An improved neuronal energy model that better captures of dynamic property of neuronal activity

Among the theories of neural information coding, the neural energy coding is more accessible to global coding features than traditional neural encoding. According to the shortcomings existing in the neuronal energy model, that is, the non-smooth nature of the energy curve, we proposed an improved neuronal energy model in this paper. The modified energy model is a good choice for establishment of the global model of brain function. And it is also the basis of energy calculation for functional cognitive neural networks in the future.     

Games with guaranteed scores and waiting moves

Combinatorial scoring games, with the property ‘extra pass moves for a player will do him no harm’, are characterized. The characterization involves an order embedding of Conway’s normal play games, and we call our class the universe of Guaranteed scoring games. Also, we give a theorem for comparing guaranteed games with scores (numbers) which extends Ettinger’s work on Dicot scoring games.     

Optimization of Fishing Strategies in Space and Time as a Non-convex Optimal Control Problem

The behavior of a fishing fleet and its impact onto the biomass of fish can be described by a nonlinear parabolic diffusion–reaction equation. Looking for an optimal fishing strategy leads to a non-convex optimal control problem with a bilinear control action. In this work, we present such an optimal control formulation, prove its well-posedness and derive first- and second-order optimality conditions. These results provide a basis for tailored finite element discretization as well as for Newton type optimization algorithms. First numerical test problems show typical features as so-called No-Take-Zones and maximal fishing quota (total allowable catches) as parts of an optimal fishing strategy.     

Higher dimensional Apollonian packings,revisited

The Apollonian circle and sphere packings are well known objects that have attracted the attention of mathematicians throughout the ages. The historically natural generalization of the procedure for generating the packing breaks down in higher dimensions, as it leads to overlapping hyperspheres. There is, however, an alternative interpretation that allows one to extend the concept to higher dimensions and in a unique way. For relatively small dimensions (2 through at least 8), those packings can be thought of as ample cones for classes of K3 surfaces. We describe the packings in some detail for dimensions 4 (with plenty of pictures), 5, and 6.     

On the Strong Convergence of Subgradients of Convex Functions

In this paper, results on the strong convergence of subgradients of convex functions along a given direction are presented; that is, the relative compactness (with respect to the norm) of the union of subdifferentials of a convex function along a given direction is investigated.     

Reflections on and of minor-closed classes of multisorted operations

The minor relation of functions is generalized to multisorted functions. Pippenger’s Galois theory for minor-closed sets of functions is extended to multisorted functions and multisorted relation pairs. Reflections of minor-closed sets are again minor-closed, and the effect of reflections on the invariant relation pairs of minor-closed sets of multisorted functions is described.     

Another refinement of the right-hand side of the Hermite–Hadamard inequality for simplices

We establish a new refinement of the right-hand side of the Hermite–Hadamard inequality for convex functions of several variables defined on simplices.     

Constrained Extremum Problems and Image Space Analysis–Part I: Optimality Conditions

Image space analysis is a new tool for studying scalar and vector constrained extremum problems as well as generalized systems. In the last decades, the introduction of image space analysis has shown that the image space associated with the given problem provides a natural environment for the Lagrange theory of multipliers and that separation arguments turn out to be a fundamental mathematical tool for explaining, developing and improving such a theory. This work, with its 3 parts, aims at contributing to describe the state-of-the-art of image space analysis for constrained optimization and to stress that it allows us to unify and generalize the several topics of optimization. In this 1st part, after a short introduction of such an analysis, necessary and sufficient optimality conditions are treated. Duality and penalization are the contents of the 2nd part. The 3rd part deals with generalized systems, in particular, variational inequalities and Ky Fan inequalities. Some further developments are discussed in all the parts.     

Consistency requirements and pattern methods in cost sharing problems with technological cooperation

Using the discrete cost sharing model with technological cooperation, we investigate the implications of the requirement that demand manipulations must not affect the agents’ shares. In a context where the enforcing authority cannot prevent agents (who seek to reduce their cost shares) from splitting or merging their demands, the cost sharing methods used must make such artifices unprofitable. The paper introduces a family of rules that are immune to these demand manipulations, the pattern methods. Our main result is the characterization of these methods using the above requirement. For each one of these methods, the associated pattern indicates how to combine the technologies in order to meet the agents’ demands. Within this family, two rules stand out: the public Aumann–Shapley rule, which never rewards technological cooperation; and the private Aumann–Shapley rule, which always rewards technology providers. Fairness requirements imposing natural bounds (for the technological rent) allow to further differentiate these two rules.