We prove uniform Hölder regularity estimates for a transport-diffusion equation with a fractional diffusion operator and a general advection field in of bounded mean oscillation, as long as the order of the diffusion dominates the transport term at small scales; our only requirement is the smallness of the negative part of the divergence in some critical Lebesgue space. In comparison to a celebrated result by Silvestre, our advection field does not need to be bounded. A similar result can be obtained in the supercritical case if the advection field is Hölder continuous. Our proof is inspired by Kiselev and Nazarov and is based on the dual evolution technique. The idea is to propagate an atom property (i.e., localisation and integrability in Lebesgue spaces) under the dual conservation law, when it is coupled with the fractional diffusion operator.

In this paper, we consider a generalized Pólya urn model with multiple drawings and time-dependent reinforcements. Suppose an urn initially contains w white and r red balls. At the nth action, m balls are drawn at random from the urn, say k white and m−k red balls, and then replaced in the urn along with cnk white and cn(m − k) red balls, where {cn} is a given sequence of positive integers. Repeat the above procedure ad infinitum. Let Xn be the proportion of the white balls in the urn after the nth action. We first show that Xn converges almost surely to a random variable X. Next, we give a necessary and sufficient condition for X to have a Bernoulli distribution with parameter w/(w + r). Finally, we prove that X is absolutely continuous if {cn} is bounded.

This article presents an overview of the published results for planetary nebulae based on images and spectroscopy from the PACS, SPIRE, and HIFI instruments on board the Herschel satellite.

In this paper, we reconsider two models of noncooperative oligopoly in general equilibrium proposed by Busetto et al. ((2008), (2011)): a version of the Shapley's window model for mixed exchange economies a la Shitovitz and its reformulation a la Cournot-Walras. We introduce the assumption that preferences of the traders belonging to the atomless part are represented by Cobb-Douglas utility functions. This assumption permits us to prove the existence of a Cournot-Nash equilibrium of the Shapley's window model - called Cobb-Douglas-Cournot-Nash equilibrium - without introducing further assumptions on atoms' endowments and preferences previously used by Busetto et al. (2011). Then, we show that the set of the Cobb-Douglas-Cournot-Nash equilibrium allocations coincides with the set of the Cournot-Walras equilibrium allocations.

In this paper a $T(1)$ theorem for Calder\'on--Zygmund operators (CZOs) valid for non-doubling measures with atoms is proved. In the classical Calder\'on--Zigmund theory an essential assumption is the doubling property of the underlying measure $\mu$ in ${\mathbb R}^n$. Some recent results have shown that a big part of the classical theory can be extended to the case of non-doubling measures, with only some mild `growth' condition on the measure $\mu$. In the particular case of the Cauchy transform $Cf(x) = \int f(y)/(y-x) \, d\mu(y)$, the main result of the paper can be stated easily: the Cauchy transform is bounded in $L^2(\mu)$ if and only if it is bounded in $L^2(\mu)$ over characteristic functions of squares. The proof of the main result is based on the use of the Haar basis with random dyadic lattices, following an idea of Nazarov, Treil and Volberg (`Cauchy integral and Calder\'on--Zygmund operators on nonhomogeneous spaces', {\em Internat. Math. Res. Notices} (1997) 15, 703--726), where a $T(1)$ theorem for non-doubling continuous measures was obtained. The case of a measure $\mu$ with atoms studied in the present paper is quite different from the case of continuous measures. For example, if $\mu$ has atoms, the $L^2(\mu)$ boundedness of a CZO does not imply its $L^p(\mu)$ boundedness for some $p\neq2$. The differences arise because a measure $\mu$ with atoms does not satisfy the essential `growth' condition that is involved in the arguments for proving the $L^2$ boundedness of CZOs for continuous measures. 2000 Mathematics Subject Classification: 42B20.

We establish several first- or second-order properties of models of first-order theories by considering their elements as atoms of a new universe of set theory and by extending naturally any structure of Boolean model on the atoms to the whole universe. For example, complete f-rings are “boundedly algebraically compact” in the language (+, −, ·, ∧, ∨, ≤), and the positive cone of a complete l-group with infinity adjoined is algebraically compact in the language (+, ∨, ≤). We also give an example with any first-order language. The proofs can be translated into “naive set theory” in a uniform way.