Mathematical motivators: using the history of mathematics to enrich the curriculum

This article provides a summary of the MSOR funded mini project Mathematical Motivators. Mathematics is usually, and of course correctly, presented ‘ready-made’ to students, with techniques and applications presented systematically and in a logical order. However, like any other academic subject, mathematics has a history which is rich in astonishing breakthroughs, false starts, misattributions, confusions and dead-ends. This history gives a narrative and human context which adds colour and context to the discipline

A Few Words about the Recently Published Anglo-Saxon Poem, the First Edition of Beowulf.

Translated and Annotate

Quantitative analysis of the mechanics of fibrillar fribronectin

Thesis (Ph.D.)--Boston UniversityThe information exchange between cells and their environment is a key mediator of cell behavior that will result in disease or dysfunction if disrupted. A thorough understanding of the in vivo cell environment is critical to relating cell behaviors observed in vitro to cell behaviors in pathogenesis and homeostasis. In addition to neighboring cells, the extracellular matrix (ECM) defines the local cell environment in the body. The protein fibronectin (Fn) is a prominent component of the ECM and a key cell adhesive ligand. Fn is assembled by cells into an extremely extensible, fibrous network through which cells migrate. Fn is also an integral part of the signaling machinery that instructs cell behavior. Cells may bind to Fn through a large number of receptors, in addition Fn binds and presents growth factors to cells, regulating their proliferative and migratory behavior. Stretch, applied to Fn fibers has been demonstrated to alter properties like binding site availability and fiber stiffness. In order to understand how molecular conformations and mechanical stretch regulate these cell instructive properties of Fn fibers, one must build a quantitative understanding of the intermolecular architecture of Fn fibers. In this study we have characterized the physical characteristics of fibronectin, its density, stiffness, extensibility, and viscoelasticity with respect to the extension of the Fn fiber. We have quantified conformational changes within the molecule that regulate both its mechanical properties and the availability of binding sites. In addition, we determined that the configuration of the molecular crosslinks strongly influences the fiber's physical properties. By taking measurements of the Fn fibers under constant tension we have shown that fibronectin is a highly viscoelastic material with extremely slow response times, indicating that in the slow pulling regime of cell tractions Fn material properties may deviate significantly from measurements made at higher pulling rates. A strong quantitative understanding of fibronectin's properties opens the door to new insights into disease and new approaches to creating engineered tissue constructs

A Few More Words about Beowulf

In September of 1815, Grundtvig published his second article about Beowulf, “Nok et Par Ord om Bjovulfs Drape” [A Few More Words about the Heroic Poem of Beowulf] in Nyeste Skilderie af Kjøbenhavn [The Latest Scenes ofCopenhagen].i The article was intended to end once and for all the literary feud that had developed between him, Grímur Jónson Thorkelin, who had published the first modern edition of Beowulf, and Peter Erasmus Müller, who had anonymouslyreviewed the edition.ii The article is notable for two main reasons: first, it reveals Grundtvig far ahead of his contemporaries as a scholar of the poem and,second, it displays in full Grundtvig’s aggressive tone when in the heat of debate.In September of 1815, Grundtvig published his second article about Beowulf,“Nok et Par Ord om Bjovulfs Drape” [A Few More Words about the HeroicPoem of Beowulf] in Nyeste Skilderie af Kjøbenhavn [The Latest Scenes ofCopenhagen].iThe article was intended to end once and for all the literary feudthat had developed between him, Grímur Jónson Thorkelin, who had publishedthe first modern edition of Beowulf, and Peter Erasmus Müller, who had anonymously reviewed the edition.iiThe article is notable for two main reasons: first, itreveals Grundtvig far ahead of his contemporaries as a scholar of the poem and,second, it displays in full Grundtvig’s aggressive tone when in the heat of debate

Using molecular mechanics to predict bulk material properties of fibronectin fibers

The structural proteins of the extracellular matrix (ECM) form fibers with finely tuned mechanical properties matched to the time scales of cell traction forces. Several proteins such as fibronectin (Fn) and fibrin undergo molecular conformational changes that extend the proteins and are believed to be a major contributor to the extensibility of bulk fibers. The dynamics of these conformational changes have been thoroughly explored since the advent of single molecule force spectroscopy and molecular dynamics simulations but remarkably, these data have not been rigorously applied to the understanding of the time dependent mechanics of bulk ECM fibers. Using measurements of protein density within fibers, we have examined the influence of dynamic molecular conformational changes and the intermolecular arrangement of Fn within fibers on the bulk mechanical properties of Fn fibers. Fibers were simulated as molecular strands with architectures that promote either equal or disparate molecular loading under conditions of constant extension rate. Measurements of protein concentration within micron scale fibers using deep ultraviolet transmission microscopy allowed the simulations to be scaled appropriately for comparison to in vitro measurements of fiber mechanics as well as providing estimates of fiber porosity and water content, suggesting Fn fibers are approximately 75% solute. Comparing the properties predicted by single molecule measurements to in vitro measurements of Fn fibers showed that domain unfolding is sufficient to predict the high extensibility and nonlinear stiffness of Fn fibers with surprising accuracy, with disparately loaded fibers providing the best fit to experiment. This work shows the promise of this microstructural modeling approach for understanding Fn fiber properties, which is generally applicable to other ECM fibers, and could be further expanded to tissue scale by incorporating these simulated fibers into three dimensional network models

Sensitivity to horizontal and vertical corrugations defined by binocular disparity

AbstractSensitivity to corrugations defined by binocular disparity differs as a function of the modulation frequency. Such functions have proved to be useful descriptive and analytical tools in the study of the mechanisms involved in disparity processing. Indeed, given certain assumptions, these sensitivity functions can be used to predict certain perceptual outcomes. Given their importance, it is surprising that there is no comprehensive data set of disparity sensitivity functions (DSF) for a range of observers over a broad range of spatial frequencies and orientations. Here we report DSFs for six observers over an eight octave range of sinusoidal corrugations in disparity (0.0125–3.2 cpd). Multi-cycle, low frequency surfaces were used to assess the degree to which the fall-off in sensitivity at low corrugation frequencies is attributable to the decreasing number of cycles displayed. The data was found to form a continuous function despite the different number of cycles displayed. We conclude that the fall off in sensitivity is due to the spatial interactions in disparity processing. We also determined DSFs for the same observers to both vertically and horizontally oriented sinusoidal disparity corrugations in order to characterise the extent of the stereoscopic anisotropy In general, the best thresholds for detecting vertically oriented disparity corrugations were higher (∼4 arc sec) than for horizontally oriented corrugations (∼2 arc sec). Moreover, the functions were shifted toward the high spatial frequency end of the spectrum

Overarching framework between Gaussian quantum discord and Gaussian quantum illumination

We cast the problem of illuminating an object in a noisy environment into a communication protocol. A probe is sent into the environment, and the presence or absence of the object constitutes a signal encoded on the probe. The probe is then measured to decode the signal. We calculate the Holevo information and bounds to the accessible information between the encoded and received signal with two different Gaussian probes---an Einstein-Podolsky-Rosen (EPR) state and a coherent state. We also evaluate the Gaussian discord consumed during the encoding process with the EPR probe. We find that the Holevo quantum advantage, defined as the difference between the Holevo information obtained from the EPR and coherent state probes, is approximately equal to the discord consumed. These quantities become exact in the typical illumination regime of low object reflectivity and low probe energy. Hence we show that discord is the resource responsible for the quantum advantage in Gaussian quantum illumination.Comment: 12 pages, 8 figure