Interval linear systems: the state of the art

Summary  Linear systems represent the computational kernel of many models that describe problems arising in the field of social, economic as well as technical and scientific disciplines. Therefore, much effort has been devoted to the development of methods, algorithms and software for the solution of linear systems. Finite precision computer arithmetics makes rounding error analysis and perturbation theory a fundamental issue in this framework (Higham 1996). Indeed, Interval Arithmetics was firstly introduced to deal with the solution of problems with computers (Moore 1979, Rump 1983), since a floating point number actually corresponds to an interval of real numbers. On the other hand, in many applications data are affected by uncertainty (Jerrell 1995, Marino & Palumbo 2002), that is, they are only known to lie within certain intervals. Thus, bounding the solution set of interval linear systems plays a crucial role in many problems. In this work, we focus on the state of the art of theory and methods for bounding the solution set of interval linear systems. We start from basic properties and main results obtained in the last years, then we give an overview on existing methods.     

Hyper-heuristics: a survey of the state of the art

Hyper-heuristics comprise a set of approaches that are motivated (at least in part) by the goal of automating the design of heuristic methods to solve hard computational search problems. An underlying strategic research challenge is to develop more generally applicable search methodologies. The term hyper-heuristic is relatively new; it was first used in 2000 to describe heuristics to choose heuristics in the context of combinatorial optimisation. However, the idea of automating the design of heuristics is not new; it can be traced back to the 1960s. The definition of hyper-heuristics has been recently extended to refer to a search method or learning mechanism for selecting or generating heuristics to solve computational search problems. Two main hyper-heuristic categories can be considered: heuristic selection and heuristic generation. The distinguishing feature of hyper-heuristics is that they operate on a search space of heuristics (or heuristic components) rather than directly on the search space of solutions to the underlying problem that is being addressed. This paper presents a critical discussion of the scientific literature on hyper-heuristics including their origin and intellectual roots, a detailed account of the main types of approaches, and an overview of some related areas. Current research trends and directions for future research are also discussed.     

Problem solving in the mathematics classroom: the German perspective

In Germany, problem solving has important roots that date back at least to the beginning of the twentieth century. However, problem solving was not primarily an aspect of mathematics education but was particularly influenced by cognitive psychologists. Above all, the Gestalt psychology developed by researchers such as Köhler (Intelligenzprüfungen an Anthropoiden. Verlag der Königlichen Akademie des Wissens, Berlin, 1917; English translation: The mentality of apes. Harcourt, Brace, New York, 1925), Duncker (Zur Psychologie des produktiven Denkens. Springer, Berlin, 1935), Wertheimer (Productive thinking. Harper, New York, 1945), and Metzger (Schöpferische Freiheit. Waldemar Kramer, Frankfurt, 1962) made extensive use of mathematical problems in order to describe their specific problem-solving theories. However, this research had hardly any influence on mathematics education—neither as a scientific discipline nor as a foundation for mathematics instruction. In the German mathematics classroom, problem solving, which is according to Halmos (in Am Math Mon 87:519–524, 1980) the “heart of mathematics,” did not attract the interest it deserved as a genuine mathematical topic. There is some evidence that this situation may change. In the past few years, nationwide standards for school mathematics have been introduced in Germany. In these standards, problem solving is specifically addressed as a process-oriented standard that should be part of the mathematics classroom through all grades. This article provides an overview on problem solving in Germany with reference to psychology, mathematics, and mathematics education. It starts with a presentation of the historical roots but gives also insights into contemporary developments and the classroom practice.     

Network hub location problems: The state of the art

Hubs are special facilities that serve as switching, transshipment and sorting points in many-to-many distribution systems. The hub location problem is concerned with locating hub facilities and allocating demand nodes to hubs in order to route the traffic between origin–destination pairs. In this paper we classify and survey network hub location models. We also include some recent trends on hub location and provide a synthesis of the literature.     

Problem solving: a personal perspective from Brazil

In this paper, I do a historical review of the concept of Problem Solving, and make some considerations about the State of the Art nowadays. A very brief notice of the art of Problem Solving in Latin America is also presented. Finally, I present some reflections on the future of the Art of Problem Solving.     

Elastic waveguides: History and the state of the art. I

We present a brief review of dispersion characteristics of normal modes in an elastic slab and a cylinder. Key topics of the 125-year history of the problem and its modern reflection in the global information space are elucidated.     

