Symmetric approximations of pseudo-Boolean functions with applications to influence indexes

We introduce an index for measuring the influence of the k-th smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear combinations of order statistic functions. We give explicit expressions for both the index and the approximation and discuss some properties of the index. Finally, we show that this index subsumes the concept of system signature in engineering reliability and that of cardinality index in decision making

Conical square function estimates in UMD Banach spaces and applications to H-infinity functional calculi

We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-McIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator (A) with certain off-diagonal bounds, such that (A) always has a bounded (H^{\infty})-functional calculus on these spaces. This provides a new way of proving functional calculus of (A) on the Bochner spaces (L^p(\R^n;X)) by checking appropriate conical square function estimates, and also a conical analogue of Bourgain's extension of the Littlewood-Paley theory to the UMD-valued context. Even when (X=\C), our approach gives refined (p)-dependent versions of known results.Comment: 28 pages; submitted for publicatio

Comparison of shoulder kinematic chain models and their influence on kinematics and kinetics in the study of manual wheelchair propulsion

Several kinematic chains of the upper limbs have been designed in musculoskeletal models to investi- gate various upper extremity activities, including manual wheelchair propulsion. The aim of our study was to compare the effect of an ellipsoid mobilizer formulation to describe the motion of the scapu- lothoracic joint with respect to regression-based models on shoulder kinematics, shoulder kinetics and computational time, during manual wheelchair propulsion activities. Ten subjects, familiar with manual wheelchair propulsion, were equipped with reflective markers and performed start-up and propulsion cycles with an instrumented field wheelchair. Kinematic data obtained from the optoelectronic system and kinetic data measured by the sensors on the wheelchair were processed using the OpenSim software with three shoulder joint modeling versions (ellipsoid mobilizer, regression equations or fixed scapula) of an upper-limb musculoskeletal model. As expected, the results obtained with the three versions of the model varied, for both segment kinematics and shoulder kinetics. With respect to the model based on regression equations, the model describing the scapulothoracic joint as an ellipsoid could capture the kinematics of the upper limbs with higher fidelity. In addition, the mobilizer formulation allowed to com- pute consistent shoulder moments at a low computer processing cost. Further developments should be made to allow a subject-specific definition of the kinematic chain

Uniformity in the Wiener-Wintner theorem for nilsequences

We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.Comment: v3: 18 p., proof that the cube construction produces compact homogeneous spaces added, measurability issues in the proof of Theorem 1.5 addressed. We thank the anonymous referees for pointing out these gaps in v

Prescribing the Jacobian in critical spaces

International audienceWe consider the Sobolev space $X=W^{s,p}({\mathbb S}^m ; {\mathbb S}^{k-1})$. We prove the existence of a robust distributional Jacobian $Ju$ for $u\in X$ provided $sp\ge k-1$. This generalizes a result of Bourgain, Brezis and the second author (Comm. Pure Appl. Math. 2005), where the case $m=k$ is considered. In the critical case where $sp=k-1$, we identify the image of the map $X\ni u\mapsto Ju$. This extends a result of Alberti, Baldo and Orlandi (J. Eur. Math. Soc. 2003) for $s=1$ and $p=k-1$. We also present a new, analytical, dipole construction method