Linear semigroups with coarsely dense orbits

Let $S$ be a finitely generated abelian semigroup of invertible linear operators on a finite dimensional real or complex vector space $V$. We show that every coarsely dense orbit of $S$ is actually dense in $V$. More generally, if the orbit contains a coarsely dense subset of some open cone $C$ in $V$ then the closure of the orbit contains the closure of $C$. In the complex case the orbit is then actually dense in $V$. For the real case we give precise information about the possible cases for the closure of the orbit.Comment: We added comments and remarks at various places. 14 page

Grain-size characterization of reworked fine-grained aeolian deposits

After a previous review of the grain-size characteristics of in situ (primary) fine-grained aeolian deposits, reworked (secondary) aeolian deposits, as modified in lacustrine environments and by alluvial and pedogenic processes, are discussed in this paper. As a reference, the grain-size characteristics of primary loess deposits are shortly described. Commonly, pedogenesis and weathering of primary loess may lead to clay neoformation and thus to an enrichment in grain diameters of 4-8 mu m, a size which is comparable to the fine background loess. Remarkably, the modal grain-size values of primary loess are preserved after re -deposition in lakes and flood plains. But, secondary lacustrine settings show a very characteristic admixture with a clayey population of 1-2,5 mu m diameter due to the process of settling in standing water. Similarly, alluvial settings show often an addition with coarse-grained sediment supplied by previously eroded sediment. However, floodplain settings show also often the presence of pools and other depressions which behave similarly to lacustrine environments. As a result, alluvial secondary loess sediments are characterized by the poorest grain-size sorting when compared with the other secondary loess and primary loess. Despite the characteristic texture of each of these deposits, grain-size characteristics of the described individual sediment categories are not always fully diagnostic and thus grain-size analysis should be complemented by other information, as sedimentary structures and fauna or flora, to reliably reconstruct the sedimentary processes and environments

Coercivity and stability results for an extended Navier-Stokes system

In this article we study a system of equations that is known to {\em extend} Navier-Stokes dynamics in a well-posed manner to velocity fields that are not necessarily divergence-free. Our aim is to contribute to an understanding of the role of divergence and pressure in developing energy estimates capable of controlling the nonlinear terms. We address questions of global existence and stability in bounded domains with no-slip boundary conditions. Even in two space dimensions, global existence is open in general, and remains so, primarily due to the lack of a self-contained $L^2$ energy estimate. However, through use of new $H^1$ coercivity estimates for the linear equations, we establish a number of global existence and stability results, including results for small divergence and a time-discrete scheme. We also prove global existence in 2D for any initial data, provided sufficient divergence damping is included.Comment: 29 pages, no figure

Augmented reality-based remote family visits in nursing homes

During the COVID-19 pandemic, many nursing homes had to restrict visitations. This had a major negative impact on the wellbeing of residents and their family members. In response, residents and family members increasingly resorted to mediated communication to maintain social contact. To facilitate high-quality mediated social contact between residents in nursing homes and remote family members, we developed an augmented reality (AR)-based communication tool. In this study, we compared the user experience (UX) of AR-communication with that of video calling, for 10 pairs of residents and family members. We measured enjoyment, spatial presence and social presence, attitudes, behavior and conversation duration. In the AR-communication condition, residents perceived a 3D projection of their remote family member onto a chair placed in front of them. In the video calling condition, the family member was shown using 2D video. In both conditions, the family member perceived the resident in the video calling mode on a 2D screen. While residents reported no differences in their UX between both conditions, family members reported higher spatial presence for the AR-communication condition compared to video-calling. Conversation durations were significantly longer during AR-communication than during video calling. We tentatively suggest that there may be (unconscious) differences in UX during AR-based communication compared to video calling

The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments

Expansion in SL_d(Z/qZ), q arbitrary

Let S be a fixed finite symmetric subset of SL_d(Z), and assume that it generates a Zariski-dense subgroup G. We show that the Cayley graphs of pi_q(G) with respect to the generating set pi_q(S) form a family of expanders, where pi_q is the projection map Z->Z/qZ

Boundedness of Pseudodifferential Operators on Banach Function Spaces

We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$, then a pseudodifferential operator $\operatorname{Op}(a)$ is bounded on $X(\mathbb{R}^n)$ whenever the symbol $a$ belongs to the H\"ormander class $S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$, $0\le\delta<1$ or to the the Miyachi class $S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)$ with $0\le\delta\le\rho\le 1$, $0\le\delta0$. This result is applied to the case of variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$.Comment: To appear in a special volume of Operator Theory: Advances and Applications dedicated to Ant\'onio Ferreira dos Santo

The Baum-Connes Conjecture via Localisation of Categories

We redefine the Baum-Connes assembly map using simplicial approximation in the equivariant Kasparov category. This new interpretation is ideal for studying functorial properties and gives analogues of the assembly maps for all equivariant homology theories, not just for the K-theory of the crossed product. We extend many of the known techniques for proving the Baum-Connes conjecture to this more general setting