We recover Reidemeister’s theorem using $\mathcal {C}^{\infty }$ functions and transversality.

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work by Goldberg and Keen [The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift. Invent. Math. 101(2) (1990), 335–372] on degree two rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further is that, despite the presence of the essential singularity, our new functions exhibit a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted.

Let G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f-trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points.

We prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_{q}$ with $d\leqslant q+1$, then there is an $\mathbb{F}_{q}$-line $L$ that intersects $C$ transversely. We also prove the same result for non-reflexive curves of degree $p+1$ and $2p+1$ when $q=p^{r}$.

Metric regularity theory lies in the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. The paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.

This paper explores the relationship between law and social theory through a discussion of the shifting views over the last century. The conversation is framed by the example of publication bias in private-public partnership networks, and the inability of a single social theory, including theories of morality and economic utility, to explain the situation’s complexities. The paper argues that where social theory meets law an added value can be generated in terms of legal doctrine if the precarious relationship between autonomy and interconnectedness is respected in terms of transversality, responsiveness, and self-normativity.

This paper is devoted to the general problem of reconstructing the cost from theobservation of trajectories, in a problem of optimal control. It is motivated by thefollowing applied problem, concerning HALE drones: one would like them to decide bythemselves for their trajectories, and to behave at least as a good human pilot. Thisapplied question is very similar to the problem of determining what is minimized in humanlocomotion. These starting points are the reasons for the particular classes of controlsystems and of costs under consideration. To summarize, our conclusion is that in general,inside these classes, three experiments visiting the same values of the control are neededto reconstruct the cost, and two experiments are in general not enough. The method isconstructive.

The proof of these results is mostly based upon the Thom’s transversality theory.

This study is partly supported by FUI AAP9 project SHARE, and by ANR Project GCM, program“blanche”, project number NT09-504490.

In this paper we present Thom’s transversality theorem in o-minimal structures (a generalization of semialgebraic and subanalytic geometry). There are no restrictions on the differentiability class and the dimensions of manifolds involved in comparison withthe general case.

The significance of singularities in the design and control of robot manipulators is well known, and there is an extensive literature on the determination and analysis of singularities for a wide variety of serial and parallel manipulators—indeed such an analysis is an essential part of manipulator design. Singularity theory provides methodologies for a deeper analysis with the aim of classifying singularities, providing local models and local and global invariants. This paper surveys applications of singularity-theoretic methods in robot kinematics and presents some new results.

It is still an open question whether a compact embedded hypersurface in the Euclidean space $\mathbb{R}^{n+1}$ with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in $\mathbb{R}^3$. In a recent paper, Alías and Malacarne (Rev. Mat. Iberoamericana18 (2002), 431–442) have shown that this is true for the case of hypersurfaces in $\mathbb{R}^{n+1}$ with constant scalar curvature, and more generally, hypersurfaces with constant higher-order $r$-mean curvature, when $r\geq2$. In this paper we deal with some aspects of the classical problem above, by considering it in a more general context. Specifically, our starting general ambient space is an orientable Riemannian manifold $\bar{M}$, where we will consider a general geometric configuration consisting of an immersed hypersurface into $\bar{M}$ with boundary on an oriented hypersurface $P$ of $\bar{M}$. For such a geometric configuration, we study the relationship between the geometry of the hypersurface along its boundary and the geometry of its boundary as a hypersurface of $P$, as well as the geometry of $P$ as a hypersurface of $\bar{M}$. Our approach allows us to derive, among others, interesting results for the case where the ambient space has constant curvature (the Euclidean space $\mathbb{R}^{n+1}$, the hyperbolic space $\mathbb{H}^{n+1}$, and the sphere $\mathbb{S}^{n+1}$). In particular, we are able to extend the symmetry results given in the recent paper mentioned above to the case of hypersurfaces with constant higher-order $r$-mean curvature in the hyperbolic space and in the sphere.

In singularity theory, J. Damon gave elegant versions of the unfolding and determinacy theorems for geometric subgroups of . and . In this work, we propose a unified treatment of the smooth stability of germs and the structural stability of versai unfoldings for a large class of such subgroups.