Mathematical Expressions - University of Texas at Austin
WEBThe WebTeX Examples include more examples of superscripts and subscripts, as well as an example of tensor notation. Prescripts. Superscripts and subscripts may also be placed before an expression. These can be created with the \multiscripts{}{}{} command, a generalization of \tensor.Here, the second argument is the base, while the first argument contains the prescripts and the third argument ...
Introduction to Categorical Probability | The n-Category Café
WEBAug 13, 2024 · An ETCS category is a nontrivial topos with three further properties: A function is determined by its effect on elements. That is, if f, g: X → Y f, g: X \to Y and f x = g x f x = g x for all x: 1 → X x: 1 \to X then f = g f = g. There’s a natural numbers object. The axiom of choice: every surjection has a section.
Markdown: Syntax - University of Texas at Austin
WEBMarkdown’s syntax is intended for one purpose: to be used as a format for writing for the web. Markdown is not a replacement for HTML, or even close to it. Its syntax is very small, corresponding only to a very small subset of HTML tags. The idea is not to create a syntax that makes it easier to insert HTML tags.
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.
WEBOverview 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 terms of volumes of moduli spaces of
Octonions and the Standard Model (Part 1) | The n-Category Café
WEBJul 17, 2020 · U→v ˉ × U→w = U(→v ˉ × →w) Now we’ve got all the machinery to describe the octonions in a way that makes them look a lot like quaternions. The quaternions were built from real scalars and real vectors: ℍ ≅ ℝ ⊕ ℝ3. So now we’ll build the octonions from complex scalars and complex vectors: 𝕆 ≅ ℂ ⊕ ℂ3.
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Timing, Span (Graph) and Cospan (Graph) | The n-Category Café
WEBAug 1, 2022 · This is a category whose objects are graphs, morphisms are cospans X → Z ← Y of graphs and composition is given by pushout, which exist for Graph. Cospan(Graph) is a symmetric monoidal category due to the cocartesian monoidal structure on Graph. For a set A, define the graph 𝒱(A) ≡ (A, ∅) to contain no edges.
The Icosidodecahedron | The n-Category Café
WEBNov 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 …
The Uniform Measure | The n-Category Café - University of Texas …
WEBNov 24, 2020 · μX = lim t → ∞μt. Conceptually, the difference between the “most spread out” measures μt and the uniform measure μX is that μt depends on the scale factor t (as in the example of the line segment), but μX doesn’t. The uniform measure is independent of scale: μuX = μX for all u> 0.