Geometry of Dirac Operators

Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$. Then \[ \textsc{Atiyah Singer: } \quad \text{Ind } \mathsf{D}= \int_M \hat{\mathcal{A}}(TM)\wedge \text{ch}(\mathcal{V}) \] where $\mathcal{V} =\text{Hom}_{\mathbb{C}l(TM)}(\slashed{\mathsf{S}},S)$. We prove the above statement with the means of the heat kernel of the heat semigroup $e^{-tD^2}$. The first outstanding result is the McKean-Singer theorem that describes the index in terms of the supertrace of the heat kernel. The trace of heat kernel is obtained from local geometric information. Moreover, if we use the asymptotic expansion of the kernel we will see that in the computation of the index only one term matters. The Berezin formula tells us that the supertrace is nothing but the coefficient of the Clifford top part, and at the end, Getzler calculus enables us to find the integral of these top parts in terms of characteristic classes.

Coagent Modified Polypropylene Prepared by Reactive Extrusion: A New Look Into the Structure-Property Relations of Injection Molded Parts

Peroxide-mediated reactive extrusion of linear isotactic polypropylene (L-PP) was conducted in the presence of trimethylolpropane trimethacrylate (TMPTMA) and triallyl trimesate (TAM) coagents, using a twin screw extruder. The resulting coagent-modified polypropylenes (CM-PP) had higher viscosities and elasticities, as well as increased crystallization temperature compared to PP reacted only with peroxide (DCP-PP). Additionally, deviations from terminal flow, and strain hardening were observed in PP modified with TAM, signifying the presence of long chain branching (LCB). The CM-PP formulations retained the modulus and tensile strength of the parent L-PP, in spite of their lower molar mass and viscosities, whereas their elongation at break and the impact strength were better. This was attributed to the finer spherulitic structure of these materials, and to the disappearance of the skin-core layer in the injection molded specimens.

Degeneracy of velocity constraints in rigid body systems

The equations governing the dynamics of rigid body systems with velocity constraints are singular at degenerate configurations in the constraint distribution. In this report, we describe the causes of singularities in the constraint distribution of interconnected rigid body systems with smooth configuration manifolds. A convention of defining primary velocity constraints in terms of orthogonal complements of one-dimensional subspaces is introduced. Using this convention, linear maps are defined and used to describe the space of allowable velocities of a rigid body. Through the definition of these maps, we present a condition for non-degeneracy of velocity constraints in terms of the one dimensional subspaces defining the primary velocity constraints. A method for defining the constraint subspace and distribution in terms of linear maps is presented. Using these maps, the constraint distribution is shown to be singular at configuration where there is an increase in its dimension.