In this paper, we construct an explicit isomorphism between Deeley $\mathbb{R}/\mathbb{Z}$-K-homology and flat K-homology. We also describe $\mathbb{R}/\mathbb{Z}$-K-homology out of $\mathbb{Z}/k\mathbb{Z}$-bordism theories.

We show the necessary part of the following theorem : a finitely generated, residually finite group has property $PL^p$ (i.e. it admits a proper isometric affine action on some $L^p$ space) if, and only if, one (or equivalently, all) of its box spaces admits a fibred coarse embedding into some $L^p$ space (sufficiency is due to [CWW13]). We also prove that coarse embeddability of a box space of a group into a $L^p$ space implies property $PL^p$ for this group.

For every group $ \mathrm{PSL}(2,q)$, $q$ a prime power, we classify all two-transitive pairs $(U,U_0)$ consisting of a subgroup $U$ of $ \mathrm{PSL}(2,q)$ and a subgroup $U_0$ of $U$ such that the action of $U$ on the cosets of $U_0$ is two-transitive. We obtain twenty classes up to conjugacy in $ \mathrm{PSL}(2,q)$ or fusion in $P\Gamma L (2,q)$ except for two cases in which we don't have that control.

In this paper we show when a bijection on a set $X$ can be made either an isometry or an expansive map with respect to a non-discrete metric on $X$. As a corollary we obtain that any bijection on an infinite set can be made biLipschitz by a non-discrete metric.

We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. We also prove that the free space over a countable compact metric space has the Schur property. These results answer questions by G. Godefroy.

A curve is, in this paper, the image of the circle $S^1$ under an immersion $f$ into $S^2$, $\mathbb{R}^2$ or the real projective plane $P_2(\mathbb{R})$, such that every multiple point of $f$ is an ordinary double point. Such a curve $C$ is double point-homogeneous or DP-homogeneous when the group of diffeomorphisms (of $S^2, \mathbb{R}^2$ or $P_2(\mathbb{R})$) preserving $C$ has a transitive action on the set of its double points. The orbits of DP-homogeneous curves in $S^2$ are totally determined; using combinatorial methods, we prove that they fall into five countably infinite families ; the description of every family is illustrated by drawings of some representatives with a small number of double points. As a corollary, we obtain a similar classification of the DP-homogeneous curves in $\mathbb{R}^2$. We also propose a conjecture about the classification of DP-homogeneous curves in $P_2(\mathbb{R})$.

Let $\Omega$ be a Carathéodory domain in the complex plane $\mathbb C$, $A^{-\infty}(\Omega)$ the space of functions that are holomorphic in $\Omega$ with polynomial growth near the boundary $\partial\Omega$, and $A^\infty(\complement\Omega)$ the space of holomorphic functions in the interior of $\complement\Omega:=\overline{\mathbb C}\setminus\Omega$, vanishing at infinity and being in $C^\infty(\complement\Omega)$. We prove that the Cauchy transformation of analytic functionals establishes a mutual duality between spaces $A^{-\infty}(\Omega)$ and $A^\infty(\complement\Omega)$. This result, together with those of [3], gives a solution to duality problem for the space $A^{-\infty}(\Omega)$ in both one and several complex variables.

The aim of the present paper is the classification of real hypersurfaces $M$, whose Jacobi structure Operator is \emph{generalized $\xi-$parallel}. The notion of generalized $\xi-$parallel Jacobi structure Operator is rather new and much weaker than $\xi-$ parallel Jacobi structure Operator which has been studied so far.

We look at value functions of primes in simple Artinian rings and associate arithmetical pseudo-valuations to Dubrovin valuation rings which, in the Noetherian case, are $\mathbb{Z}$-valued. This allows a divisor theory for bounded Krull orders.

In this paper, we study the surfaces whose geodesics are slant curves. We show that a unit speed curve $\gamma$ in the 3-dimensional Euclidean space is a slant helix if and only if it is a geodesic of a helix surface. We prove that the striction line of a helix surface is a general helix; as a consequence, slant helices are characterized as geodesics of the tangent surface of a general helix. Finally, we provide two methods for constructing slant helices in helix surfaces.

Dans cet article, nous construisons le morphisme exponentiel $e^D:A\longrightarrow A$ associé à une dérivation localement nilpotente $D$ sur un monoïde $A$ dans une catégorie monoïdale symétrique. De plus, nous montrons que le noyau de $D$ coïncide avec le sous-objet de $A$ fixé par $e^D$. Enfin, nous étudions comment on peut prolonger une dérivation localement nilpotente sur $A$ à sa localisation $A_S$, où $S\subseteq Hom_{A-Mod}(A,A)$ est un sous-ensemble stable par multiplication.

In this paper, we construct the exponential morphism $e^D:A\longrightarrow A$ associated to a locally nilpotent derivation $D$ on a monoid $A$ in a symmetric monoidal category. Further, we show that the kernel of $D$ is identical to the subobject of $A$ invariant under $e^D$. Finally, we study how to extend a locally nilpotent derivation on $A$ to its localization $A_S$, where $S\subseteq Hom_{A-Mod}(A,A)$ is a multiplicatively closed subset.