properties of Runge-Kutta-Rosenbrock methods when applied to nonlinear stiff systems. It reports some instructive examples and numerical experiments performed with a number of simple 2-stage schemes and the Rosenbrock code ROW4A. Attention is paid to the conversion of non-autonomous problems to the autonomous form. An important conclusion is that this conversion may lead to a significant loss in accuracy.

In this paper by a constructive method we show that the node connectivity of a hypercube of dimension n and base r is 2 n . Other interesting properties of the graph are also presented.

The paper describes a hybrid variable penalty method (in which the penalty functions are finite on the boundary of the constrained set) for solving a general nonlinear programming problem. The method combines two types of variable penalty functions to obtain a hybrid formulation in such a way that the error in the approximation of the Hessian matrix resulting from using only the first derivatives of the constraints, are minimized both in the feasible and infeasible domains. The hybrid algorithm poses a sequence of unconstrained optimization problems with mechanism to control the quality of the approximation for the Hessian matrix. The unconstrained problems are solved using a modified Newton's algorithm. The hybrid method is found to exhibit the following characteristics: (i) Admit small values of the penalty weight r to start SUMT (i.e. r = 1 × 10−3 or smaller) with no apparent ill-conditioning; (ii) Stimulate faster rate of convergence for a single r; (iii) Permit feasible or infeasible initial starting points.The numerical effectiveness of the algorithm is demonstrated on a relatively large set of test problems.

The generic term fuzzy quantifier is employed in this paper to denote the collection of quantifiers in natural languages whose representative elements are: several, most, much, not many, very many, not very many, few, quite a few, large number, small number, close to five, approximately ten, frequently, etc. In our approach, such quantifiers are treated as fuzzy numbers which may be manipulated through the use of fuzzy arithmetic and, more generally, fuzzy logic.A concept which plays an essential role in the treatment of fuzzy quantifiers is that of the cardinality of a fuzzy set. Through the use of this concept, the meaning of a proposition containing one or more fuzzy quantifiers may be represented as a system of elastic constraints whose domain is a collection of fuzzy relations in a relational database. This representation, then, provides a basis for inference from premises which contain fuzzy quantifiers. For example, from the propositions “Most U's are A's” and “Most A's are B's,” it follows that “Most2 U's are B's,” where most2 is the fuzzy product of the fuzzy proportion most with itself.The computational approach to fuzzy quantifiers which is described in this paper may be viewed as a derivative of fuzzy logic and test-score semantics. In this semantics, the meaning of a semantic entity is represented as a procedure which tests, scores and aggregates the elastic constraints which are induced by the entity in question.

An oft-heard argument for Natural Language interfaces is their promise of reducing the effort an infrequent user would have to exert in using a computer system. This viewpoint has led to a research concentration on removing “artificial” constraints on a user's freedom of expression, allowing it to move closer to everyday speech. This paper discusses two complementary directions for extending Natural Language interfaces to data bases, which can make them more useful systems. These extensions make Natural Language interfaces less simply “windows” through which data can be called into view, and more “articulate experts” on the data base system and what it represents. The two directions involve (1) broadening the range of query types that can be handled and (2) extending the range of responses that can be provided.