I updated the YouTube playlists for Compressible flow. There are now two version of the class I taught for almost ten years. The first was recorded live in 2024 in front of an in person and online audience. YouTube Playlist – Fall 2024 Recording The original was recorded in one of my offices and edited …
Category Archives: Mathematics
Compressible Flow Notes
I finally published my compressible flow notes online. They are under classes – comp. flow. They are the result of teaching the course over nine years at University of Florida. I compressed the file a bit to conserve my website bandwidth. Enjoy!
Socially Sanctioned Incompetence
In America, there is a funny, and revealing, version of mathematics. One popular joke is that, at a dinner party, the only socially acceptable thing to be bad at is mathematics. Someone recalls high school and says they are terrible at math, another says, me too, and everybody laughs. This is not so common in …
High-Order CFD for Validating Analytical Solution of the Navier-Stokes Equations – ‘BlackJack’
The Computational Fluid Dynamics (CFD) code, ‘BlackJack,’ was created with a singular purpose: to generate extremely accurate numerical solutions of the Navier-Stokes equations in support of a broader program to develop new analytical solutions to those same equations. Unlike general-purpose CFD tools designed for industrial applications or engineering approximations, BlackJack CFD is a research code, …
Framework for Analytical Solutions of the Navier-Stokes Equations for Hyperbolic Boundary Value Problems in the Aerodynamic Near-Field
Abstract: A framework to create new specific analytical solutions of the equations of motion for hyperbolic boundary value problems is presented. The method relies on a closed-form integral equation for mass density, involving a term that combines sources, geometry, ambient values, and radiation. Products of the density integral result in new more complicated solutions. The …
Navier-Stokes Equations and Practicality
Because an effort is likely impossible and impractical does not mean it is not worth attempting. The Navier-Stokes equations and turbulent flow represent the last great classical problem in physics. Since the time of Leonard Euler and Jean-Baptiste le Rond d’Alembert, many have devoted much of their lives to working on these problems. Although they …
Deming and Statistics
In God we trust. All others must bring data. — W. Edwards Deming Deming revolutionized quality management with his emphasis on data-driven decision-making. His 1950s lectures on Statistical Product Quality Administration in Japan were instrumental in Japan’s post-war economic growth, helping it become the world’s second-largest economy. Deming was awarded the National Medal of Technology …
Gödel and Time
$\mathrm{LL}$ cosmological solutions with non-vanishing density of matter known at present ${ }^1$ have the common property that, in a certain sense, they contain an “absolute” time coordinate, ${ }^2$ owing to the fact that there exists a one-parametric system of three-spaces everywhere orthogonal on the world lines of matter. It is easily seen that …
Underlying Assumptions
How Can Any Part of Mathematics Be Proven? The answer lies in demonstrating that a mathematical statement must be true if the underlying simpler math is also true. It is a great difficulty to show the increment from 1 to 2. Between 1910 and 1913, a three-volume work was published on this subject. Titled Principia …
Boole and Laws of Thought
George Boole, in the 1840s, proposed that variables could represent more than just numbers. Boole’s work, published in “An Investigation of the Laws of Thought” (1854), introduced algebra with two values: 1 (true) and 0 (false). Instead of traditional algebraic operations, Boolean algebra uses AND, OR, and NOT, also known as conjunction, disjunction, and complement. …