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 …
Girard on the Limitations of Categories | The n-Category Café
2008年7月23日 · Re: Girard on the Limitations of Categories. Mauro Piccolo and Claudia Faggian have studied the connection between (a fragment) Ludics and models of computation in detail in the paper. Ludics is a Model for the finitary Linear Pi-Calculus. The close connection between Pi-Calculus and game semantics (see the paper Processes and Games by Honda ...
The Icosidodecahedron | The n-Category Café
2022年11月16日 · 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 ...
Semi-Simplicial Types, Part I: Motivation and History
2024年3月9日 · A semi-simplicial set is defined to consist of sets Xn for n ≥ 0, along with face maps ∂k: Xn → Xn − 1, for k ∈ {0, …, n}, satisfying the relations: ∂k ∘ ∂l = ∂l − 1 ∘ ∂k for k <l. One thinks of Xn as the set of n -simplices, and of the face maps as giving the boundary components of a given n -simplex. For example, X0 ...
Dimensional Analysis in Algebra and Geometry
2024年8月8日 · Ancient Greek mathematicians, so I’ve heard, understood multiplication geometrically. To reason about A × B A \times B they would draw line segments of length A A and length B B and form a rectangle with those as sides. This has area A × B A \times B.. Fine and dandy. But then what does an equation like 2 x 2 = x + 3 2x^2 = x + 3 mean? 2 x 2 2x^2 has units of area, x x has units of length ...
Category Theory Course Notes | The n-Category Café
2016年3月10日 · Meas. \mathrm {Meas}. Various kinds of categories, including monoids, groupoids, groups, preorders, equivalence relations and posets. The definition of a functor. Doing mathematics inside a category: isomorphisms, monomorphisms and epimorphisms. Christina Osborne’s notes.
A Look at the Mathematical Origins of Western Musical Scales
2010年2月26日 · 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.
Stirling's Formula from Statistical Mechanics | The n-Category Café
2024年8月25日 · 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 ...
Galois Theory | The n-Category Café - University of Texas at Austin
2024年8月15日 · About 40 short explanatory videos. A large collection of problems. Nearly 500 multiple choice questions. I’m a little bemused by the popularity of the Galois theory notes. I’ve made quite a few sets of course notes public before, e.g.: Fourier analysis. General topology. Linear algebra. Category theory.
Topoi from Lawvere Theories | The n-Category Café
2019年12月5日 · Topoi from Lawvere Theories. Posted by John Baez. Often we get a classifying topos by taking a finite limits theory T T and forming the category of presheaves T^ = SetTop \hat {T} = Set^ {T^ {op}}. For example, in their book, Mac Lane and Moerdijk get the classifying topos for commutative rings from the finite limits theory for rings this way.