Multi-step quasi-Newton methods for optimization

Quasi-Newton methods update, at each iteration, the existing Hessian approximation (or its inverse) by means of data deriving from the step just completed. We show how “multi-step” methods (employing, in addition, data from previous iterations) may be constructed by means of interpolating polynomials, leading to a generalization of the “secant” (or “quasi-Newton”) equation. The issue of positive-definiteness in the Hessian approximation is addressed and shown to depend on a generalized version of the condition which is required to hold in the original “single-step” methods. The results of extensive numerical experimentation indicate strongly that computational advantages can accrue from such an approach (by comparison with “single-step” methods), particularly as the dimension of the problem increases.     

The Borda rule and Pareto stability: a further comment

In their paper “The Borda rule and Pareto stability: a comment” published in 1979 by Econometrica, Farkas and Nitzan revealed the “intimate relationship” between the Borda rule and the Pareto criterion. The idea was the following: in a profile of total orders, when there is a candidate who obviously wins under unanimous agreement of the voters, that candidate should be in the choice set. In a profile where there is no obvious winner, the candidates that are the closest to unanimity should be chosen. According to this principle, they defined a choice rule called “closeness to unanimity” and they showed that it is equivalent to the Borda rule. In our paper, we give an equivalent result for a ranking rule. Then we try to obtain similar results when aggregating profiles of tournaments, weak orders, semiorders, fuzzy relations, … and we show that the definition of an obvious winner is no more obvious.     

Classes of pretopological spaces closed under the formation of final structures

We investigate coreflective subconstructs of the construct Prtop of pretopological spaces and continuous maps and in particular the inclusion “order” between these subconstructs. We describe the smallest, second and third coreflective class and then all minimal elements that are strictly larger. Using these minimal elements we obtain a “partition” of the whole conglomerate of coreflective subconstructs of Prtop. The results dealing with classes in one member of this partition have an immediate interpretation in the framework of reflexive relations.     

One-dimensional, two-fluid modelling of turbulent premixed flames

The “two-fluid” mathematical model for turbulent combustion is applied to a one-dimensional, premixed, stabilized ducted flame. The flame is assumed to consist of two interspersed fluids (“reactants” and “products”), each characterized by its own properties and interacting through the exchange of mass, heat, and momentum. The distributions of pressure, densities, velocities, and volume fractions across the duct were successfully simulated. From a parametric study on the effects of the empirical constants involved in the interfluid relations, the significant dependence of the system on the parameters that characterize the mass transfer rate and the relative effect of mass transfer to momentum transfer was confirmed. The application of the model to transient states proved its ability to predict system oscillations.     

Fuzzy logic in measurements

Our main interest in this paper is to translate from “natural language” into “system theoretical language”. This is of course important since a statement in system theory can be analyzed mathematically or computationally. We assume that, in order to obtain a good translation, “system theoretical language” should have great power of expression. Thus we first propose a new frame of system theory, which includes the concepts of “measurement” as well as “state equation”. And we show that a certain statement in usual conversation, i.e., fuzzy modus ponens with the word “very”, can be translated into a statement in the new frame of system theory. Though our result is merely one example of the translation from “natural language” into “system theoretical language”, we believe that our method is fairly general.     

The algebraic structure of relative twisted vertex operators

Twisted vertex operators based on rational lattices have had many applications in vertex operator algebra theory and conformal field theory. In this paper, “relativized” twisted vertex operators are constructed in a general context based on isometries of rational lattices, and a generalized twisted Jacobi identity is established for them. This result generalizes many previous results. Relatived untwisted vertex operators had been studied in a monograph by the authors. The present paper includes as a special case the proof of the main relations among twisted vertex operators based on even lattices announced some time ago by the second author.     

Stratification method in the theory of thin shells : PMM vol. 43, no. 6, 1979, pp. 1058–1064

It is shown that on the basis of certain simplifications induced in the physical and geometrical dependences, such a “stratification” of a shell can be achieved for which the fibers of each of two layers will be deformed just as thin rods whose axes agree with the lines of principal curvature of the shell middle surface. The approach to analyzing shells on the basis of the relationships to be obtained below is called the “stratification method”.     

An experimental comparison of four graph drawing algorithms

In this paper we present an extensive experimental study comparing four general-purpose graph drawing algorithms. The four algorithms take as input general graphs (with no restrictions whatsoever on connectivity, planarity, etc.) and construct orthogonal grid drawings, which are widely used in software and database visualization applications. The test data (available by anonymous ftp) are 11,582 graphs, ranging from 10 to 100 vertices, which have been generated from a core set of 112 graphs used in “real-life” software engineering and database applications. The experiments provide a detailed quantitative evaluation of the performance of the four algorithms, and show that they exhibit trade-offs between “aesthetic” properties (e.g., crossings, bends, edge length) and running time.     

