Isometric group actions on Hilbert spaces: growth of cocycles

We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with "controlled" Folner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growth.Comment: 26 pages, no figure. To appear in Geom. Funct. Ana

The Howe-Moore property for real and p-adic groups

We consider in this paper a relative version of the Howe-Moore Property, about vanishing at infinity of coefficients of unitary representations. We characterize this property in terms of ergodic measure-preserving actions. We also characterize, for linear Lie groups or p-adic Lie groups, the pairs with the relative Howe-Moore Property with respect to a closed, normal subgroup. This involves, in one direction, structural results on locally compact groups all of whose proper closed characteristic subgroups are compact, and, in the other direction, some results about the vanishing at infinity of oscillatory integrals.Comment: 25 pages, no figur

Isometric group actions on Hilbert spaces: structure of orbits

Our main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.Comment: 12 pages, to appear in Canadian Math.

Isometric Group Actions on Hilbert Spaces: Growth of Cocycles

Abstract.: We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into a Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with "controlled” Følner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growt

A review of user needs to drive the development of lower limb prostheses

Background: The development of bionic legs has seen substantial improvements in the past years but people with lower-limb amputation still suffer from impairments in mobility (e.g., altered balance and gait control) due to significant limitations of the contemporary prostheses. Approaching the problem from a human-centered perspective by focusing on user-specific needs can allow identifying critical improvements that can increase the quality of life. While there are several reviews of user needs regarding upper limb prostheses, a comprehensive summary of such needs for those affected by lower limb loss does not exist. Methods: We have conducted a systematic review of the literature to extract important needs of the users of lower-limb prostheses. The review included 56 articles in which a need (desire, wish) was reported explicitly by the recruited people with lower limb amputation (N = 8149). Results: An exhaustive list of user needs was collected and subdivided into functional, psychological, cognitive, ergonomics, and other domain. Where appropriate, we have also briefly discussed the developments in prosthetic devices that are related to or could have an impact on those needs. In summary, the users would like to lead an independent life and reintegrate into society by coming back to work and participating in social and leisure activities. Efficient, versatile, and stable gait, but also support to other activities (e.g., sit to stand), contribute to safety and confidence, while appearance and comfort are important for the body image. However, the relation between specific needs, objective measures of performance, and overall satisfaction and quality of life is still an open question. Conclusions: Identifying user needs is a critical step for the development of new generation lower limb prostheses that aim to improve the quality of life of their users. However, this is not a simple task, as the needs interact with each other and depend on multiple factors (e.g., mobility level, age, gender), while evolving in time with the use of the device. Hence, novel assessment methods are required that can evaluate the impact of the system from a holistic perspective, capturing objective outcomes but also overall user experience and satisfaction in the relevant environment (daily life)

Isometric group actions on Banach spaces and representations vanishing at infinity

Our main result is that the simple Lie group $G=Sp(n,1)$ acts properly isometrically on $L^p(G)$ if $p>4n+2$. To prove this, we introduce property ({\BP}_0^V), for $V$ be a Banach space: a locally compact group $G$ has property ({\BP}_0^V) if every affine isometric action of $G$ on $V$, such that the linear part is a $C_0$-representation of $G$, either has a fixed point or is metrically proper. We prove that solvable groups, connected Lie groups, and linear algebraic groups over a local field of characteristic zero, have property ({\BP}_0^V). As a consequence for unitary representations, we characterize those groups in the latter classes for which the first cohomology with respect to the left regular representation on $L^2(G)$ is non-zero; and we characterize uniform lattices in those groups for which the first $L^2$-Betti number is non-zero.Comment: 28 page

Extracorporeal Membrane Oxygenation for Severe Acute Respiratory Distress Syndrome associated with COVID-19: An Emulated Target Trial Analysis.

RATIONALE: Whether COVID patients may benefit from extracorporeal membrane oxygenation (ECMO) compared with conventional invasive mechanical ventilation (IMV) remains unknown. OBJECTIVES: To estimate the effect of ECMO on 90-Day mortality vs IMV only Methods: Among 4,244 critically ill adult patients with COVID-19 included in a multicenter cohort study, we emulated a target trial comparing the treatment strategies of initiating ECMO vs. no ECMO within 7 days of IMV in patients with severe acute respiratory distress syndrome (PaO2/FiO2 <80 or PaCO2 ≥60 mmHg). We controlled for confounding using a multivariable Cox model based on predefined variables. MAIN RESULTS: 1,235 patients met the full eligibility criteria for the emulated trial, among whom 164 patients initiated ECMO. The ECMO strategy had a higher survival probability at Day-7 from the onset of eligibility criteria (87% vs 83%, risk difference: 4%, 95% CI 0;9%) which decreased during follow-up (survival at Day-90: 63% vs 65%, risk difference: -2%, 95% CI -10;5%). However, ECMO was associated with higher survival when performed in high-volume ECMO centers or in regions where a specific ECMO network organization was set up to handle high demand, and when initiated within the first 4 days of MV and in profoundly hypoxemic patients. CONCLUSIONS: In an emulated trial based on a nationwide COVID-19 cohort, we found differential survival over time of an ECMO compared with a no-ECMO strategy. However, ECMO was consistently associated with better outcomes when performed in high-volume centers and in regions with ECMO capacities specifically organized to handle high demand. This article is open access and distributed under the terms of the Creative Commons Attribution Non-Commercial No Derivatives License 4.0 (http://creativecommons.org/licenses/by-nc-nd/4.0/)