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.     

A note on ‘on antipodal graphs’

A lemma stated in our earlier paper has been corrected without affecting the major content of that paper. An independent proof for the succeeding lemma in that paper has also been produced. It is be noted that given a graph G, the antipodal graph A(G) is uniquely defined, but given A(G), there exist more than one graph H such that A(H) = A(G).     

A note on the selection of Benders’ cuts

A new cut selection criterion for Benders’ cuts is proposed and computationally analyzed. The results show that the new criterion is more robust—and often considerably faster—than the standard ones.     

A note on the regularity criterion of the two-dimensional Newton–Boussinesq equations

In this paper, we consider the two-dimensional Newton–Boussinesq equations with the incompressibility condition. We obtain a regularity criterion for the Newton–Boussinesq equations by virtue of the commutator estimate.     

A note on q-difference equations for Ramanujan’s integrals

The Ramanujan Journal - This short paper derives the relationship between solutions of q-difference equations and generating functions for q-orthogonal polynomials. The key of the method is to...     

A note on Δ 2 0 -Spectra of linear orderings and degree spectra of the successor relation

In this paper we construct linear orderings whoseΔ ₂ ⁰ -spectra coincide with classes of all high₀ and high₁ degrees, respectively. We also prove that there exists a computable linear ordering such that its degree spectrum of the successor relation coincides with a fixed nonempty class of degrees which represents a Σ ₁ ⁰ -spectrum of some Ø′-computable linear ordering.     

‘Minesweeper’ and spectrum of discrete Laplacians

The article is devoted to a problem inspired by the ‘Minesweeper’ computer game. It is shown that certain configurations of open cells guarantee the existence and the uniqueness of solution. Mathematically the problem is reduced to some spectral properties of discrete differential operators. It is shown how the uniqueness can be used to create a new game which preserves the spirit of ‘Minesweeper’ but does not require a computer.     

A note on stochastic semilinear equations and their associated Fokker–Planck equations

The main purpose of this paper is to prove existence and uniqueness of (probabilistically weak and strong) solutions to stochastic differential equations (SDE) on Hilbert spaces under a new approximation condition on the drift, recently proposed in [6] to solve Fokker–Planck equations (FPE), extended in this paper to a considerably larger class of drifts. As a consequence we prove existence of martingale solutions to the SDE (whose time marginals then solve the corresponding FPE). Applications include stochastic semilinear partial differential equations with white noise and a non-linear drift part which is the sum of a Burgers-type part and a reaction diffusion part. The main novelty is that the latter is no longer assumed to be of at most linear, but of at most polynomial growth. This case so far had not been covered by the existing literature. We also give a direct and more analytic proof for existence of solutions to the corresponding FPE, extending the technique from [6] to our more general framework, which in turn requires to work on a suitable Gelfand triple rather than just the Hilbert state space.     

A note on the equilibria of the unbounded traveler’s dilemma

When there is no upward limit on admissible claims, the traveler’s dilemma admits a continuum of symmetric mixed strategy equilibria in addition to the pure strategy equilibrium in which both players ask and obtain the minimum. The payoff of any of these equilibria exceeds the payoff of the pure strategy one and any claim represents an attainable payoff. If the distinction between a large and an unbounded action set is fuzzy, this result can explain some puzzling stylized facts on the behavior of experimental subjects in the game.     

A supplement to ‘A note on invariant measure semigroups’

A unified definition of measure semigroup is given and it is shown how the types of invariance for the measure are formulated and what effect they have on the embeddability of an Abelian semigroup in a group. The online version of the original article can be found under doi:.     

A note on maximizing the permanent of a positive definite hermitian matrix,given the eigenvalues ∗

As a step toward understanding the unsolved problem of determining how large the permanent of a positive semi-definite matrix can be, given the eigenvalues, we note that a necessary condition for A to be a permanent maximizing matrix is that A commute with its permanental adjoint.     

A note on uniqueness of the ground state for nonlinear Schrödinger equations

We are concerned with the following nonlinear Schrödinger equation where , . For small enough and a class of , we show the uniqueness of the positive ground state under certain assumptions on asymptotic behavior of and its first derivatives. Here our results are suitable for a kind of which has different increasing rates at different directions.