Statistical learning control of uncertain systems: theory and algorithms

It has recently become clear that many control problems are too difficult to admit analytic solutions. New results have also emerged to show that the computational complexity of some “solved” control problems is prohibitive. Many of these control problems can be reduced to decidability problems or to optimization questions. Even though such questions may be too difficult to answer analytically, or may not be answered exactly given a reasonable amount of computational resources, researchers have shown that we can “approximately” answer these questions “most of the time”, and have “high confidence” in the correctness of the answers.     

Cellular automata study of high burn-up structures

We propose a new approach, based on the cellular automata, for processing and modeling the structure dynamics of UO₂ at different cross-section averaged burn-ups. Micrographs of the material surface, subjected to both “as-polished” and “as-etched” treatments, with a magnification of 1250× have been used in our study. It has been shown that this approach provides efficient tools for investigation of the surface structure dynamics both at local and global levels.     

Construction of the value function in a pursuit-evasion game with three pursuers and one evader

A differential pursuit-evasion game is considered with three pursuers and one evader. It is assumed that all objects (players) have simple motions and that the game takes place in a plane. The control vectors satisfy geometrical constraints and the evader has a superiority in control resources. The game time is fixed. The value functional is the distance between the evader and the nearest pursuer at the end of the game. The problem of determining the value function of the game for any possible position is solved.

Three possible cases for the relative arrangement of the players at an arbitrary time are studied: “one-after-one”, “two-after-one”, “three-after-one-in-the-middle” and “three-after-one”. For each of the relative arrangements of the players a guaranteed result function is constructed. In the first three cases the function is expressed analytically. In the fourth case a piecewise-programmed construction is presented with one switchover, on the basis of which the value of the function is determined numerically. The guaranteed result function is shown to be identical with the game value function. When the initial pursuer positions are fixed in an arbitrary manner there are four game domains depending on their relative positions. The boundary between the “three-after-one-in-the-middle” domain and the “three-after-one” domain is found numerically, and the remaining boundaries are interior Nicomedean conchoids, lines and circles. Programs are written that construct singular manifolds and the value function level lines.     

The Dempster–Shafer calculus for statisticians

The Dempster–Shafer (DS) theory of probabilistic reasoning is presented in terms of a semantics whereby every meaningful formal assertion is associated with a triple (p, q, r) where p is the probability “for” the assertion, q is the probability “against” the assertion, and r is the probability of “don’t know”. Arguments are presented for the necessity of “don’t know”. Elements of the calculus are sketched, including the extension of a DS model from a margin to a full state space, and DS combination of independent DS uncertainty assessments on the full space. The methodology is applied to inference and prediction from Poisson counts, including an introduction to the use of join-tree model structure to simplify and shorten computation. The relation of DS theory to statistical significance testing is elaborated, introducing along the way the new concept of “dull” null hypothesis.     

Planar and axially symmetric configurations which are circumvented with the maximum critical mach number

The structure of planar and axially symmetric configurations which, by satisfying a number of geometrical constraints, are circumvented in a boundless space or in a cylindrical channel by an ideal (non-viscous and non-thermally conducting) gas with a maximal critical Mach number M* is found. The analysis is carried out using the “rectilinearity property” of a sonic line in “subsonic” flows (SF), the “principle of a maximum” for an SF and “comparison theorems” which are either taken from /1/ or serve as a generalization of the corresponding assertions from /1/. Following /1/, configurations are considered which have a plane or axis of symmetry parallel to the velocity V_∞ of the approach stream, while flows in which (including the boundary) the Mach number M 1 are said to be “subsonic”. As usual, by M* we mean a value of M_∞ such that the inequality M1, which is satisfied in the whole stream when M_∞ M*, is violated when M_∞>M*.

The configurations investigated include closed bodies and the leading (trailing) parts of a semi-infinite plate or a circular cylinder in an unbounded flow and in a channel as well as lattices of symmetric profiles. Both in /1/, where the structure of closed planar and axially symmetric bodies was found, as well as in /2/, where such bodies were constructed numerically, the generatrices of all the configurations investigated contain the end planes or the segments replacing them of the maximum permissible slope (in modulus) and the “free” streamlines with M 1. Now, however, unlike in /1, 2/, segments of the horizontals are added to it in the general case. Furthermore, in the case of flows in channels and lattices, the configurations which have been found can be circumvented with the development of finite domains of advancing sonic flow.     

