Rigidity and volume preserving deformation on degenerate simplices

Given a degenerate $(n+1)$-simplex in a $d$-dimensional space $M^d$ (Euclidean, spherical or hyperbolic space, and $d\geq n$), for each $k$, $1\leq k\leq n$, Radon's theorem induces a partition of the set of $k$-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in $M^d$ for $d=n$, and the volumes of $k$-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all $k$-faces; and this property still holds in $M^d$ for $d\geq n+1$ if an invariant $c_{k-1}(\alpha^{k-1})$ of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant $c_k(\omega)$ we discovered for any $k$-stress $\omega$ on a cell complex in $M^d$. We introduce a characteristic polynomial of the degenerate simplex by defining $f(x)=\sum_{i=0}^{n+1}(-1)^{i}c_i(\alpha^i)x^{n+1-i}$, and prove that the roots of $f(x)$ are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case.Comment: 27 pages, 2 figures. To appear in Discrete & Computational Geometr

A Long, Loving Look at the Real: An Experiential Ignatian Approach to Immersion

International travel is a popular and widespread practice among higher education institutions, but the pedagogy and approach to these programs varies widely. The Center for Global Education and Experience at Augsburg University facilitates international immersion programs for U.S. students, faculty, and staff focused on social justice and solidarity; this approach is particularly attractive to Jesuit institutions and has led to fruitful collaborations. This article offers an experienced immersion facilitator’s personal reflections on designing and implementing immersion programs and applies key themes of Ignatian spirituality and pedagogy to the experience of international immersion travel

Locked and Unlocked Chains of Planar Shapes

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle less than 90 degrees admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof details. (Fixed crash-induced bugs in the abstract.

Numerical analysis of four-wave mixing between 2 ps mode-locked laser pulses in a tensile-strained bulk SOA

A numerical model of four-wave mixing between 2-ps pulses in a tensile-strained bulk semiconductor optical amplifier is presented. The model utilizes a modified Schrodinger equation to model the pulse propagation. The Schrodinger equation parameters such as the material gain first and second order dispersion, linewidth enhancement factors and optical loss coefficient are obtained using a previously developed steady-state model. The predicted four-wave mixing pulse characteristics show reasonably good agreement with experimental pulse characteristics obtained using frequency resolved optical gating

Numerical analysis of pulse pedestal and dynamic chirp formation on picosecond modelocked laser pulses after propagation through a semiconductor optical amplifier

A numerical analysis, based on a modified Schrodinger equation, of the formation of pulse pedestals and dynamic chirp formation on picosecond pulses after propagation through a semiconductor optical amplifier is presented. The numerical predictions are confirmed by an experiment that utilises the frequency resolved optical gating technique for the amplified pulse characterisation

Triple-wavelength fiber ring laser based on a hybrid gain medium actively mode-locked at 10 GHz

A fiber ring laser based on a hybrid gain medium that produces three simultaneously mode-locked wavelength channels is presented. The lithium niobate based modulator used to actively mode-lock the laser cavity at 10 GHz is birefringence compensated to reduce its polarization sensitivity. A Lyot filter defines the lasers multiwavelength spectrum which has a wavelength spacing of 1 nm. The polarization sensitive nature of the laser cavity and its affect on the performance of the laser is discussed

Directing cell migration using micropatterned and dynamically adhesive polymer brushes

This work was funded by the BBSRC, New Investigator Award (BB/J000914/1)

Preferences of children in grades two through eight in social studies subject areas

Thesis (Ed.M.)--Boston Universit

Characterizing the universal rigidity of generic frameworks

A framework is a graph and a map from its vertices to E^d (for some d). A framework is universally rigid if any framework in any dimension with the same graph and edge lengths is a Euclidean image of it. We show that a generic universally rigid framework has a positive semi-definite stress matrix of maximal rank. Connelly showed that the existence of such a positive semi-definite stress matrix is sufficient for universal rigidity, so this provides a characterization of universal rigidity for generic frameworks. We also extend our argument to give a new result on the genericity of strict complementarity in semidefinite programming.Comment: 18 pages, v2: updates throughout; v3: published versio