8,087 research outputs found
Global stability for the prion equation with general incidence
We consider the so-called prion equation with the general incidence term
introduced in [Greer et al., 2007], and we investigate the stability of the
steady states. The method is based on the reduction technique introduced in
[Gabriel, 2012]. The argument combines a recent spectral gap result for the
growth-fragmentation equation in weighted spaces and the analysis of a
nonlinear system of three ordinary differential equations
Single-equation tests for cointegration with GLS detrended data
We provide GLS-based versions of two widely used approaches for testing whether or not non-stationary economic time series are cointegrated: single-equation static re- gression or residual-based tests and single-equation conditional error correction model (ECM) based tests. Our approach is to consider nearly optimal tests for unit roots and apply them in the cointegration context. Our GLS versions of the tests do in- deed provide substantial improvements over their OLS counterparts. We derive the local asymptotic power functions of all tests considered for a DGP with weakly ex- ogenous regressors. This allows obtaining the relevant non-centrality parameter to quasi-di§erence the data. We investigate the e§ect of non-weakly exogenous regressors via simulations. With weakly exogenous regressors strongly correlated with the depen- dent variable, the ECM tests are clearly superior. When the regressors are potentially non-weakly exogenous, the residuals-based tests are clearly preferred
Residuals-based tests for cointegration with generalized least-squares detrended data
We provide generalized least-squares (GLS) detrended versions of single-equation static regression or residuals-based tests for testing whether or not non-stationary time series are cointegrated. Our approach is to consider nearly optimal tests for unit roots and to apply them in the cointegration context. We derive the local asymptotic power functions of all tests considered for a triangular data-generating process, imposing a directional restriction such that the regressors are pure integrated processes. Our GLS versions of the tests do indeed provide substantial power improvements over their ordinary least-squares counterparts. Simulations show that the gains in power are important and stable across various configurations
Optimal growth for linear processes with affine control
We analyse an optimal control with the following features: the dynamical
system is linear, and the dependence upon the control parameter is affine. More
precisely we consider , where
and are matrices with some prescribed structure. In the
case of constant control , we show the existence of an
optimal Perron eigenvalue with respect to varying under some
assumptions. Next we investigate the Floquet eigenvalue problem associated to
time-periodic controls . Finally we prove the existence of an
eigenvalue (in the generalized sense) for the optimal control problem. The
proof is based on the results by [Arisawa 1998, Ann. Institut Henri Poincar\'e]
concerning the ergodic problem for Hamilton-Jacobi equations. We discuss the
relations between the three eigenvalues. Surprisingly enough, the three
eigenvalues appear to be numerically the same
Bayesian Modeling of a Human MMORPG Player
This paper describes an application of Bayesian programming to the control of
an autonomous avatar in a multiplayer role-playing game (the example is based
on World of Warcraft). We model a particular task, which consists of choosing
what to do and to select which target in a situation where allies and foes are
present. We explain the model in Bayesian programming and show how we could
learn the conditional probabilities from data gathered during human-played
sessions.Comment: 30th international workshop on Bayesian Inference and Maximum
Entropy, Chamonix : France (2010
Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate
We are interested in the large time behavior of the solutions to the
growth-fragmentation equation. We work in the space of integrable functions
weighted with the principal dual eigenfunction of the growth-fragmentation
operator. This space is the largest one in which we can expect convergence to
the steady size distribution. Although this convergence is known to occur under
fairly general conditions on the coefficients of the equation, we prove that it
does not happen uniformly with respect to the initial data when the
fragmentation rate in bounded. First we get the result for fragmentation
kernels which do not form arbitrarily small fragments by taking advantage of
the Dyson-Phillips series. Then we extend it to general kernels by using the
notion of quasi-compactness and the fact that it is a topological invariant
High-order WENO scheme for Polymerization-type equations
Polymerization of proteins is a biochimical process involved in different
diseases. Mathematically, it is generally modeled by
aggregation-fragmentation-type equations. In this paper we consider a general
polymerization model and propose a high-order numerical scheme to investigate
the behavior of the solution. An important property of the equation is the mass
conservation. The fifth-order WENO scheme is built to preserve the total mass
of proteins along time
Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts
We study the asymptotic behaviour of the following linear
growth-fragmentation equation and prove that under fairly general assumptions on the division
rate its solution converges towards an oscillatory function,explicitely
given by the projection of the initial state on the space generated by the
countable set of the dominant eigenvectors of the operator. Despite the lack of
hypo-coercivity of the operator, the proof relies on a general relative entropy
argument in a convenient weighted space, where well-posedness is obtained
via semigroup analysis. We also propose a non-dissipative numerical scheme,
able to capture the oscillations
Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems
We study a growth maximization problem for a continuous time positive linear
system with switches. This is motivated by a problem of mathematical biology
(modeling growth-fragmentation processes and the PMCA protocol). We show that
the growth rate is determined by the non-linear eigenvalue of a max-plus
analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the
ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the
solutions or subsolutions of which yield Barabanov and extremal norms,
respectively. We exploit contraction properties of order preserving flows, with
respect to Hilbert's projective metric, to show that the non-linear eigenvector
of the operator, or the "weak KAM" solution of the HJ equation, does exist. Low
dimensional examples are presented, showing that the optimal control can lead
to a limit cycle.Comment: 8 page
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