Recognizing polygonal parts from width measurements

Automatic recognition of parts is an important problem in many industrial applications. One model of the problem is: given a finite set of polygonal parts, use a set of “width” measurements taken by a parallel-jaw gripper to determine which part is present. We study the problem of computing efficient strategies (“grasp plans”), with the goal to minimize the number of measurements necessary in the worst case. We show that finding a minimum length grasp plan is -hard, and give a polynomial time approximation algorithm that is simple and produces a solution that is within a log factor from optimal.     

Factors Influencing the Outcomes of Collaborative Mathematical Problem Solving: An Introduction

This study is an investigation of the factors that influence the effectiveness of collaboration on open-ended mathematical tasks. Students in Grades 3, 6, and 9 worked in groups of 3 on 2 chance and data tasks-1 related to fair dice and the other related to associations among variables presented on data cards. The groups' outcomes and the types of collaboration observed are investigated in relation to issues raised in the literature. Various phenomena are identified that influence cognitive “lifting,” “hovering,” and “falling,” that is, improvement, no change, and reduction in levels of functioning, respectively. These phenomena include cognitive factors, social or interpersonal factors, and external factors.     

Chinese and English Mathematics Language: The Relation Between Linguistic Clarity and Mathematics Performance

In Study 1, 48 judges rated the clarity of Chinese, English, and “Chinglish” (Chinese words translated into English) mathematical words-for example, the Chinglish version of the Chinese word for quadrilateral is “four-side-shape.” Native Chinese-speaking judges achieve greater agreement on the relative clarity of Chinese words than do native English-speaking judges on the relative clarity of English words. More Chinese words are rated clear than are English. Chinglish mathematical words tend to be rated more clear than English. The inherent compound word structure of the Chinese language seems well suited to portray mathematical ideas.

In Study 2, we examined the relations among the clarity of Chinese mathematical terms, U.S. urban junior high school students' Chinese reading ability, and their mathematics performance. There is a strong correlation between Chinese reading ability and performance on test items with mathematics words rated clear by Chinese judges. The relative clarity of mathematical terms in the Chinese language may contribute to Chinese-speaking students' understanding of mathematics and to superior mathematics performance.     

Quasilinear conflict-controlled processes with additional restrictions

A class of conflict-controlled processes [1–3] with additional (“phase” type) restrictions on the state of the evader is considered. A similar unrestricted problem was considered in [4]. Unlike [5, 6] the boundary of the “phase” restrictions is not a “death line” for the evader. Sufficient conditions for the solvability of the pursuit and evasion problems are obtained, which complement a range of well-known results [5–10].^‡     

A constrained optimization approach to finite element mesh smoothing

The quality of a finite element solution has been shown to be affected by the quality of the underlying mesh. A poor mesh may lead to unstable and/or inaccurate finite element approximations. Mesh quality is often characterized by the “smoothness” or “shape” of the elements (triangles in 2-D or tetrahedra in 3-D). Most automatic mesh generators produce an initial mesh where the aspect ratio of the elements are unacceptably high. In this paper, a new approach to produce acceptable quality meshes from a topologically valid initial mesh is presented. Given an initial mesh (nodal coordinates and element connectivity), a “smooth” final mesh is obtained by solving a constrained optimization problem. The variables for the iterative optimization procedure are the nodal coordinates (excluding, the boundary nodes) of the finite element mesh, and appropriate bounds are imposed on these to prevent an unacceptable finite element mesh. Examples are given of the application of the above method for 2- and 3-D meshes generated using automatic mesh generators. Results indicate that the new method not only yields better quality elements when compared with the traditional Laplacian smoothing, but also guarantees a valid mesh unlike the Laplacian method.     

Rule-based models of multivariable functions

This study is devoted to fuzzy rule based modelling of multiple-input single-output nonlinear numerical relationships. The model under investigation is viewed as a collection of conditional statements “if state Ω_i then y = g_i(x, a_i)”, i = 1, 2,…, N with Ω_i being a fuzzy relation defined in the space of the input variables. In contrast to the commonly encountered identification approach that is dwelled upon discrete experimental data, the one proposed in this study is concerned with explicitly articulated nonlinear input-output relationship. The main thrust is in the development of a fuzzy partition of the input variables completed through a sequence of fuzzy relations rather than Cartesian products of fuzzy sets. This approach allows us to maintain the number of necessary rules under control and avoid a combinatorial explosion otherwise inevitable in situations of highly multivariable functions. Introduced are criteria of separability and function variability whose objective is to guide a distribution and granularity of the linguistic labels forming the condition part of the rules.     

Extension of Newman's formula for the Sylvester equation to the generic case

The Sylvester “inivilator” equation AX - XB=C is solved in finite terms when A and B are quite general. Newman's formula for the case of A and B periodic is recovered by a limiting process. Some applications are discussed.