Rediscovering the Artistic Side of Mathematics

Welcome to the inaugural issue of the LASER, a journal devoted to the problems at the interface of math and art. The terms ’math’ and ’art’ are to be broadly construed to encompass all quantitative sciences and forms of art. The journal’s name, acronym for Linking Art and Science through Education and Research, suggests our interest in the theory, practice and pedagogy of this interdisciplinary subject

Orientation of Rigid Bodies Freefalling in Newtonian and Non-Newtonian Fluids

This thesis deals with the subject of terminal orientations of rigid bodies, sedimenting in Newtonian and non-Newtonian liquids. It is a well established fact that homogeneous bodies of revolution around an axis ('a') with fore-aft symmetry will orient themselves with respect to the direction of gravity ('g') depending upon their shape and upon the nature of the fluid in which they are immersed. If, for instance, we are considering an ellipsoidal object falling in a Newtonian fluid such as water, then the body falls with 'a' eventually becoming perpendicular to the direction of 'g'. However, if the same body falls in a viscoelastic fluid where the inertial effects can be disregarded, then 'a' will eventually become parallel to 'g'. It has also been noted that long bodies falling in fluids with certain polymeric concentrations can take on angles between the horizontal and vertical orientations. These intermediate angles are referred to as tilt angles. The objective of this thesis is the explanation of this orientation phenomenon in different liquids.Our approach to the problem has been three-fold, experimental, mathematical and also numerical. We perform several experiments on sedimentation of particles in a variety of viscoelastic and Newtonian liquids to verify and fill gaps in the previous observations. A second set of experiments that we perform involves a modified flow chamber setup where the particle is fixed at the center of the chamber while water flows past it. We are able able to replicate previous experiments at low and intermediate Re, with both these experiments. The equations to describe the problem of freefall of a rigid body of arbitrary shape, in a liquid, are obtained from a frame attached to the body and is formulated for any general fluid model. In addition, we also obtain the equations for the body, since the problem we are dealing with is one of fluid-structure interaction. We establish well-posedness of the equations by showing the exitence and uniqueness of steady solutions to the problem of sedimentation in a Second order fluid, with Re=0 and arbitrary material parameters using the Banach fixed point theorem. In order to explain the terminal orientation assumed by the body, we consider the effect of torques imposed by different components of the liquid such as inertia, viscoelasticiy and shear-thinning. The equilibrium resulting from the competition of the different torques should reveal the terminal angle. Guided by the fact that the orientation phnomenon is observed at very small Re and We, we formulate the torque equations at first order in these material parameters. The calculation is performed for four different liquid models, Newtownian, Power-law, Second order fluid and a modified Second order model which we introduce here for the first time. The different orientation observations seen in experiments is well explained by these models. Finally, a simple quasi-steady stability argument is used to establish stability of the equilibrium states. For this final argument, we numerically evaluate the torque imposed by the individual components of the liquid upon a sedimenting prolate ellipsoid in an unbounded three dimensional fluid domain surrounding the body

Creativity as an Emergent Property of Complex Educational System

The importance of creativity in education has been discussed often in the literature. While there remains no agreed-upon definition of creativity, the psychological literature points to traits of a creative person. These include the ability to think outside the box, make connections between seemingly disparate ideas, and question norms. The literature provides several examples of classroom experiments to help foster creativity in the classroom. In science and mathematics, we can start by getting students to recognize mathematics and the sciences as being creative endeavors. While these attempts are noteworthy, they are not necessarily aligned with instructional practices. In this article, we propose that to promote creative thinking in our classrooms, we need to see our educational system as a complex system or a network of connections between different disciplines. The 20th century notion that school and college education is rooted in discipline-based reductionism and that learning leads to specialization caters to a few, leaving a large number of students to fail out of the system. The American liberal arts educational model prides itself on giving students a holistic perspective by exposing them to various disciplines. However, merely exposing students to different ideas without having them realize the deep, underlying connections is like expecting interesting dynamics in a collection of disconnected nodes. We propose that the education system is a complex system composed of various nodes, representing different disciplines with the edges representing the flow of unifying ideas between them. Connections between the nodes allow for flow in these paths, resulting in greater opportunity for creativity, which is an emergent property of such a network. The abstract notions discussed above are illustrated by deliberate attempts (ambitious though small) made at the authors’ institution to build an educational experience focused on creativity

Vortex Induced Oscillations of Cylinders

This article submitted to the APS-DFD 2008 conference, accompanies the fluid dynamics video depicting the various orientational dynamics of a hinged cylinder suspended in a flow tank. The different behaviors displayed by the cylinder range from steady orientation to periodic oscillation and even autorotation. We illustrate these features using a phase diagram which captures the observed phenomena as a function of Reynolds number and reduced inertia. A hydrogen bubble flow visualization technique is also used to show vortex shedding structure in the cylinder's wake which results in these oscillations.Comment: 3 page

Inspiring Mathematical Creativity through Juggling

The goal of the Creativity in Mathematics and Science project, funded by the National Science Foundation’s [NSF’s] Improving Undergraduate STEM Educa- tion program, is to reconsider how we teach mathematics at the collegiate level. Over the last three years, we have developed interdisciplinary modules that seek to encourage students, including non-STEM majors, to see mathematics in unexpected places, make connections to their own interests and disciplines, and explore creativity in mathematics. Relying on traits of creativity such as the ability to connect ideas, be inquisitive, question norms, and have flexibility [1], we encouraged students to participate and understand mathematics in unconventional ways. The scheduling of a professional juggling company’s performance at our on-campus theater inspired us to create a module connecting mathematics and juggling for both a general education mathematics course and a mechanics course. We drew from research on the mathematics of juggling [2, 3] to develop a module that encouraged students to explore the patterns, notations, and mathematical elements of juggling in a variety of ways. Their final projects, representing further explorations, were displayed in our theater’s lobby and featured interactive displays and demonstrations. In this paper we describe our experiences developing and implementing this juggling module, students’ experiences with the modules, and their development of final projects

