The core and nucleolus of games: A note on a paper by Göthe-Lundgren et al.

In the paper “On the nucleolus of the basic vehicle routing game”, Mathematical Programming 72, 83–100 (1996), G?the-Lundgren et al. develop a constraint generation method to compute the pre-nucleolus of a game. Their method assumes that constraints that are redundant in the representation of the core can be ignored in the computation of the pre-nucleolus. We provide an example that shows that for a game with an empty core such an assumption is, in general, not valid. Further, we show that a statement made by G?the-Lundgren et al. about an intuitive interpretation of the pre-nucleolus is misleading. Received: January 1996 / Accepted: February 2000?Published online January 17, 2001     

A note on the paper ‘On dynamical multi-team Cournot game in exploitation of a renewable resource’

In a recent paper (Asker, 2007) [1] a dynamic Cournot oligopoly game is proposed and it is claimed that this model represents competition among firms that exploit a common access natural resource. According to the author’s claim, the feature that relates the model with renewable natural resource harvesting is given by the presence of a particular cost function where the total cost of each fisherman is proportional to the square of the own quantity of harvesting and inversely proportional to the total harvesting quantity. In contrast, the usual function used in the literature on the exploitation of natural resources (such as fisheries) is inversely proportional to the available resource stock, and not to the total harvesting. This, in some sense, assumes exactly the opposite (as the available resource is inversely proportional to the total harvesting). So, we believe that the paper (Asker, 2007) [1] contains an error which is probably due to a misunderstanding or a misreading and misinterpretation of the (well-established) literature on bioeconomic modelling, but nevertheless misleading to researchers interested in bioeconomic modelling. The aim of this short note is to explain the mistake and to summarize the correct derivation and interpretation of the cost function. Our goal is to avoid the propagation of a subtle (but nevertheless misleading) error.     

Nonlinear differential systems in the 3–space: A note on periodic solutions by the analysis of two examples

In this note, we address the problem of the existence and location of periodic solutions in nonlinear differential systems in the 3–space. Our main motivation is the study, via bifurcation theory, of periodic solutions (especially limit cycles). We study this problem in two simple polynomial (chaotic) systems: The first one, due to Muthuswamy and Chua, is the mathematical model to the simplest chaotic circuit, and the second is due to Sprott et al.     

Addendum to a note on the zeros of solutions of w″+P(z)w=0 where P is a polynomial

In the original paper [1], it was shown that the zeros of solutions of w″ + P(z)w = 0, where P(z) is a polynomial of degree n ≥ 1, must approach certain rays. This was proved by first obtaining asymptotic formulas for a fundamental set of solutions in sectors, and then using them to derive estimates on the rate at which the nearby zeros approach the ray. The estimates derived in [1] for the rate of approach were rough estimates which were sufficient to prove the main result but simple enough to avoid unnecessary complications in the proof. The present note is intended to give the best estimate which can be derived from the asymptotic formulas for the rate of approach of the zeros. The main reason for deriving these estimates is that they show that for many equations (e.g., the Titchmarsh equation) the rate of approach is actually much faster than that indicated by the rough estimate in [1]. In fact, we show that the estimate dramatically improves whenever P(z) has the property that the translate P(z ? c) which eliminates the term of degree n ? 1 also eliminates the term of degree n ? 2.     

Comment on “Effects of the slip boundary condition on non-Newtonian flows in a channel” [Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1377–1384]

Recently, Ellahi [1] discussed the slip effects on the flows of an Oldroyd 8-constant fluid using the homotopy analysis method. Crucial flaws in [1] are pointed out in this comment. The present paper provides an exact solution and a numerical solution by shooting method using Runge–Kutta algorithm of the flow problems considered in [1] with the correct nonlinear boundary conditions.     

Comment on: “Fundamental flows with nonlinear slip conditions: exact solutions”, by R. Ellahi, T. Hayat, F. M. Mahomed and A. Zeeshan, Z. Angew. Math. Phys. 61 (2010) 877–888

In their article (Fundamental flows with nonlinear slip conditions: exact solutions, R. Ellahi, T. Hayat, F. M. Mahomed and A. Zeeshan, Z. Angew. Math. Phys. 61 (2010) 877–888.), the authors considered three simple cases of the steady flow of a third grade fluid between parallel plates with slip conditions; namely, Couette flow, Poiseuille flow, and generalized Couette flow. They obtained exact solutions, which were utilized in a way that did not lead to useful results. Their conclusion that the Couette flow cannot be obtained from the generalized Couette flow, by dropping the pressure gradient, is incorrect. Meaningful results based on their solution are herein presented.     

Comment on: “Fundamental flows with nonlinear slip conditions: exact solutions”, by R. Ellahi,T. Hayat,F. M. Mahomed and A. Zeeshan,Z. Angew. Math. Phys. 61 (2010) 877–888

In their article (Fundamental flows with nonlinear slip conditions: exact solutions, R. Ellahi, T. Hayat, F. M. Mahomed and A. Zeeshan, Z. Angew. Math. Phys. 61 (2010) 877–888.), the authors considered three simple cases of the steady flow of a third grade fluid between parallel plates with slip conditions; namely, Couette flow, Poiseuille flow, and generalized Couette flow. They obtained exact solutions, which were utilized in a way that did not lead to useful results. Their conclusion that the Couette flow cannot be obtained from the generalized Couette flow, by dropping the pressure gradient, is incorrect. Meaningful results based on their solution are herein presented.     

Correction to “Some Theorems on the Spaces of Quasiconstant Curvature” (Vol.3 (1983) No.1,1-16)

In view of V≠O, the relation (P. 6, line 16) should be replaced by with f_0=1/f, and(3.12) should be changed to the form ds~2=1/U~2[sum from a=1 to n-1 ((dx~a)~2+V~2(dx~n)~2]) where V and U are connected by the relation (*). After altering the formulation in such a way, we are justified to consider the limiting case f_0=0, which has     

A note on “Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality” [Positivity. 18, 449–473(2014)]

The note points out that the sufficiency of proposition 2.1 in Anh (Positivity 18:449–473, 2014) is erroneous and we provide an example to illustrate it. Also the proof of proposition 2.2 in Anh (Positivity 18:449–473, 2014) is incorrect and we give a new proof.     

On the Scientific,Pedagogic, and Public Activities of Academician Mikhail Alekseevich Lavrent'ev at the Ukrainian Academy of Sciences (1939–1949) (on the 100th Anniversary of the Birth of M. A. Lavrent'ev)

Ukrainian Mathematical Journal -