An updated survey on the linear ordering problem for weighted or unweighted tournaments

In this paper, we survey some results, conjectures and open problems dealing with the combinatorial and algorithmic aspects of the linear ordering problem. This problem consists in finding a linear order which is at minimum distance from a (weighted or not) tournament. We show how it can be used to model an aggregation problem consisting of going from individual preferences defined on a set of candidates to a collective ranking of these candidates.     

An axiomatic look at the Erdős-Trost problem

The Erd?s-Trost problem can be formulated in the following way: “If the triangle XY Z is inscribed in the triangle ABC—with X, Y, and Z on the sides BC, CA, and AB, respectively—then one of the areas of the triangles BXZ, CXY , AY Z is less than or equal to the area of the triangle XY Z.” There are many different solutions for this problem. In this note we take up a very elementary proof (due to Szekeres) and deduce that the class of ordered translation planes is the level in the hierarchy of affine planes where the Erd?s-Trost statement still holds true. We also look at the conditions an absolute plane needs to satisfy for the validity of the Erd?s-Trost statement.     

An investigation into the use of microcomputers to teach mathematical problem‐solving skills to 13 year olds

This is an account of an experiment to teach mathematical problem‐solving skills to 13‐year‐old students with the aid of microcomputers. Twenty‐one children worked through six microcomputer program‐generated problem situations in two and a half hours. Nine of the children worked ‘hands on’ with microcomputers. Twelve of the children were class taught, with a single microcomputer used for demonstration purposes. A comparison of pre‐test and post‐test scores on tests indicated that no significant gains in achievement were made by either group. Informal observations suggest a positive affective response. The findings are discussed at length and possible explanations for the lack of achievement are conjectured.     

How relevant is the lack of reciprocity in pairwise comparisons? An experiment with AHP

We consider the 1-median problem with euclidean distances with uncertainty in the weights, expressed as possible changes within given bounds and a single budget constraint on the total cost of change. The upgrading (resp. downgrading) problem consists of minimizing (resp. maximizing) the optimal 1-median objective value over these weight changes. The upgrading problem is shown to belong to the family of continuous single facility location-allocation problems, whereas the downgrading problem reduces to a convex but highly non-differentiable optimization problem. Several structural properties of the optimal solution are proven for both problems, using novel planar partitions, the knapsack Voronoi diagrams, and lead to polynomial time solution algorithms.     

Are inconsistent decisions better? An experiment with pairwise comparisons

This paper tries to contribute to the well-known and lengthy discussion on whether we should allow decision makers to be inconsistent, or, to put it in another way, if intransitive decisions better represent their preferences. We will focus specifically on the multiple-criteria decision making problem, since it is one of the most usual stances where this intransitivity appears. Results from an experiment carried out with graduate students shows that, when intransitivities are removed, the preferences of decision makers are not better represented, and therefore that the effort to do so may not be justified.     

An improvement on Waring?CGoldbach problem for unlike powers

Let p _i be prime numbers. In this paper, it is proved that for any integer k?R5, with at most $O\big(N^{1-\frac{1}{3k\times2^{k-2}}+\varepsilon}\big)$ exceptions, all positive even integers up to N can be expressed in the form $p_{2}^{2}+p_{3}^{3}+p_{5}^{5}+p_{k}^{k}$ . This improves the result $O\big(\frac{N}{\log^{c}N}\big)$ for some c>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1?C4] for the problem in the form $p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}$ . This method can also be used for some other similar forms.     

“Are the genre and the Geschlecht one and the same number?” An inquiry into Alfred Clebsch's Geschlecht

This article is aimed at throwing new light on the history of the notion of genus, whose paternity is usually attributed to Bernhard Riemann while its original name Geschlecht is often credited to Alfred Clebsch. By comparing the approaches of the two mathematicians, we show that Clebsch's act of naming was rooted in a projective geometric reinterpretation of Riemann's research, and that his Geschlecht was actually a different notion than that of Riemann. We also prove that until the beginning of the 1880s, mathematicians clearly distinguished between the notions of Clebsch and Riemann, the former being mainly associated with algebraic curves, and the latter with surfaces and Riemann surfaces. In the concluding remarks, we discuss the historiographic issues raised by the use of phrases like “the genus of a Riemann surface”—which began to appear in some works of Felix Klein at the very end of the 1870s—to describe Riemann's original research.     

An application of the fixed point theorem to the inverse sturm–liouville problem

The Sturm–Liouville operators −y″ + v(x)y on [0, 1] with Dirichlet boundary conditions y(0) = y(1) = 0 are considered. For any 1 ≤ p < ∞, a short proof of the characterization theorem for the spectral data corresponding to v ∈ L^p(0, 1) is given. Bibliography: 10 titles.     

An open problem on the optimality of an asymptotic solution to Duffing’s nonlinear oscillation problem

Based on the renormalization group method, Kirkinis (2012) [8] obtained an asymptotic solution to Duffing’s nonlinear oscillation problem. Kirkinis then asked if the asymptotic solution is optimal. In this paper, an affirmative answer to the open problem is given by means of the homotopy analysis method.     

An ant colony optimization approach for the multidimensional knapsack problem

Ant colony optimization is a metaheuristic that has been applied to a variety of combinatorial optimization problems. In this paper, an ant colony optimization approach is proposed to deal with the multidimensional knapsack problem. It is an extension of Max Min Ant System which imposes lower and upper trail limits on pheromone values to avoid stagnation. In order to choose the lower trail limit, we provide a new method which takes into account the influence of heuristic information. Furthermore, a local search procedure is proposed to improve the solutions constructed by ants. Computational experiments on benchmark problems are carried out. The results show that the proposed algorithm can compete efficiently with other promising approaches to the problem.     

An inventory–routing problem with the objective of travel time minimization

In this paper, we consider an inventory–routing problem (IRP) in a large petroleum and petrochemical enterprise group. Compared to many other IRPs, the problem in this paper includes some special aspects due to the operational constraints, such as hours-of-service regulations of the company and the industry. Also, in some cases, it is more important to avoid stock out for any station, rather than purely focusing on transportation cost minimization. The objective is to minimize the maximum of the route travel time, which is not addressed in the literature so far. We present a tabu search algorithm to tackle the problem, which builds in an efficient and effective procedure to improve the search quality in each iteration. Moreover, lower bounds of reasonable sized problems, which are intractable in the formulated mathematical model by existing optimization software, are obtained via Lagrangian relaxation technique. Computational results indicate that the lower bounds are tight and the tabu search is capable of providing near optimal, close-to-lower-bound solutions in a computational time effective manner.     

An approximation for the nonlinear filtering problem,with error bound †

For the standard continuous-time nonlinear filtering problem an approximation approach is derived. The approximate filter is given by the solution to an appropriate discrete-time approximating filtering problem that can be explicitly solved by a finite-dimensional procedure. Furthermore an explicit upper bound for the approximation error is derived. The approximating problem is obtained by first approximating the signal and then using measure transformation to express the original observation process in terms of the approximating signal     

An improved algorithm for the $$L_2$$ – $$L_p$$ minimization problem

Mathematical Programming - In this paper we consider a class of non-Lipschitz and non-convex minimization problems which generalize the $$L_2$$ – $$L_p$$ minimization problem. We propose an...