A Modified Stokes-Einstein Equation for A Beta Aggregation

Background: In all amyloid diseases, protein aggregates have been implicated fully or partly, in the etiology of the disease. Due to their significance in human pathologies, there have been unprecedented efforts towards physiochemical understanding of aggregation and amyloid formation over the last two decades. An important relation from which hydrodynamic radii of the aggregate is routinely measured is the classic Stokes-Einstein equation. Here, we report a modification in the classical Stokes-Einstein equation using a mixture theory approach, in order to accommodate the changes in viscosity of the solvent due to the changes in solute size and shape, to implement a more realistic model for A beta aggregation involved in Alzheimer\u27s disease. Specifically, we have focused on validating this model in protofibrill lateral association reactions along the aggregation pathway, which has been experimentally well characterized. Results: The modified Stokes-Einstein equation incorporates an effective viscosity for the mixture consisting of the macromolecules and solvent where the lateral association reaction occurs. This effective viscosity is modeled as a function of the volume fractions of the different species of molecules. The novelty of our model is that in addition to the volume fractions, it incorporates previously published reports on the dimensions of the protofibrils and their aggregates to formulate a more appropriate shape rather than mere spheres. The net result is that the diffusion coefficient which is inversely proportional to the viscosity of the system is now dependent on the concentration of the different molecules as well as their proper shapes. Comparison with experiments for variations in diffusion coefficients over time reveals very similar trends. Conclusions: We argue that the standard Stokes-Einstein\u27s equation is insufficient to understand the temporal variations in diffusion when trying to understand the aggregation behavior of A beta 42 proteins. Our modifications also involve inclusion of improved shape factors of molecules and more appropriate viscosities. The modification we are reporting is not only useful in A beta aggregation but also will be important for accurate measurements in all protein aggregation systems

A modified Stokes-Einstein equation for Aβ aggregation

<p>Abstract</p> <p>Background</p> <p>In all amyloid diseases, protein aggregates have been implicated fully or partly, in the etiology of the disease. Due to their significance in human pathologies, there have been unprecedented efforts towards physiochemical understanding of aggregation and amyloid formation over the last two decades. An important relation from which hydrodynamic radii of the aggregate is routinely measured is the classic Stokes-Einstein equation. Here, we report a modification in the classical Stokes-Einstein equation using a mixture theory approach, in order to accommodate the changes in viscosity of the solvent due to the changes in solute size and shape, to implement a more realistic model for A<it>β</it> aggregation involved in Alzheimer’s disease. Specifically, we have focused on validating this model in protofibrill lateral association reactions along the aggregation pathway, which has been experimentally well characterized.</p> <p>Results</p> <p>The modified Stokes-Einstein equation incorporates an effective viscosity for the mixture consisting of the macromolecules and solvent where the lateral association reaction occurs. This effective viscosity is modeled as a function of the volume fractions of the different species of molecules. The novelty of our model is that in addition to the volume fractions, it incorporates previously published reports on the dimensions of the protofibrils and their aggregates to formulate a more appropriate shape rather than mere spheres. The net result is that the diffusion coefficient which is inversely proportional to the viscosity of the system is now dependent on the concentration of the different molecules as well as their proper shapes. Comparison with experiments for variations in diffusion coefficients over time reveals very similar trends.</p> <p>Conclusions</p> <p>We argue that the standard Stokes-Einstein’s equation is insufficient to understand the temporal variations in diffusion when trying to understand the aggregation behavior of A<it>β</it>42 proteins. Our modifications also involve inclusion of improved shape factors of molecules and more appropriate viscosities. The modification we are reporting is not only useful in A<it>β</it> aggregation but also will be important for accurate measurements in all protein aggregation systems.</p

Fatty Acid Concentration and Phase Transitions Modulate Aβ Aggregation Pathways

Aggregation of amyloid β (Aβ) peptides is a significant event that underpins Alzheimer disease (AD) pathology. Aβ aggregates, especially the low-molecular weight oligomers, are the primary toxic agents in AD and hence, there is increasing interest in understanding their formation and behavior. Aggregation is a nucleation-dependent process in which the pre-nucleation events are dominated by Aβ homotypic interactions. Dynamic flux and stochasticity during pre-nucleation renders the reactions susceptible to perturbations by other molecules. In this context, we investigate the heterotypic interactions between Aβ and fatty acids (FAs) by two independent tool-sets such as reduced order modelling (ROM) and ensemble kinetic simulation (EKS). We observe that FAs influence Aβ dynamics distinctively in three broadly-defined FAconcentration regimes containing non-micellar, pseudo-micellar or micellar phases. While the non-micellar phase promotes on-pathway fibrils, pseudo-micellar and micellar phases promote predominantly off-pathway oligomers, albeit via subtly different mechanisms. Importantly off-pathway oligomers saturate within a limited molecular size, and likely with a different overall conformation than those formed along the on-pathway, suggesting the generation of distinct conformeric strains of Aβ, which may have profound phenotypic outcomes. Our results validate previous experimental observations and provide insights into potential influence of biological interfaces in modulating Aβ aggregation pathways