We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2-regularity (a certain kind of second-order regularity) for a once differentiable mapping whose derivative is Lipschitz continuous. Under this 2-regularity condition, we obtain the representation theorem and the covering theore...

let $fm(x)$ be the  space of  all finite regular borel measures on $x$. a general measure algebra is a subspace  of$fm(x)$,which is an $l$-space and has a multiplication preserving the probability measures. let $clsubseteqfm(x)$ be a general measure algebra on a locallycompact space $x$. in this paper, we investigate the relation between arensregularity of $cl$ and the topology of $x$. we  find...

This paper deals with the spatial and temporal regularity of the unique Hilbert space valued mild solution to a semilinear stochastic parabolic partial differential equation with nonlinear terms that satisfy global Lipschitz conditions and certain linear growth bounds. It is shown that the mild solution has the same optimal regularity properties as the stochastic convolution. The proof is eleme...

A new proof of a pathwise uniqueness result of Krylov and Röckner is given. It concerns SDEs with drift having only certain integrability properties. In spite of the poor regularity of the drift, pathwise continuous dependence on initial conditions may be obtained, by means of this new proof. The proof is formulated in such a way to show that the only major tool is a good regularity theory for ...

The Szemerédi Regularity Lemma states that any sufficiently large graph G can be partitioned into a bounded (independent of the size of the graph) number of regular, or “random-looking,” components. The resulting partition can be viewed as a regularity graph R. The Key Lemma shows that under certain conditions, the existence of a subgraph H in R implies its existence in G. We prove the Regulari...