2,115 research outputs found
Efficiency in the cake-eating problem with quasi-geometric discounting
This paper shows that any equilibrium allocation in the cake-eating problem with quasi-geometric discounting is not Pareto efficient. However, efficiency can be established by introducing a planner who controls the initial endowment and makes transfers over time. It is shown than any Pareto efficient allocation can be supported by a perfect equilibrium with transfers.Pareto efficiency
Exact goodness-of-fit testing for the Ising model
The Ising model is one of the simplest and most famous models of interacting
systems. It was originally proposed to model ferromagnetic interactions in
statistical physics and is now widely used to model spatial processes in many
areas such as ecology, sociology, and genetics, usually without testing its
goodness of fit. Here, we propose various test statistics and an exact
goodness-of-fit test for the finite-lattice Ising model. The theory of Markov
bases has been developed in algebraic statistics for exact goodness-of-fit
testing using a Monte Carlo approach. However, finding a Markov basis is often
computationally intractable. Thus, we develop a Monte Carlo method for exact
goodness-of-fit testing for the Ising model which avoids computing a Markov
basis and also leads to a better connectivity of the Markov chain and hence to
a faster convergence. We show how this method can be applied to analyze the
spatial organization of receptors on the cell membrane.Comment: 20 page
An inequality of Kostka numbers and Galois groups of Schubert problems
We show that the Galois group of any Schubert problem involving lines in
projective space contains the alternating group. Using a criterion of Vakil and
a special position argument due to Schubert, this follows from a particular
inequality among Kostka numbers of two-rowed tableaux. In most cases, an easy
combinatorial injection proves the inequality. For the remaining cases, we use
that these Kostka numbers appear in tensor product decompositions of
sl_2(C)-modules. Interpreting the tensor product as the action of certain
commuting Toeplitz matrices and using a spectral analysis and Fourier series
rewrites the inequality as the positivity of an integral. We establish the
inequality by estimating this integral.Comment: Extended abstract for FPSAC 201
Spontaneous nucleation of structural defects in inhomogeneous ion chains
Structural defects in ion crystals can be formed during a linear quench of
the transverse trapping frequency across the mechanical instability from a
linear chain to the zigzag structure. The density of defects after the sweep
can be conveniently described by the Kibble-Zurek mechanism. In particular, the
number of kinks in the zigzag ordering can be derived from a time-dependent
Ginzburg-Landau equation for the order parameter, here the zigzag transverse
size, under the assumption that the ions are continuously laser cooled. In a
linear Paul trap the transition becomes inhomogeneous, being the charge density
larger in the center and more rarefied at the edges. During the linear quench
the mechanical instability is first crossed in the center of the chain, and a
front, at which the mechanical instability is crossed during the quench, is
identified which propagates along the chain from the center to the edges. If
the velocity of this front is smaller than the sound velocity, the dynamics
becomes adiabatic even in the thermodynamic limit and no defect is produced.
Otherwise, the nucleation of kinks is reduced with respect to the case in which
the charges are homogeneously distributed, leading to a new scaling of the
density of kinks with the quenching rate. The analytical predictions are
verified numerically by integrating the Langevin equations of motion of the
ions, in presence of a time-dependent transverse confinement. We argue that the
non-equilibrium dynamics of an ion chain in a Paul trap constitutes an ideal
scenario to test the inhomogeneous extension of the Kibble-Zurek mechanism,
which lacks experimental evidence to date.Comment: 19 pages, 5 figure
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