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The use of inflatable structures in aerospace applications is becoming increasingly widespread. In order to monitor the inflation status and overall health of these inflatables, an accurate means of shape sensing is required. To this end, we investigated two existing methods for measuring simple curvature, or curvature in one-dimension. The first method utilizes a pair of strain sensing Fiber Bragg Gratings (FBGs) separated by a known distance; dividing the difference in strain by the separation distance yields an experimental value for the one-dimensional curvature at a point. The second method makes use of conductive ink-based flex sensors, which give a variable resistance based on curvature. We used the latter was in a design for a Curvature-Based Inflation Controller (CBIC). While the controller successfully inflated a test body, its overall utility is limited by the simplicity of its sensors. To improve the shape sensing capabilities of the controller, we investigated the use of FBGs in a multidimensional array. We fabricated a curvature-sensing FBG pair on an inflatable membrane and tested its accuracy as the membrane was shaped into a known radius of curvature. This work reports on the assembly of three such curvature-sensing FBG pairs into a two-dimensional Curvature-Sensing Rosette (CSR). The goal is to use this rosette to measure the curvature of a surface in multiple directions at a single point. A 3-D printed surface with saddle geometry was used to calibrate the curvature-sensing rosette. Presented will be methods of extracting values for the tensor of curvature for the surface at a point using the curvature-sensing rosette, along with experimental verification. This essentially defines the local geometry about the rosette, measured in real time. By employing an array of such rosettes across the surface of an inflatable structure, the local curvature of the inflatable could be known at every point. Combining these curvature measurements can yield an accurate depiction of the global geometry. Thus, the inflation status of the inflatable space structure could be monitored in real time.

AN ABSTRACT OF THE THESIS OF TARUN KUMAR DONGA, for the Master of Science degree in Civil Engineering, Presented on August 29th 2017, at Southern Illinois University Carbondale. TITLE: EFFECT OF TEMPERATURE INCREASE ON BUCKLING OF AXIALLY RESTRAINED BEAM-COLUMNS WITH DOUBLE CURVATURE MAJOR PROFESSOR: Dr. Aslam Kassimali, Ph.D. The main objective of this research was to study the buckling and post-buckling response of axially restrained beam-columns under thermal loading. Also the effects of slenderness ratios on pre-buckling and post-buckling behavior which is neglected in (AISC 2010 specification) for Structural Steel Buildings, was examined. The results of this study indicate that: a) The deflection and end moment amplification factors are significantly smaller for the restrained beam-columns under temperature increase than the corresponding unrestrained beam-columns under mechanical axial loads. - b) The deflection amplification and moment amplification factors tend to increase with increasing ratio of end moments. c) Thermal amplification factors, even in the pre buckling range, were found to be smaller than the AISC (2010) specification values, the difference was more significant for longer beam columns as compared to shorter ones.

