Well-posedness,stability and numerical results for the thermoelastic behavior of a coupled joint-beam PDE-ODE system modeling the transverse motions of the antennas of a space structure

A mathematical model for both axial and transverse motions of two beams with cylindrical cross-sections coupled through a joint is presented and analyzed. The motivation for this problem comes from the need to accurately model damping and joint dynamics for the next generation of inflatable/rigidizable space structures. Thermo-elastic damping is included in the two beams and the motions are coupled through a joint which includes an internal moment. Thermal response in each beam is modeled by two temperature fields. The first field describes the circumferentially averaged temperature along the beam, and is linked to the axial deformation of the beam. The second describes the circumferential variation and is coupled to transverse bending. The resulting equations of motion consist of four, second-order in time, partial differential equations, four, first-order in time, partial differential equations, four second order ordinary differential equations, and certain compatibility boundary conditions. The system is written as an abstract differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections and temperature fields, and a finite-dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove that the system is well-posed, and that with positive damping parameters the resulting semigroup is exponentially stable. Steady states are characterized and several numerical approximation results are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Imaging cytoplasmic cAMP in mouse brainstem neurons

Background  

cAMP is an ubiquitous second messenger mediating various neuronal functions, often as a consequence of increased intracellular Ca²⁺ levels. While imaging of calcium is commonly used in neuroscience applications, probing for cAMP levels has not yet been performed in living vertebrate neuronal tissue before.     

Understanding Distributions by Modeling Them

In current curriculum materials for middle school students in the US, data and chance are considered as separate topics. They are then ideally brought together in the minds of high school or university students when they learn about statistical inference. In recent studies we have been attempting to build connections between data and chance in the middle school by using a modeling approach made possible by new software capabilities that will be part of TinkerPlots 2.0 (TinkerPlots is published by Key Curriculum Press and has been developed with grants from the National Science Foundation (ESI-9818946, REC-0337675, ESI-0454754). Opinions expressed here are our own and not necessarily those of the Foundation.). Using a new Sampler object, students build “factories” to model not only prototypical chance events, but also distributions of measurement errors and of heights of people. We provide the rationale for having students model a wide range of phenomena using a single software tool and describe how we are using this capability to help young students develop a robust, statistical perspective.     

A simple model for a two-stage chemical reaction with diffusion

We analyse the evolution of a two-stage chemical reaction betweentwo neighbouring plumes of reactants. Under the assumption thatthe plumes are approximately Gaussian we derive a system ofordinary differential equations for the total amount, the centroidand the variance of each reactant. We compare the solution ofthese equations with full numerical simulation of the reaction.Excellent agreement is obtained, with solution of the near-Gaussianmodel requiring considerably less computational effort thanthe full simulations. Of key importance is the yield of thereaction, and we discuss this feature in particular.