Boundedness of maximal functions on non-doubling manifolds with ends

Let $M$ be a manifold with ends constructed in \cite{GS} and $\Delta$ be the Laplace-Beltrami operator on $M$. In this note, we show the weak type $(1,1)$ and $L^p$ boundedness of the Hardy-Littlewood maximal function and of the maximal function associated with the heat semigroup \M_\Delta f(x)=\sup_{t> 0} |\exp (-t\Delta)f(x)| on $L^p(M)$ for $1 < p \le \infty$. The significance of these results comes from the fact that $M$ does not satisfies the doubling condition

AIDA: ab initio domain assembly server.

AIDA: ab initio domain assembly server, available at http://ffas.burnham.org/AIDA/ is a tool that can identify domains in multi-domain proteins and then predict their 3D structures and relative spatial arrangements. The server is free and open to all users, and there is an option for a user to provide an e-mail to get the link to result page. Domains are evolutionary conserved and often functionally independent units in proteins. Most proteins, especially eukaryotic ones, consist of multiple domains while at the same time, most experimentally determined protein structures contain only one or two domains. As a result, often structures of individual domains in multi-domain proteins can be accurately predicted, but the mutual arrangement of different domains remains unknown. To address this issue we have developed AIDA program, which combines steps of identifying individual domains, predicting (separately) their structures and assembling them into multiple domain complexes using an ab initio folding potential to describe domain-domain interactions. AIDA server not only supports the assembly of a large number of continuous domains, but also allows the assembly of domains inserted into other domains. Users can also provide distance restraints to guide the AIDA energy minimization