Developmental processes involving symmetry-breaking of homogeneous cell populations into leaders and followers are found in many important contexts. Cells constrained by culture on two-dimensional scaffolds, as well as in three-dimensional shapes, appear to respond to convex curves with an increasing propensity to protrude, while concave curves in contrast appear to inhibit protrusion. This has interesting implications in terms of a potential positive feedback loop. This feedback may act in symmetry-breaking, through amplification of initial stochastic differences in cell shape, and also in collective migration, through reinforcing and directing the coherent movement of collectives. In this study, epithelial cells were cultured on two-dimensional micropatterns with variable curvatures to examine the effect of edge geometry and other variables on the likelihood of protrusions forming. This platform allowed the quantification of F-actin-based protrusions at the periphery of multicellular epithelial clusters, in segments defined by cluster edge curvature. The initial observations confirmed reports in the literature of preferential localisation of protrusions at more convex regions, and relative inhibition at more concave regions. A previously-published work has postulated a role for secreted modulators of motility, with the shape of a group of cells determining the concentration of diffusing morphogen each individual cell is exposed to. To test this hypothesis, a low-shear flow culture chamber was used to disrupt the putative gradients. Despite theoretical and empirical support for the sufficiency of the flow condition to disrupt autocrine signalling, micropatterned cells cultured under flow showed no significant differences from the control condition. These findings form the basis of a manuscript which has been accepted for publication by the Journal of Anatomy. The results of an Atomic Force Microscopy (AFM) study carried out by collaborators were suggestive of a role for cellular mechanotransduction in sensing and responding to micropattern curvature. Differential calcium channel mechanoactivation was hypothesised as being one potential mechanism underlying the response to curvature, given the known involvement of mechanosensitive ion channels in cellular responses to force and substrate stiffness, and the multiple roles of calcium in cellular motility. Artificially increasing cytosolic calcium levels with Ionomycin reduced protrusion rates at convex curves. However, treatment with BAPTA-AM to sequester intracellular calcium had no effect on protrusion rates. ROCK inhibitor, in contrast, increased protrusion rates at concave curves, and Blebbistatin increased protrusion rates globally. These results together are suggestive of differential control of myosin depending on local curvature: cyclic and driven by calcium-activation of MLCK in the convex regions (with lamellipodia undergoing protrusion-retraction cycles), versus sustained and controlled by ROCK in the concave regions (where lamellipodia are inhibited). The unexpected finding that protrusions at convex regions were resistant to the actin cytoskeleton-disrupting drug Cytochalasin D may point to a role for a tropomyosin isoform in defining the differing mechanical characteristics of the actin cytoskeleton in response to local curvature. In addition, the previously-noted lack of effect of BAPTA-AM treatment (which has been shown to interfere with dynamic microtubules) is suggestive of a role for stabilised microtubules in protrusions at convex regions. These indications of unique characteristics to the protrusions promoted by convex curvature give added support to the curvature-protrusion feedback model, and its relevance to tissue morphogenesis. In summary, this work provides evidence against a previously-published suggested mechanism for the curvature-protrusion feedback loop that is proposed to act during epithelial morphogenesis, and evidence in support of a role for a calcium-based mechanism in driving the initiation and maintenance of leader cells in migrating epithelial sheets. Further work is called for in characterising the protrusions promoted by convex curvature, and the mechanisms controlling them. This area is of significance in gaining greater understanding of tissue morphogenesis in pathogenesis and development, and of potential value in tissue engineering applications.

An important part of stereo vision consists of finding and matching points in two images which correspond to the same physical element in the scene. We show that zeros of curvature of curves are perspective projection invariants and can therefore be used to find corresponding points. They can be used to help solve the registration problem (Longuet-Higgins, 1982) and to obtain the correct depth when a curve enters the forbidden zone (Krol and van de Grind, 1982). They are also relevant to theories for representing image curves. We consider the stability of these zeros of curvature.

