This class covers the analysis and modeling of stochastic processes. Topics include …
This class covers the analysis and modeling of stochastic processes. Topics include measure theoretic probability, martingales, filtration, and stopping theorems, elements of large deviations theory, Brownian motion and reflected Brownian motion, stochastic integration and Ito calculus and functional limit theorems. In addition, the class will go over some applications to finance theory, insurance, queueing and inventory models.
Movement of ions in and out of cells is crucial to maintaining …
Movement of ions in and out of cells is crucial to maintaining homeostasis within the body and ensuring that biological functions run properly. The natural movement of molecules due to collisions is called diffusion. Several factors affect diffusion rate: concentration, surface area, and molecular pumps. This activity demonstrates diffusion, osmosis, and active transport through 12 interactive models.
This course continues the content covered in 18.100 Analysis I. Roughly half …
This course continues the content covered in 18.100 Analysis I. Roughly half of the subject is devoted to the theory of the Lebesgue integral with applications to probability, and the other half to Fourier series and Fourier integrals.
This course develops and applies scaling laws and the methods of continuum …
This course develops and applies scaling laws and the methods of continuum and statistical mechanics to biomechanical phenomena over a range of length scales, from molecular to cellular to tissue or organ level.
In this activity, students interact with 12 models to observe emergent phenomena …
In this activity, students interact with 12 models to observe emergent phenomena as molecules assemble themselves. Investigate the factors that are important to self-assembly, including shape and polarity. Try to assemble a monolayer by "pushing" the molecules to the substrate (it's not easy!). Rotate complex molecules to view their structure. Finally, create your own nanostructures by selecting molecules, adding charges to them, and observing the results of self-assembly.
This course describes the processes by which mass, momentum, and energy are …
This course describes the processes by which mass, momentum, and energy are transported in plasmas, with special reference to magnetic confinement fusion applications. The Fokker-Planck collision operator and its limiting forms, as well as collisional relaxation and equilibrium, are considered in detail. Special applications include a Lorentz gas, Brownian motion, alpha particles, and runaway electrons. The Braginskii formulation of classical collisional transport in general geometry based on the Fokker-Planck equation is presented. Neoclassical transport in tokamaks, which is sensitive to the details of the magnetic geometry, is considered in the high (Pfirsch-Schluter), low (banana) and intermediate (plateau) regimes of collisionality.
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