### A Bound for the Fixed-Point Index of an Area-Preserving Map with Applications to Mechanics.

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We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map ${f}_{*}{|}_{Y}:Y\subseteq {J}^{r}(D,M)\to {J}^{r}(D,N)$ is generically (for $f:M\to N$) transverse to a submanifold $Z\subseteq {J}^{r}(D,N)$. We apply this to study transversality properties of a restriction of a fixed map $g:M\to P$ to the preimage ${\left({j}^{s}f\right)}^{-1}\left(A\right)$ of a submanifold $A\subseteq {J}^{s}(M,N)$ in terms of transversality properties of the original map $f$. Our main result is that for a reasonable class of submanifolds $A$ and a generic map $f$ the restriction ${g|}_{{\left({j}^{s}f\right)}^{-1}\left(A\right)}$ is also generic. We also present an example of $A$ where the...

We present an example of an o-minimal structure which does not admit ${C}^{\infty}$ cellular decomposition. To this end, we construct a function $H$ whose germ at the origin admits a ${C}^{k}$ representative for each integer $k$, but no ${C}^{\infty}$ representative. A number theoretic condition on the coefficients of the Taylor series of $H$ then insures the quasianalyticity of some differential algebras ${\mathcal{A}}_{n}\left(H\right)$ induced by $H$. The o-minimality of the structure generated by $H$ is deduced from this quasianalyticity property.

We calculate the mapping $H*(BO;\mathbb{Z}\u2082)\to H*({K}^{1,0};\mathbb{Z}\u2082)$ and obtain a generating system of its kernel. As a corollary, bounds on the codimension of fold maps from real projective spaces to Euclidean space are calculated and the rank of a singular bordism group is determined.

This is mainly a survey on the theory of caustics and wave front propagations with applications to differential geometry of hypersurfaces in Euclidean space. We give a brief review of the general theory of caustics and wave front propagations, which are well-known now. We also consider a relationship between caustics and wave front propagations which might be new. Moreover, we apply this theory to differential geometry of hypersurfaces, getting new geometric properties.