2,256 research outputs found
Multidimensional extension of the Morse--Hedlund theorem
A celebrated result of Morse and Hedlund, stated in 1938, asserts that a
sequence over a finite alphabet is ultimately periodic if and only if, for
some , the number of different factors of length appearing in is
less than . Attempts to extend this fundamental result, for example, to
higher dimensions, have been considered during the last fifteen years. Let
. A legitimate extension to a multidimensional setting of the notion of
periodicity is to consider sets of \ZZ^d definable by a first order formula
in the Presburger arithmetic . With this latter notion and using a
powerful criterion due to Muchnik, we exhibit a complete extension of the
Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of
$\ZZ^d$ definable in in terms of some functions counting recurrent
blocks, that is, blocks occurring infinitely often
Values on regular games under Kirchhoff’s laws
In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework leading to applications to games with communication graphs, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shapley value proprosed by Faigle and Kern, and Bilbao and Edelman still work. Our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff’s laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which is satisfied by the proposed value.Regular set systems; regular games; Shapley value; probabilistic efficient values; regular values; Kirchhoff’s laws.
Interaction transform for bi-set functions over a finite set
Set functions appear as a useful tool in many areas of decision making and operations research, and several linear invertible transformations have been introduced for set functions, such as the Möbius transform and the interaction transform. The present paper establish similar transforms and their relationships for bi-set functions, i.e. functions of two disjoint subsets. Bi-set functions have been recently introduced in decision making (bi-capacities) and game theory (bi-cooperative games), and appear to open new areas in these fields.Set function; Bi-set function; Möbius transform; Interaction transform
Games on lattices, multichoice games and the Shapley value: a new approach
Multichoice games have been introduced by Hsiao and Raghavan as a generalization of classical cooperative games. An important notion in cooperative game theory is the core of the game, as it contains the rational imputations for players. We propose two definitions for the core of a multichoice game, the first one is called the precore and is a direct generalization of the classical definition. We show that the precore coincides with the definition proposed by Faigle, and that it contains unbounded imputations, which makes its application questionable. A second definition is proposed, imposing normalization at each level, causing the core to be a convex closed set. We study its properties, introducing balancedness and marginal worth vectors, and defining the Weber set and the pre-Weber set. We show that the classical properties of inclusion of the (pre)core into the (pre)-Weber set as well as their equality remain valid. A last section makes a comparison with the core defined by van den Nouweland et al.multichoice game ; lattice ; core
Values on regular games under Kirchhoff's laws
In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework, namely regular set systems, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shaphey value proposed by Faigle and Kern and Bilbao and Edelman still work. our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff's laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which is satisfied by the proposed value.Regular set systems, regular games, Shapley value, probabilistic efficient values, regular values, Kirchhoff's laws.
Super-sample covariance approximations and partial sky coverage
Super-sample covariance (SSC) is the dominant source of statistical error on
large scale structure (LSS) observables for both current and future galaxy
surveys. In this work, we concentrate on the SSC of cluster counts, also known
as sample variance, which is particularly useful for the self-calibration of
the cluster observable-mass relation; our approach can similarly be applied to
other observables, such as galaxy clustering and lensing shear. We first
examined the accuracy of two analytical approximations proposed in the
literature for the flat sky limit, finding that they are accurate at the 15%
and 30-35% level, respectively, for covariances of counts in the same redshift
bin. We then developed a harmonic expansion formalism that allows for the
prediction of SSC in an arbitrary survey mask geometry, such as large sky areas
of current and future surveys. We show analytically and numerically that this
formalism recovers the full sky and flat sky limits present in the literature.
We then present an efficient numerical implementation of the formalism, which
allows fast and easy runs of covariance predictions when the survey mask is
modified. We applied our method to a mask that is broadly similar to the Dark
Energy Survey footprint, finding a non-negligible negative cross-z covariance,
i.e. redshift bins are anti-correlated. We also examined the case of data
removal from holes due to, for example bright stars, quality cuts, or
systematic removals, and find that this does not have noticeable effects on the
structure of the SSC matrix, only rescaling its amplitude by the effective
survey area. These advances enable analytical covariances of LSS observables to
be computed for current and future galaxy surveys, which cover large areas of
the sky where the flat sky approximation fails.Comment: 14 pages, 10 figures. Updated to match version published in Astronomy
& Astrophysic
Algorithmic Aspects of Switch Cographs
This paper introduces the notion of involution module, the first
generalization of the modular decomposition of 2-structure which has a unique
linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm
and we take advantage of the involution modular decomposition tree to state
several algorithmic results. Cographs are the graphs that are totally
decomposable w.r.t modular decomposition. In a similar way, we introduce the
class of switch cographs, the class of graphs that are totally decomposable
w.r.t involution modular decomposition. This class generalizes the class of
cographs and is exactly the class of (Bull, Gem, Co-Gem, C_5)-free graphs. We
use our new decomposition tool to design three practical algorithms for the
maximum cut, vertex cover and vertex separator problems. The complexity of
these problems was still unknown for this class of graphs. This paper also
improves the complexity of the maximum clique, the maximum independant set, the
chromatic number and the maximum clique cover problems by giving efficient
algorithms, thanks to the decomposition tree. Eventually, we show that this
class of graphs has Clique-Width at most 4 and that a Clique-Width expression
can be computed in linear time
Polymer chain generation for coarse-grained models using radical-like polymerization
An innovative method is proposed to generate configurations of coarse grained
models for polymer melts. This method, largely inspired by chemical ``radical
polymerization'', is divided in three stages: (i) nucleation of radicals
(reacting molecules caching monomers); (ii) growth of chains within a solvent
of monomers; (iii) termination: annihilation of radicals and removal of
residual monomers. The main interest of this method is that relaxation is
performed as chains are generated. Pure mono and poly-disperse polymers melts
are generated and compared to the configurations generated by the Push Off
method from Auhl et al.. A detailed study of the static properties (gyration
radius, mean square internal distance, entanglement length) confirms that the
radical-like polymerization technics is suitable to generate equilibrated
melts. The method is flexible, and can be adapted to generate nano-structured
polymers, namely diblock and triblock copolymers.Comment: 9 pages, 12 figure
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