You Could Have Invented De Bruijn Indices | The n-Category Café
WEBAug 30, 2021 · You Could Have Invented De Bruijn Indices. One of the most curious things about formal treatments of type theory is the attention paid to dummy variables. The point is simple enough to explain to a calculus student: just as the integrals ∫2 0x3dx and ∫2 0t3dt are equal, so are the functions λx. x3 and λt. t3 — the name of the variable ...
Stirling's Formula from Statistical Mechanics | The n-Category Café
WEBAug 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 ...
The Pi Calculus | The n-Category Café - University of Texas at Austin
WEBAug 15, 2009 · Bigraphs have evolved from process calculi, especially the calculus of Mobile Ambients (invented by L Cardelli and A Gordon) and the Pi calculus. Over-simplifying a little, the Ambient calculus models spatial reconfiguration, while the Pi calculus models reconfiguration of connectivity. Ubiquitous systems need both.
April 2020 Archives | The n-Category Café - University of Texas at …
WEBApr 28, 2020 · A couple years ago I blogged about the polycategory of multivariable adjunctions and how it embeds in the 2-Chu construction Chu (Cat, Set) Chu(Cat,Set).After my talk at the ACT@UCR seminar this week, some folks were hanging out at the category theory zulip, and Reid Barton asked. The theory of model categories is self-dual. Is there some useful way to embed {model categories + Quillen ...
A Look at the Mathematical Origins of Western Musical Scales
WEBFeb 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.
A Discussion on Notions of Lawvere Theories
WEBMay 1, 2017 · The Kan Extension Seminar II continues with a discussion of the paper Notions of Lawvere Theory by Stephen Lack and Jirí Rosický. In his landmark thesis, William Lawvere introduced a method to the study of universal algebra that was vastly more abstract than those previously used. This method actually turns certain mathematical stuff ...
The Tenfold Way (Part 5) | The n-Category Café - University of …
WEBNov 22, 2020 · The tenfold way, Notices Amer. Math. Soc. 67 (November 2020), 1599–1601. The tenfold way became important in physics around 2010: it implies that there are ten fundamentally different kinds of matter. But it goes back to 1964, when C. T. C. Wall classified real ‘super division algebras’. He found that …
Dynamical Systems and Their Steady States | The n-Category Café
WEBBaez–Pollard’s approach The category Dynam \mathbf{Dynam} of open dynamical systems. Let’s start by introducing the main players. A dynamical system is usually defined as a manifold M M whose points are “states”, together with a smooth vector field on M M saying how these states evolve in time. Since the motivation in this paper comes from chemistry, our manifolds will be euclidean ...
Bernoulli Numbers and the Harmonic Oscillator
WEBAug 17, 2024 · The Bernoulli numbers can be defined like this: x ex − 1 = B0 + B1x + B2x2 2! + B3x3 3! + ⋯. and if you grind them out, you get. B0 = 1 B1 = −1 2 B2 = 1 6 B3 = 0 B4 = − 1 30. and so on. The pattern is quite strange. Bernoulli numbers are connected to hundreds of interesting things.
Guillotine Partitions and the Hipparchus Operad
WEBDec 26, 2022 · Wikipedia, Guillotine partition. There are 6 types of guillotine partition where you make 2 cuts, so we say the 2nd Schröder number is 6: The first few Schröder numbers Sn S_n go like this: etc. For example, S3 = 22 S_3 = 22 since there are 22 types of guillotine partition of a square with 3 cuts, as shown below: