General dd-position sets

The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of ${\rm gp}_d(G)$ with respect to the suitable values of $d$. We show that the decision problem concerning finding ${\rm gp}_d(G)$ is NP-complete for any value of $d$. The value of ${\rm gp}_d(G)$ when $G$ is a path or a cycle is computed and a structural characterization of general $d$-position sets is shown. Moreover, we present some relationships with other topics including strong resolving graphs and dissociation sets. We finish our exposition by proving that ${\rm gp}_d(G)$ is infinite whenever $G$ is an infinite graph and $d$ is a finite integer.Comment: 16 page

Local orientations of fluctuating fluid interfaces

Thermal fluctuations cause the local normal vectors of fluid interfaces to deviate from the vertical direction defined by the flat mean interface position. This leads to a nonzero mean value of the corresponding polar tilt angle which renders a characterization of the thermal state of an interface. Based on the concept of an effective interface Hamiltonian we determine the variances of the local interface position and of its lateral derivatives. This leads to the probability distribution functions for the metric of the interface and for the tilt angle which allows us to calculate its mean value and its mean square deviation. We compare the temperature dependences of these quantities as predicted by the simple capillary wave model, by an improved phenomenological model, and by the microscopic effective interface Hamiltonian derived from density functional theory. The mean tilt angle discriminates clearly between these theoretical approaches and emphasizes the importance of the variation of the surface tension at small wave lengths. Also the tilt angle two-point correlation function is determined which renders an additional structural characterization of interfacial fluctuations. Various experimental accesses to measure the local orientational fluctuations are discussed.Comment: 29 pages, 12 figure

Link Monotonic Allocation Schemes

A network is a graph where the nodes represent players and the links represent bilateral interaction between the players. A reward game assigns a value to every network on a fixed set of players. An allocation scheme specifies how to distribute the worth of every network among the players. This allocation scheme is link monotonic if extending the network does not decrease the payoff of any player. We characterize the class of reward games that have a link monotonic allocation scheme. Two allocation schemes for reward games are studied, the Myerson allocation scheme and the position allocation scheme, which are both based on allocation rules for communication situations. We introduce two notions of convexity in the setting of reward games and with these notions of convexity we characterize the classes of reward games where the Myerson allocation scheme and the position allocation scheme are link monotonic. As a by-product we find a characterization of the Myerson value and the position value on the class of reward games using potentials.network;reward game;monotonic allocation scheme

Marginality and the position value

We present a new characterization of the position value, one of the most prominent allocation rules for communication situations (graph-games or games with restricted communication). This characterization includes the PL-marginality property, an extension for communications situations of the classic marginality for TU-games, as well as component efficiency and balanced link contributions for necessary players

The role of oxygen vacancies on the structure and the density of states of iron doped zirconia

In this paper we study, both with theoretical and experimental approach, the effect of iron doping in zirconia. Combining density functional theory (DFT) simulations with the experimental characterization of thin films, we show that iron is in the Fe3+ oxidation state and accordingly that the films are rich in oxygen vacancies (VO). VO favor the formation of the tetragonal phase in doped zirconia (ZrO2:Fe) and affect the density of state at the Fermi level as well as the local magnetization of Fe atoms. We also show that the Fe(2p) and Fe(3p) energy levels can be used as a marker for the presence of vacancies in the doped system. In particular the computed position of the Fe(3p) peak is strongly sensitive to the VO to Fe atoms ratio. A comparison of the theoretical and experimental Fe(3p) peak position suggests that in our films this ratio is close to 0.5. Besides the interest in the material by itself, ZrO2:Fe constitutes a test case for the application of DFT on transition metals embedded in oxides. In ZrO2:Fe the inclusion of the Hubbard U correction significantly changes the electronic properties of the system. However the inclusion of this correction, at least for the value U = 3.3 eV chosen in the present work, worsen the agreement with the measured photo-emission valence band spectra.Comment: 24 pages, 8 figure

Game positions of multiple hook removing game

Multiple Hook Removing Game (MHRG for short) introduced in [1] is an impartial game played in terms of Young diagrams. In this paper, we give a characterization of the set of all game positions in MHRG. As an application, we prove that for t ∈ Z≥0 and m, n ∈ N such that t ≤ m ≤ n, and a Young diagram Y contained in the rectangular Young diagram Yt,n of size t × n, Y is a game position in MHRG with Ym,n the starting position if and only if Y is a game position in MHRG with Yt,n−m+t the starting position, and also that the Grundy value of Y in the former MHRG is equal to that in the latter MHRG

Endogeneously arising network allocation rules

In this paper we study endogenously arising network allocation rules. We focus on three allocation rules: the Myerson value, the position value and the component-wise egalitarian solution. For any of these three rules we provide a characterization based on component efficiency and some balanced contribution property. Additionally, we present three mechanisms whose equilibrium payoffs are well defined and coincide with the three rules under consideration if the underlying value function is monotonic. Nonmonotonic value functions are shown to deal with allocation rules applied to monotonic covers. The mechanisms are inspired by the implementation of the Shapley value by P´erez-Castrillo and Wettstein (2001). We conclude with some comments on this implementation of the Shapley value