We describe the notion of a weakly Lipschitz mapping on a $C^q$ stratification. We also distinguish a class of regularity conditions that are in some sense invariant under definable, locally Lipschitz and weakly bi-Lipschitz homeomorphisms. This class includes the Whitney (B) condition and the Verdier condition.

The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, this framework gives an efficient algorithm to calculate all tautological equations using only finite-dimensional linear algebra. Other applications include the proofs of Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy and Virasoro conjecture for target manifolds with conformal semisimple quantum cohomology, both for genus up to...

Let $𝔤$ be a complex, semisimple Lie algebra, with an involutive automorphism $\vartheta$ and set $𝔨=\mathrm{Ker}(\vartheta -I)$, $𝔭=\mathrm{Ker}(\vartheta +I)$. We consider the differential operators, $𝒟(𝔭{)}^K$, on $𝔭$ that are invariant under the action of the adjoint group $K$ of $𝔨$. Write $\tau :𝔨\to \mathrm{Der}𝒪\left(𝔭\right)$ for the differential of this action. Then we prove, for the class of symmetric pairs $(𝔤,𝔨)$ considered by Sekiguchi, that $\left\{d\in 𝒟\left(𝔭\right):d\left(𝒪(𝔭{)}^K\right)=0\right\}=𝒟\left(𝔭\right)\tau \left(𝔨\right)$. An immediate consequence of this equality is the following result of Sekiguchi: Let $({𝔤}_0,{𝔨}_0)$ be a real form of one of these symmetric pairs $(𝔤,𝔨)$, and suppose that $T$ is a $K_0$-invariant...

In this paper we develop fundamental tools and methods to study meromorphic functions in an equivariant setup. As our main result we construct quotients of Rosenlicht-type for Stein spaces acted upon holomorphically by complex-reductive Lie groups and their algebraic subgroups. In particular, we show that in this setup invariant meromorphic functions separate orbits in general position. Applications to almost homogeneous spaces and principal orbit types are given. Furthermore, we use the main result...

We investigate the action of the automorphism group of a closed Riemann surface of genus at least two on its set of theta characteristics (or spin structures). We give a characterization of those surfaces admitting a non-trivial automorphism fixing either all of the spin structures or just one. The case of hyperelliptic curves and of the Klein quartic are discussed in detail.

This paper presents a parametrization of a feed-forward control based on structures of subspaces for a non-interacting regulation. With advances in technological development, robotics is increasingly being used in many industrial sectors, including medical applications (e. g., micro-manipulation of internal tissues or laparoscopy). Typical problems in robotics and general mechanisms may be mathematically formalized and analyzed, resulting in outcomes so general that it is possible to speak of structural...

In this article, we complete the interpretation of groups of classes of invariant divisors on a complex toric variety X of dimension n in terms of suitable (co-) homology groups. In [BBFK], we proved the following result (see Satz 1 below): Let $ClDi{v}_C^{}\left(X\right)$ and $ClDi{v}_W^{}\left(X\right)$ denote the groups of classes of invariant Cartier resp. Weil divisors on X. If X is non degenerate (i.e., not equivariantly isomorphic to the product of a toric variety and a torus of positive dimension), then the natural homomorphisms $ClDi{v}_C^{}\left(X\right)\to H^2\left(X\right)$ and $ClDi{v}_W^{}\left(X\right)\to H_{2n-2}^{cld}\left(X\right)$ are...