Semistability and Simple Connectivity at Infinity of Finitely Generated Groups with a Finite Series of Commensurated Subgroups

A subgroup $H$ of a group $G$ is $commensurated$ in $G$ if for each $g\in G$, $gHg^{-1}\cap H$ has finite index in both $H$ and $gHg^{-1}$. If there is a sequence of subgroups $H=Q_0\prec Q_1\prec ...\prec Q_{k}\prec Q_{k+1}=G$ where $Q_i$ is commensurated in $Q_{i+1}$ for all $i$, then $Q_0$ is $subcommensurated$ in $G$. In this paper we introduce the notion of the simple connectivity at infinity of a finitely generated group (in analogy with that for finitely presented groups). Our main result is: If a finitely generated group $G$ contains an infinite, finitely generated, subcommensurated subgroup $H$, of infinite index in $G$, then $G$ is 1-ended and semistable at $\infty$. If additionally, $H$ is finitely presented and 1-ended, then $G$ is simply connected at $\infty$. A normal subgroup of a group is commensurated, so this result is a strict generalization of a number of results, including the main theorems of G. Conner and M. Mihalik \cite{CM}, B. Jackson \cite{J}, V. M. Lew \cite{L}, M. Mihalik \cite{M1}and \cite{M2}, and J. Profio \cite{P}.Comment: 21 pages, 3 figures. arXiv admin note: text overlap with arXiv:1201.296

The Relationship between Force Platform Measures and Total Body Center of Mass

The ability of a person to maintain stable posture is essential for activities of daily living. Research in this field has evolved to include sensitive assessment technology including force platforms and 3-dimensional kinematic motion analysis systems. Although many studies have investigated postural stability under the auspice of posturography and the use of force platforms, relatively few have incorporated kinematic motion analysis techniques. Furthermore, of the studies that have utilized a multivariate research model, none have sought to identify the relationship between force platform measures including both the variation of movement of the x- and y-coordinates of the center of pressure (COP), and the 3-dimensional coordinates of the total body center of mass (COM). This study used a descriptive design to evaluate the relationship between force platform measures and the kinematic measures dealing with the total body COM in 14 healthy participants (height = 1.70 ± 0.09 m, mass = 67.7 ± 9.9 kg; age = 24.9 ± 3.8 yrs). Intraclass correlations (ICC) and standard error of measurements (SEM) were determined for common variables of interest used in standard posturography models. The results suggest that the variation of the excursion of the COP coordinates best represent the variation of the total body COM in the x- and y-directions. There was a force platform measure that correlated significantly with the vertical component of total body COM in only 3 of the 8 conditions. The ICC values obtained when analyzing individual conditions revealed that the variation in the force measurements were much more reliable than those representing the variation in movement of the COP, suggesting a need for the development of higher order methods of modeling 3-dimensional COM information from force platforms

Convergence and Optimality of Adaptive Mixed Methods on Surfaces

In a 1988 article, Dziuk introduced a nodal finite element method for the Laplace-Beltrami equation on 2-surfaces approximated by a piecewise-linear triangulation, initiating a line of research into surface finite element methods (SFEM). Demlow and Dziuk built on the original results, introducing an adaptive method for problems on 2-surfaces, and Demlow later extended the a priori theory to 3-surfaces and higher order elements. In a separate line of research, the Finite Element Exterior Calculus (FEEC) framework has been developed over the last decade by Arnold, Falk and Winther and others as a way to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can be obtained by solving finite dimensional subproblems. In 2011, Holst and Stern merged these two lines of research by developing a framework for variational crimes in abstract Hilbert complexes, allowing for application of the FEEC framework to problems that violate the subcomplex assumption of Arnold, Falk and Winther. When applied to Euclidean hypersurfaces, this new framework recovers the original a priori results and extends the theory to problems posed on surfaces of arbitrary dimensions. In yet another seemingly distinct line of research, Holst, Mihalik and Szypowski developed a convergence theory for a specific class of adaptive problems in the FEEC framework. Here, we bring these ideas together, showing convergence and optimality of an adaptive finite element method for the mixed formulation of the Hodge Laplacian on hypersurfaces.Comment: 22 pages, no figures. arXiv admin note: substantial text overlap with arXiv:1306.188