A note on “Methods for constructing distance matrices and the inverse eigenvalue problem”

In this paper, a symmetric nonnegative matrix with zero diagonal and given spectrum, where exactly one of the eigenvalues is positive, is constructed. This solves the symmetric nonnegative eigenvalue problem (SNIEP) for such a spectrum. The construction is based on the idea from the paper Hayden, Reams, Wells, “Methods for constructing distance matrices and the inverse eigenvalue problem”. Some results of this paper are enhanced. The construction is applied for the solution of the inverse eigenvalue problem for Euclidean distance matrices, under some assumptions on the eigenvalues.     

A note on “branch-and-price approach for the multi-skill project scheduling problem”

In a recent paper published in Optimization Letters, Montoya et al. (Optim Lett 8:1721–1734, 2014) proposed a branch-and-price approach for a multi-skill project scheduling problem. In that paper, an integer linear programming formulation was first introduced which, unfortunately, has a number of inconsistences. At the interest of mathematical rigor, in this note, we refine such formulation.     

A note on Boiteux’ surplus function and dual Pareto efficiency

The purpose of this note is to study first a notion of a surplus function that originates in the work of [Boiteux, M., 1951. Le Revenu Distribuable et les Pertes Économiques. Econometrica 112–133] and to rely upon this notion to study dual Pareto efficiency in an exchange economy. This function, which we call the Boiteux’ surplus function, measures how many units of income an individual must be given to move from a reference utility level, to another utility level. We prove several properties of the Boiteux’ surplus function, and study in particular its links with the expenditure and the indirect utility functions. With regard to dual Pareto efficiency and the Boiteux’ surplus function our results are as follows. A feasible reference price–income pair is dual Pareto efficient if and only if it zero-maximizes the sum of individual Boiteux’ surplus functions among all feasible price–income pairs. We use these results to give a new proof (for the case of an exchange economy with positive prices) of the characterization by Luenberger of dual Pareto optima.     

A brief note on analyticity and causality,and the “levels of physics”

We argue that the causality of physics has its roots in the analyticity within the two dimensions of a fractal quantum path and further show how this picture has convergence with quantum superstrings.     

A note on the Hamilton–Waterloo problem with C8-factors and Cm-factors

In this paper, we almost completely solve the Hamilton–Waterloo problem with $C_8$-factors and $C_m$-factors where the number of vertices is a multiple of $8m$.     

A note on “a dual-ascent approach to the fixed-charge capacitated network design problem”

We show by a counterexample that the dual-ascent procedure proposed by Herrmann, Ioannou, Minis and Proth in a 1996 issue of the European Journal of Operational Research is incorrect in the sense that it does not generate a valid lower bound to the optimal value of fixed-charge capacitated network design problems. We provide a correct dual-ascent procedure based on the same ideas and we give an interpretation of it in terms of a simple Lagrangean relaxation. Although correct, this procedure is not effective, as in general, it provides a less tighter bound than the linear programming relaxation.     

A note on Furstenberg’s filtering problem

This note gives a positive answer to an old question in elementary probability theory that arose in Furstenberg’s seminal article “Disjointness in Ergodic Theory.” As a consequence, Furstenberg’s filtering theorem holds without any integrability assumption.     

A note “On some primary subgroups and the structure of finite groups”

We point out an error in the paper (Monatsh Math. doi:10.1007/s00605-013-0492-3, 2013) by Kang, and give new brief proofs of the main results.     

A note on an open problem about the first Painlevé equation

In this note, we apply numerical analysis to the first equation, find the conditions for it to have oscillating solutions and therefore solve an open problem posed by Peter A. Clarkson.     

A note on Galvão's “A graph theoretical bound for the p-median problem”

Galvão (1981) proposed an algorithm for calculating a strong lower bound for the p-median problem on vertex-unweighted networks. It is shown that there is a mistake in the algorithm. An improved version of the algorithm is given.     

A note on a set-mapping problem of Hajnal and Máté

It is consistent that there exists a set mappingF with <F(, )< for + 2 >w ₂ with no uncountable free sets.Research supported by Hungarian National Research Fund No. 1805 and 1908.     

A note ‘On a single machine-scheduling problem with separated position and resource effects’

This note concerns the paper [Janiak A, Kovalyov MY, Lichtenstein M. On a single machine-scheduling problem with separated position and resource effects. Optimization; 2013. doi:10.1080/02331934.2013.804077], which presents an analysis, a counterexample and a pseudocode related with our proof of optimality for a resource allocation algorithm given in [Rudek A, Rudek R. A note on optimization in deteriorating systems using scheduling problems with the aging effect and resource allocation models. Comput. Math. Appl. 2011;62:1870–1878]. We show that the discussed analysis is based only on one part of our proof omitting its integral second part, which is the source of misunderstanding. The considered counterexample is applied for an algorithm, which was not the method presented in our paper, whereas our algorithm provides the correct result for the mentioned counterexample. The provided pseudocode of the resource allocation algorithm, which is presented as the correct method, is a pseudocode of the algorithm described in our paper. Therefore, we show that the results presented in our paper are correct.     

A note on a problem of Erd?s on right angles

Erd?s asked the following question: is it true that a set of n² points in the plane always contain 2n−2 points which do not determine a right angle? We show that, apart from a log-factor, the answer is in the affirmative.     

A note on the equal division kernel and the α‐power model

The equal division kernel and the α‐power solution are two recently proposed theories of coalition formation in n‐person characteristic function games with sidepayments. A closed form solution for payoff disbursement is derived from the α‐power model for n‐person games with only 1, n‐1, and n‐person permissible coalitions and no weak players. It is shown that for this class of games, which are frequently employed to experimentally test theories of coalition formation, the equal division kernel is a special case of the α‐power model with α = 1/2.     

A note on the probabilistic approach to Turán's problem

The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let X_s(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(X_l), given that min(X_l) ≥ 1. The following lower bound on the variance of X_s is proved:

$\text{Var(X}{}_{\mathrm{s}}\text{)?mm}\text{n?2k}\text{s?k}\text{n}\text{s}{}^{\mathrm{?1}}\text{n}\text{k}{}^1$

, where m = |E(H)| and $\text{m}\text{= (}{}_{\mathrm{k}}{}^{\mathrm{n}}\text{) ? m}$. This implies the following: putting t(k, l) = lim_n→∞T(n, l, k)(_kⁿ)^?1 then t(k, l) ≥ T(s, l, k)((_k^s) ? 1)^?1, whenever s ≥ l > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned.