Stirling's Formula from Statistical Mechanics | The n-Category Café
Aug 25, 2024 · Stirling’s formula says. N! ∼ √2πN(N e)N. where ∼ means that the ratio of the two quantities goes to 1 as N → ∞. Some proofs start with the observation that. N! = ∫∞ 0xNe − xdx. This says that N! is the Laplace transform of the function xN. Laplace transforms are important statistical mechanics. So what is this particular ...
Dimensional Analysis in Algebra and Geometry
Aug 8, 2024 · Another name for the study of categories of line objects is “dimensional analysis”. In dimensional analysis, a physical theory is described by specifying an abelian group of “dimensions” (these are the line objects) together with a commutative algebra of “quantities” (these are the sections of the line objects) which is graded by ...
A Look at the Mathematical Origins of Western Musical Scales
Feb 26, 2010 · This means that the frequency ratio is (9 / 8)3 × 256 / 243 which is precisely 3 / 2. So the fifth note is 3 / 2 times the frequency of the first note, and we finally have an explanation as to why this ratio is called a fifth. So we have now arrived at a rather rough approximation to the Western major scale.
In Praise of the Gershgorin Disc Theorem | The n-Category ...
Aug 9, 2016 · Sometimes called “Gershgorin’s Circle Theorem” Right — lots of people do call it that. But mathematicians settled long ago on using “circle” for the hollow shape and “disc” for the solid shape, and the actual circles play no particular role in Gershgorin’s theorem; it’s the discs that matter. So I prefer the name “Gershgorin disc theorem”.
Why Mathematics is Boring | The n-Category Café - University of …
Mar 28, 2024 · Why Mathematics is Boring. I don’t really think mathematics is boring. I hope you don’t either. But I can’t count the number of times I’ve launched into reading a math paper, dewy-eyed and eager to learn, only to have my enthusiasm slowly but remorselessly crushed by pages and pages of bad writing.
EGA1: The Language of Schemes | The n-Category Café
Feb 6, 2020 · EGA1: The Language of Schemes. Posted by John Baez. Tim Hosgood, Ryan Keleti and others have finished an English translation of this classic: Alexander Grothendieck, EGA1: The Language of Schemes. and the LaTeX files are now open-source on GitHub. They are working on the rest of EGA, and they could use help! Posted at February 6, 2020 1:28 AM UTC.
Octonions and the Standard Model (Part 4) | The n-Category Café
August 3, 2020. Octonions and the Standard Model (Part 4) Posted by John Baez. Last time we saw what we can do by choosing a square root of −1 in the octonions. They become a 4-dimensional complex vector space, and their automorphisms fixing this square root of −1 form the group SU(3). This is the symmetry group of the strong force —and ...
The Icosidodecahedron | The n-Category Café
Nov 16, 2022 · The icosidodecahedron can be built by truncating either a regular icosahedron or a regular dodecahedron. It has 30 vertices, one at the center of each edge of the icosahedron—or equivalently, one at the center of each edge of a dodecahedron. It is a beautiful, highly symmetrical shape. But it is just a shadow of a more symmetrical shape with ...
Overview of the general case 1 The generalized McShane’s identity gives a formula for the length L1 of a pre-specified boundary component 1 of X 2Mg; n(L1;:::;L ) as a sum of terms over all pants containing 1. 2 We integrate this formula over moduli space Mg;n(L1;:::;Ln) using Mirzakhani’s integration formula which gives us an integral in …
Schrödinger's Unified Field Theory | The n-Category Café
Dec 26, 2019 · Schrödinger’s Unified Field Theory. Posted by John Baez. Erwin Schrödinger fled Austria during World War II. In 1940 he found a position in the newly founded Dublin Institute for Advanced Studies. This allowed him to think again. He started publishing papers on unified field theories in 1943, based on earlier work of Eddington and Einstein.