Operadic categories and Duoidal Deligne's conjecture

The purpose of this paper is two-fold. In Part 1 we introduce a new theory of operadic categories and their operads. This theory is, in our opinion, of an independent value. In Part 2 we use this new theory together with our previous results to prove that multiplicative 1-operads in duoidal categories admit, under some mild conditions on the underlying monoidal category, natural actions of contractible 2-operads. The result of D. Tamarkin on the structure of dg-categories, as well as the classical Deligne conjecture for the Hochschild cohomology, is a particular case of this statement.Comment: 54 pages, to appear in Advances in Mathematic

Bar constructions for topological operads and the Goodwillie derivatives of the identity

We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper20.abs.html Version 3: Reference to Salvatore added (see footnote 3, page 834

In vivo visualization and analysis of 3-D hemodynamics in cerebral aneurysms with flow-sensitized 4-D MR imaging at 3T

Introduction: Blood-flow patterns and wall shear stress (WSS) are considered to play a major role in the development and rupture of cerebral aneurysms. These hemodynamic aspects have been extensively studied in vitro using geometric realistic aneurysm models. The purpose of this study was to evaluate the feasibility of in vivo flow-sensitized 4-D MR imaging for analysis of intraaneurysmal hemodynamics. Methods: Five cerebral aneurysms were examined using ECG-gated, flow-sensitized 4-D MR imaging at 3T in three patients. Postprocessing included quantification of flow velocities, visualization of time-resolved 2-D vector graphs and 3-D particle traces, vortical flow analysis, and estimation of WSS. Flow patterns were analyzed in relation to aneurysm geometry and aspect ratio. Results: Magnitude, spatial and temporal evolution of vortical flow differed markedly among the aneurysms. Particularly unstable vortical flow was demonstrated in a wide-necked parophthalmic ICA aneurysm (high aspect ratio). Relatively stable vortical flow was observed in aneurysms with a lower aspect ratio. Except for a wide-necked cavernous ICA aneurysm (low aspect ratio), WSS was reduced in all aneurysms and showed a high spatial variation. Conclusion: In vivo flow-sensitized 4-D MR imaging can be applied to analyze complex patterns of intraaneurysmal flow. Flow patterns, distribution of flow velocities, and WSS seem to be determined by the vascular geometry of the aneurysm. Temporal and spatial averaging effects are drawbacks of the MR-based analysis of flow patterns as well as the estimation of WSS, particularly in small aneurysms. Further studies are needed to establish a direct link between definitive flow patterns and different aneurysm geometrie