We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of S × S.

‎Let $R$ be a commutative ring with non-zero identity. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. ‎Also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎Finally, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ...

This report defines various operations and describes algorithms for weighted multi-tape automata (WMTAs). It presents, among others, a new approach to multi-tape intersection, meaning the intersection of a number of tapes of one WMTA with the same number of tapes of another WMTA, which can be seen as a generalization of transducer intersection. In our approach, multi-tape intersection is not co...

PM**£(*') f° r h£rS$2 where h ( < ^ ) is the point nearest to s* at which €(li)«<r(4j). If $(52)==<r(52), then let h~$%. For r<t2 let p(r) =p(^)+log log log fc — log log log r for Ui^r^h where #1 (<*2) is the point of intersection of y—p with y = p (£2) + log log log fe—log log log r. Let p(r)~p for r i^ rgwi . It is always possible to choose H SO large that r\<ti\. We repeat the procedure and ...

Article history: Received 14 November 2009 Received in revised form 29 April 2010 Accepted 6 May 2010

Given a simple generalized polygonA of line segments and arcs that is free to move and rotate and an oriented monotone chainB composed of smooth parametric curved edges, the positions and orientations forA to gouge-freely contactB (i.e., the contact configurations) is a C0 continuous surface in a three dimensional space R3. Past results either limitB to be polygonal or depend on the very compli...

This report defines various operations and describes algorithms for weighted multi-tape automata (WMTAs). It presents, among others, a new approach to multi-tape intersection, meaning the intersection of a number of tapes of one WMTA with the same number of tapes of another WMTA, which can be seen as a generalization of transducer intersection. In our approach, multi-tape intersection is not co...