In this paper, we consider the quantum version of a Hamiltonian model describing friction. This model consists of a particle which interacts with a bosonic reservoir representing a homogeneous medium through which the particle moves. We show that if the particle is confined, then the Hamiltonian admits a ground state if and only if a suitable infrared condition is satisfied. The latter is violated in the case of linear friction, but satisfied when the friction force is proportional to a higher power of the particle speed.

Given a semidirect product $G\,=\,N\,\rtimes \,H$ where $N$ is nilpotent, connected, simply connected and normal in $G$ and where $H$ is a vector group for which $ad(\mathfrak{h})$ is completely reducible and $\mathbf{R}$-split, let $\tau $ denote the quasiregular representation of $G$ in ${{L}^{2}}(N)$. An element $\psi \,\in \,{{L}^{2}}(N)$ is said to be admissible if the wavelet transform $f\,\mapsto \,\left\langle f,\,\tau (\cdot )\psi \right\rangle $ defines an isometry from ${{L}^{2}}(N)$ into ${{L}^{2}}(G)$. In this paper we give an explicit construction of admissible vectors in the case where $G$ is not unimodular and the stabilizers in $H$ of its action on $\hat{N}$ are almost everywhere trivial. In this situation we prove orthogonality relations and we construct an explicit decomposition of ${{L}^{2}}(G)$ into $G$-invariant, multiplicity-free subspaces each of which is the image of a wavelet transform . We also show that, with the assumption of (almost-everywhere) trivial stabilizers, non-unimodularity is necessary for the existence of admissible vectors.

We derive a weighted ${{L}^{2}}$-estimate of the Witten spinor in a complete Riemannian spin manifold $({{M}^{n}},\,g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of $M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of $M$.

Let $G$ be a locally compact group, and let ${{A}_{\text{cb}}}(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that ${{C}^{*}}(G)$ is residually finite-dimensional, we show that ${{A}_{\text{cb}}}(G)$ is operator amenable. In particular, ${{A}_{\text{cb}}}({{\mathbb{F}}_{2}})$ is operator amenable even though ${{\mathbb{F}}_{2}}$, the free group in two generators, is not an amenable group. Moreover, we show that if $G$ is a discrete group such that ${{A}_{\text{cb}}}(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$ if and only if it has an approximate identity bounded in the $\text{cb}$-multiplier norm.

In this paper we study the Chen–Ruan cohomology ring of weighted projective spaces. Given a weighted projective space ${{\mathbf{P}}^{n}}_{{{q}_{0}},\ldots ,{{q}_{n}}}$, we determine all of its twisted sectors and the corresponding degree shifting numbers. The main result of this paper is that the obstruction bundle over any 3-multisector is a direct sum of line bundles which we use to compute the orbifold cup product. Finally we compute the Chen–Ruan cohomology ring of weighted projective space $\mathbf{P}_{1,\,2,\,2,\,3,3,\,{{3}^{\centerdot }}}^{5}$.

Cubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in ${{\mathbb{R}}^{d}}$ in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris–Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.

Suppose that we have the unit Euclidean ball in ${{\mathbb{R}}^{n}}$ and construct new bodies using three operations — linear transformations, closure in the radial metric, and multiplicative summation defined by ${{\left\| x \right\|}_{K+0L}}\,=\,\sqrt{{{\left\| x \right\|}_{K}}{{\left\| x \right\|}_{L}}}$. We prove that in dimension 3 this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in ${{L}_{0}}$ that naturally extends the corresponding properties of ${{L}_{P}}$-spaces with $p\,\ne \,0$, and show that the procedure described above gives exactly the unit balls of subspaces of ${{L}_{0}}$ in every dimension. We provide Fourier analytic and geometric characterizations of spaces embedding in ${{L}_{0}}$, and prove several facts confirming the place of ${{L}_{0}}$ in the scale of ${{L}_{P}}$-spaces.

Let $\mathcal{D}$ be a division algebra over a nonarchimedean local field. Given an irreducible representation $\pi $ of $G{{L}_{2}}(\mathcal{D})$, we describe its restriction to the diagonal subgroup ${{\mathcal{D}}^{*}}\,\times \,{{\mathcal{D}}^{*}}$. The description is in terms of the structure of the twisted Jacquet module of the representation $\pi $. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on restriction also shows that $\pi $ is ${{\mathcal{D}}^{*}}\,\times \,{{\mathcal{D}}^{*}}$-distinguished if and only if $\pi $ admits a Shalika model. We further prove that if $\mathcal{D}$ is a quaternion division algebra then the twisted Jacquet module is multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to ${{\mathcal{D}}^{*}}\,\times \,{{\mathcal{D}}^{*}}$ in the quaternionic case.

Le diagramme d’Eisenbud et Neumann d’un germe est un arbre qui représente ce germe et permet d’en calculer les invariants. On donne une démonstration algébrique d’un résultat caractérisant l’ensemble des quotients jacobiens d’un germe d’application $(f,\,g)$ à partir du diagramme d’Eisenbud et Neumann de $fg$.

In this paper we study ruled surfaces which appear as exceptional surface in a succession of blowing-ups. In particular we prove that the $e$-invariant of such a ruled exceptional surface $E$ is strictly positive whenever its intersection with the other exceptional surfaces does not contain a fiber (of $E$). This fact immediately enables us to resolve an open problem concerning an intersection configuration on such a ruled exceptional surface consisting of three nonintersecting sections. In the second part of the paper we apply the non-vanishing of $e$ to the study of the poles of the well-known topological, Hodge and motivic zeta functions.