Wir untersuchen, welchen Einfluss die Verankerung von langen, hydrophilen Polymeren in Lipidmembranen auf deren elastische Eigenschaften ausübt. Theoretisch werden zwei Grenzbereiche für die spontane Krümmung der Membran erwartet: <br /> i) bei kleinen Oberflächenkonzentrationen des Polymers (Pilzgebiet) sollte die spontane Krümmung linear von der Oberflächendichte des verankerten Polymers abhängen;<br /> ii) bei hoher Bedeckung (Bürstengebiet) sollte die Abhängigkeit quadratisch sein. Wir versuchen, Vorhersagen für das Bürstengebiet zu prüfen, indem wir die morphologischen Veränderungen beobachten, die bei Riesen (Giant)- Vesikeln hervorgerufen werden. <br /> <br /> Als lange Polymere verwenden wir fluoreszenzmarkierte &#955;-Phage DNA Moleküle, die durch eine Biotin-Avidin-Biotin Verbindung an biotinhaltigen Lipidvesikeln befestigt sind. Wir kontrollieren die Oberflächenkonzentration der Anker durch Variation der Menge an biotinhaltigem Lipid in der Membran. Die Menge der an der Membran verankerten DNA wird durch Fluoreszenzmessungen quantifiziert. Änderungen in den elastischen Eigenschaften der Membran bei Anbindung der DNA, werden über eine Analyse der Vesikel-Fluktuationen kontrolliert. Die spontane Krümmung der Membran steigt mit der Oberflächenbeladung. Bei höheren Verankerungen bilden die Vesikel Knospen (budding). Die Größe der Knospen kann ebenfalls zur Bestimmung der Krümmung der Membran verwendet werden. Der Einfluss auf die Biegesteifigkeit ist Thema weiterer Untersuchungen. / We study the effect on the elastic properties of lipid membranes induced by anchoring of long hydrophilic polymers. Theoretically, two limiting regimes for the membrane spontaneous curvature are expected : <br /> i) at low surface polymer concentration (mushroom regime) the spontaneous curvature should scale linearly with the surface density of anchored polymers; <br /> ii) at high coverage (brush regime) the dependence should be quadratic. We attempt to test the predictions for the brush regime by monitoring the morphological changes induced on giant vesicles.<br /> <br /> As long polymers we use fluorescently labeled &#955;-phage DNA molecules which are attached to biotinylated lipid vesicles with a biotin-avidin-biotin linkage. By varying the amount of biotinylated lipid in the membrane we control the surface concentration of the anchors. The amount of anchored DNA to the membrane is quantified with fluorescence measurements. Changes in the elastic properties of the membrane as DNA grafts to it are monitored via analysis of the vesicle fluctuations. The spontaneous curvature of the membrane increases as a function of the surface coverage. At higher grafting concentrations the vesicles bud. The size of the buds can also be used to assess the membrane curvature. The effect on the bending stiffness is a subject of further investigation.

For a sequence of Hilbert spaces and continuous linear operators the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra. Of particular interest are those sequences for which the curvature is "small" at each step, e.g., belongs to a fixed operator ideal. In this context we elaborate the theory of Fredholm sequences and show how to introduce the Lefschetz number.

In the following, we build on previous work done on higher derivative gravity, in particular Lovelock gravity. The latter is a family of theories in higher space-time dimensions in which interactions involving higher powers of curvature are introduced, but the equations of motion remain second order in derivatives. We develop a new theory involving cubic terms in the curvature. We then show that the equations of motion for graviton fluctuations remain second order. The curvature cubed term is shown not to be a topological object, contrary to the belief that dimensionally extended Euler densities provided the only stable dimensionally continued theories of gravity (Lovelock gravity). Black hole solutions are studied in this new gravitational framework.

In the following, we build on previous work done on higher derivative gravity, in particular Lovelock gravity. The latter is a family of theories in higher space-time dimensions in which interactions involving higher powers of curvature are introduced, but the equations of motion remain second order in derivatives. We develop a new theory involving cubic terms in the curvature. We then show that the equations of motion for graviton fluctuations remain second order. The curvature cubed term is shown not to be a topological object, contrary to the belief that dimensionally extended Euler densities provided the only stable dimensionally continued theories of gravity (Lovelock gravity). Black hole solutions are studied in this new gravitational framework.

I will present the mean curvature flow in Euclidean spaces and the Lagrangian mean curvature flow. We will first study the mean curvature evolution of submanifolds in Euclidean spaces, with an emphasis on the case of hypersurfaces. Along the way we will demonstrate the basic techniques in the study of geometric flows in general (for example, various maximum principles and the treatment of singularities). After that we will move on to the study of Lagrangian mean curvature flows. We will make the relevant definitions and prove the fundamental result that the Lagrangian condition is preserved along the mean curvature flow in Kähler-Einstein manifolds, which started the extensive, and still ongoing, research on Lagrangian mean curvature flows. We will also define special Lagrangian submanifolds as calibrated submanifolds in Calabi-Yau manifolds. Finally, we will study the mean curvature flow of conormal bundles as submanifolds of C^n. Using some tools developed recently, we will show that if a surface has strictly negative curvatures, then away from the zero section, the Lagrangian mean curvature flow starting from a conormal bundle does not develop Type I singularities.