Explaining inefficiency in nonparametric production models: the state of the art

The performance of economic producers is often affected by external or environmental factors that, unlike the inputs and the outputs, are not under the control of the Decision Making Units (DMUs). These factors can be included in the model as exogenous variables and can help to explain the efficiency differentials, as well as improve the managerial policy of the evaluated units. A fully nonparametric methodology, which includes external variables in the frontier model and defines conditional DEA and FDH efficiency scores, is now available for investigating the impact of external-environmental factors on the performance. In this paper, we offer a state-of-the-art review of the literature, which has been proposed to include environmental variables in nonparametric and robust (to outliers) frontier models and to analyse and interpret the conditional efficiency scores, capturing their impact on the attainable set and/or on the distribution of the inefficiency scores. This paper develops and complements the approach of B?din et al. (2012) by suggesting a procedure that allows us to make local inference and provide confidence intervals for the impact of the external factors on the process. We advocate for the nonparametric conditional methodology, which avoids the restrictive “separability” assumption required by the two-stage approaches in order to provide meaningful results. An illustration with real data on mutual funds shows the usefulness of the proposed approach.     

Elastic waveguides: history and the state of the art. II

In this paper, the normal modes of an elastic rectangular waveguide are analyzed. We retrace the key aspects of the almost 150-year history of this problem. Using the superposition method, we have obtained an analytical solution of the problem for four types of symmetry of the wave field. In addition, we have established important differences of the dispersion characteristics of normal modes in a rectangle from the Rayleigh–Lamb modes for an infinite plate and the Pochhammer–Chree modes for a cylinder. We give also an estimate of a series of approximate theories for a rectangular waveguide.

The numerical interpretation of the results of analysis is however necessary, and it is a degree of perfection which it would be very important to give to every application of analysis to the natural sciences. So long as it is not obtained, the solutions may be said to remain incomplete and useless, and the truth which it is proposed to discover is no less hidden in the formulas of analysis than it was in the physical problem itself.

                             J. Fourier [28, Sec. 13]

     

Implementation — the bottleneck of operations research: the state of the art

With respect to the practical application of the methods and techniques of operations research, the methodology of model construction and model implementation must be considered to be the main bottleneck. During the whole OR process, from the identification of a problem to the control of an implemented model, a lot of methodological decisions must be made that affect the implementation. Therefore, for OR to give aids for decisions, the whole OR process must be considered from the point of view of implementation. In the article, the implementation problem and the implementation concept will be discussed. Recent research contributions are presented and discussed with a view to current and potential practical implications.     

Problem solving in France: didactic and curricular perspectives

In this paper, we address the issue of problem solving in classrooms in France through two different and complementary approaches: didactic research and curricular choices. These two approaches correspond to two different, but not independent perspectives on problem solving and we investigate the existing links between them. We show that in France, the solving of problems is given a central role both in didactic research and curricular choices and that problem solving, as generally understood, is an object taking controversial positions, and we try to elucidate the rationale behind such positions. This paper is structured into two main parts: the first part devoted to didactic research, the second part to curricula, and one last part, discussion, wherein we question didactic research and curricula as regards their potential for influencing the reality of problem solving in classroom practices.     

Problem solving and Working Mathematically: an Australian perspective

This article reviews “problem solving” in mathematics in Australia and how it has evolved in recent years. In particular, problem solving is examined from the perspectives of research, curricula and instructional practice, and assessment. We identify three key themes underlying observed changes in the research agenda in Australia in relation to problem solving: Obliteration, Maturation and Generalisation. Within state mathematics curricula in Australia, changes in the language and construction of the curriculum and in related policy documents have subsumed problem solving within the broader category of Working Mathematically. In relation to assessment, research in Australia has demonstrated the need for alignment of curriculum, instruction and assessment, particularly in the case of complex performances such as mathematical problem solving. Within the category of Working Mathematically, recent Australian curriculum documents appear to accept an obligation to provide both standards for mathematical problem solving and student work samples that illustrate such complex performances and how they might be assessed.     

Finding candidate singular optimal controls: A state of the art survey

Pontryagin's maximum principle gives no information about a singular optimal control if the problem is linear. This survey shows how candidate singular optimal controls may be found for linear and nonlinear problems. A theorem is given on the maximum order of a linear singular problem.This paper is based in part on the research undertaken by the author at the Hatfield Polytechnic, Hatfield, Hertfordshire, England, for the Ph.D. Degree.