Mandibular Torus : A Synthesis of New and Previously Reported Data and a Discusssion of its Cause

Using original data on 1,5000 mandibles, but mainly previously published data, I present a overview of the distribution characteristics of mandibular torus and a hypothesis concerning its cause. Pedigree studies have established that genetic factors influence torus development. Extrinsic factors are strongly implicated by other evidence: prevalence among Arctic peoples, effect of dietary change, age regression, preponderance in males and on the right side, effect of cranial deformation, concurrence with palatine torus and maxillary alveolar exostoses, and clinical evidence. I propose that the primary factor is masticatory stress. According to a mechanism suggested by orthodontic research, the horizontal component of bite force tips the lower canine, premolars and first molar so that their root apices exert pressure on the periodontal membrane, causing formation of new bone on the lingual cortical plate of the alveolar process. Thus formed, the hyperostosis is vulnerable to trauma and its periosteal covering becomes bruised causing additional deposition of bone. Genes influence torus indirectly through their effect on occlusion. A patern of increased expressivity with incidence suggests that a quasicontinuous model may provide a better fit to pedigree data than single locus models previously tested.

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.