Optimal L2-error estimates for the semidiscrete Galerkin\ud approximation to a second order linear parabolic initial and\ud boundary value problem with nonsmooth initial data

In this article, we have discussed a priori error estimate for the semidiscrete Galerkin approximation of a general second order parabolic initial and boundary value problem with non-smooth initial data. Our analysis is based on an elementary energy argument without resorting to parabolic duality technique. The proposed technique is also extended to a semidiscrete mixed method for parabolic problems. Optimal L2-error estimate is derived for both cases, when the initial data is in L2

A Complete Formulation of Baum-Conens' Conjecture for the Action of Discrete Quantum Groups

We formulate a version of Baum-Connes' conjecture for a discrete quantum group, building on our earlier work (\cite{GK}). Given such a quantum group \cla, we construct a directed family \{\cle_F \} of $C^*$-algebras ($F$ varying over some suitable index set), borrowing the ideas of \cite{cuntz}, such that there is a natural action of \cla on each \cle_F satisfying the assumptions of \cite{GK}, which makes it possible to define the "analytical assembly map", say $\mu^{r,F}_i$, $i=0,1,$ as in \cite{GK}, from the \cla-equivariant $K$-homolgy groups of \cle_F to the $K$-theory groups of the "reduced" dual \hat{\cla_r} (c.f. \cite{GK} and the references therein for more details). As a result, we can define the Baum-Connes' maps \mu^r_i : \stackrel{\rm lim}{\longrightarrow} KK_i^\cla(\cle_F,\IC) \raro K_i(\hat{\cla_r}), and in the classical case, i.e. when \cla is $C_0(G)$ for a discrete group, the isomorphism of the above maps for $i=0,1$ is equivalent to the Baum-Connes' conjecture. Furthermore, we verify its truth for an arbitrary finite dimensional quantum group and obtain partial results for the dual of $SU_q(2).$Comment: to appear in "K Theory" (special volume for H. Bass). A preliminary version was available as ICTP preprint since the early this yea

Endocranial Morphology of the Extinct North American Lion (Panthera atrox)

The extinct North American lion (Panthera atrox) is one of the largest felids (Mammalia, Carnivora) to have ever lived, and it is known from a plethora of incredibly well-preserved remains. Despite this abundance of material, there has been little research into its endocranial anatomy. CT scans of a skull of P. atrox from the Pleistocene La Brea Tar pits were used to generate the first virtual endocranium for this species and to elucidate previously unknown details of its brain size and gross structure, cranial nerves, and inner-ear morphology. Results show that its gross brain anatomy is broadly similar to that of other pantherines, although P. atrox displays less cephalic flexure than either extant lions or tigers, instead showing a brain shape that is reminiscent of earlier felids. Despite this unusual reduction in flexure, the estimated absolute brain size for this specimen is one of the largest reported for any felid, living or extinct. Its encephalization quotient (brain size as a fraction of the expected brain mass for a given body mass) is also larger than that of extant lions but similar to that of the other pantherines. The advent of CT scans has allowed nondestructive sampling of anatomy that cannot otherwise be studied in these extinct lions, leading to a more accurate reconstruction of endocranial morphology and its evolution

Optimal error estimates of a mixed finite element method for\ud parabolic integro-differential equations with non smooth initial data

In this article, a new mixed method is proposed and analyzed for parabolic integro-differential equations (PIDE) with nonsmooth initial data. Compared to mixed methods for PIDE, the present method does not bank on a reformulation using a resolvent operator. Based on energy arguments and without using parabolic type duality technique, optimal L2-error estimates are derived for semidiscrete approximations, when the initial data is in L2. Due to the presence of the integral term, it is, further, observed that estimate in dual of H(div)-space plays a role in our error analysis. Moreover, the proposed analysis follows the spirit of the proof technique used for deriving optimal error estimates of finite element approximations to PIDE with smooth data and therefore, it unifies both the theories, i.e., one for smooth data and other for nonsmooth data. Finally, the proposed analysis can be easily extended to other mixed method for PIDE with rough initial data and provides an improved result