Welcome back. This week, I am excited to delve into the Lebesgue integral, which is a more powerful alternative to the Riemann integral that we have dealt with so far. This new, more modern piece of mathematics is due to the work of Henri Lebesgue, a French mathematician who lived from 1875 to 1941. To actually define the Lebesgue integral, we will first develop some core ideas … [Read more...] about The Lebesgue Integral: A Newer and More Flexible Alternative to the Riemann Integral
Editor Blogs
Partial Differential Equations Meet Electricity & Magnetism: Maxwell’s Equations, Poisson’s Equation, and Eigenfunctions of the Laplacian
Welcome back. Over the last few weeks, we have studied what I hope are some fascinating results in mathematics. We have developed the theory behind multivariable calculus and applied it to describe well-behaved fluids. We have also put the classic Riemann integral on rigorous footing and proven the second Fundamental Theorem of Calculus. This week, we will continue our journey … [Read more...] about Partial Differential Equations Meet Electricity & Magnetism: Maxwell’s Equations, Poisson’s Equation, and Eigenfunctions of the Laplacian
Predicting the Path a Particle Will Take in a Fluid: A Brief Overview
I hope that all of my readers have had a restful Thanksgiving Break and are ready to begin their final exam studies. This week, we will discuss how to design a numerical program that will predict the path a particle will take in a real-world fluid flow. To do this, we must first outline our overall problem. Suppose that we have mounted a wing inside a wind tunnel. When we … [Read more...] about Predicting the Path a Particle Will Take in a Fluid: A Brief Overview
MDMA and Psychotherapy
Methylenedioxymethamphetamine, or MDMA, is an entactogen that increases empathy, distorts one’s perceptions, and can increase a user’s pleasure from sensory experiences (NIH). It is considered a Schedule I drug by the FDA, meaning it has no clinical uses and has a high likelihood to be abused. However, research is being conducted into possible therapeutic uses for treating PTSD … [Read more...] about MDMA and Psychotherapy
Monocular Visual Cues and VR
Monocular Cues are visual cues used for depth perception that are dependent on one eye. Several different types of monocular cues help us to estimate the distance of objects: interposition, motion parallax, relative size and clarity, texture gradient, linear perspective, and light and shadow. In this blog, I’ve gone through and described each monocular cue in detail, which are … [Read more...] about Monocular Visual Cues and VR