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In this paper, we study the third Painleve equation with parameters gamma = 0, alpha delta not equal 0. The Puiseux series formally satisfying this equation (after a certain change of variables) asymptotically approximate of Gevrey order one solutions to this equation in sectors with vertices at infinity. We present a family of values of the parameters delta = -beta(2)/2 not equal 0 such that these series are of exact Gevrey order one, and hence diverge. We prove the 1-summability of them and provide analytic functions which are approximated of Gevrey order one by these series in sectors with the vertices at infinity.

C´eline Catto¨en and Matt VisserSchool of Mathematics, Statistics, and Computer Science,Victoria University of Wellington,P.O.Box 600, Wellington, New Zealandceline.cattoen@mcs.vuw.ac.nz, matt.visser@mcs.vuw.ac.nzWe use generalized Puisieux series expansions to determine the behaviour of the scalefactor in the vicinity of typical cosmological milestones occurring in a FRW universe. Wedescribe some of the consequences of this generalized Puisieux series expansion on otherphysical observables.

We complete the stability results of the paper Bourles et al. (SIAM J Control Optim 53:2725---2761, 2015), and for this purpose use the linear time-varying (LTV) discrete-time behaviors and the exponential stability (e.s.) of this paper. In the main theorem we characterize the e.s. of an autonomous LTV system by standard spectral properties of a complex matrix connected with the system. We extend the theory of discrete-time LTV behaviors, developed in the quoted publication, from the coefficient field of rational functions to that of locally convergent Laurent series or even of Puiseux series. The stability test can and has to be applied in connection with the construction of stabilizing compensators.

Robust multivariable controllers for stable infinite-dimensional systems in the Callier-Desoer (CD) algebra are discussed. In particular, the following robust regulation problem is solved. Given reference and disturbance signals, which are linear combinations of signals of the form t/sup j/ sin(/spl omega//sub k/t+/spl phi//sub k/), j/spl ges/0, k=0, /spl middot//spl middot//spl middot/, n, find a low-order finite-dimensional controller so that the outputs asymptotically track the reference signals, asymptotically reject the disturbance signals, and the closed-loop system is stable and robust with respect to a class of perturbations in the plant. The proposed controller consists of a positive scalar gain /spl epsi/ and certain polynomial matrices K/sub k/(s) for k=0, /spl middot//spl middot//spl middot/, n. The main result of the paper shows that the matrices K/sub k/(s) must satisfy certain stability conditions involving the values of the plant transfer function only at the reference and disturbance signal frequencies /spl omega//sub k/ for k=0, /spl middot//spl middot//spl middot/, n. The controller has unstable poles on the imaginary axis. The behavior of these poles as a function of the scalar parameter /spl epsi/ in the form of a Puiseux series is given.

The authors study two person, zero sum, stochastic games with zero stop probabilities. Two distinct formulations are emphasized, (1) the infinite stage game with payoffs discounted at an interest rate close to zero and (2) the game with a large but finite number of stages. The authors give a complete theory of such games. The work implies all known existence theorems for optimal policies in Markov decision processes. It also generalizes all previous existence theorems for the value of a stochastic game. The approach differs from previous work in that it is algebraic and makes no use of the theory of Markov chains. The essential idea of this approach is to apply Tarski's Principle on real closed fields to a field of asymptotic expansions, which the authors term the field of real Puiseux series.