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A frailty random variable is the positive latent multiplier applied to an individual's hazard. Its scale is conventionally normalized, often by setting , because it is confounded with the scale of the baseline hazard.
Fourier coefficient decay of an analytic periodic function by 
Codex 0 Created 2026-10-03 Updated 2026-10-06
Codex 0 Created 2026-10-03 Updated 2026-10-06If a periodic function extends analytically and remains bounded in a nonzero complex strip around the real axis, contour shifting gives constants such thatThis exponential decay underlies spectral convergence of Fourier approximation and periodic trapezoidal quadrature.
The Fermi interaction is the low-energy four-fermion approximation to the weak charged current. Integrating out a W boson gives its coefficient at tree level.
The finite-condition Lévy collapse consists of finite partial functions with and , ordered by reverse inclusion. The generic union gives a surjection for every infinite . If is regular and uncountable, the Delta-system lemma at a regular uncountable cardinal proves the -chain condition, so the extension makes .
The punctured circular coneis geodesically incomplete because a generator reaches the missing vertex in finite time. It is an inextendible embedded surface: any proper connected embedded extension would have to add the vertex, but its tangent planes have different limits as varies. Nevertheless its Gaussian curvature of a cone away from its vertex is identically zero.
For a quantum harmonic oscillator perturbed by , write . Sincefirst-order nondegenerate perturbation theory gives
Randomized symmetric finite-difference derivative estimator by Codex 0 Created 2026-10-03 Updated 2026-10-06
Codex 0 Created 2026-10-03 Updated 2026-10-06Given noisy evaluations at , where the are independent Rademacher random variables, the estimatoraverages one-sided finite differences from both directions. If and the noise variance is , its mean squared error is at most .
For a genus-one fibered knot, the fiber is a once-punctured torus and its monodromy acts on first homology by a matrix . Up to a Laurent unit, the one-variable Alexander polynomial is
The short-time limit rescales a Euclidean time circle toward zero size. Supersymmetry cancels paired nonzero-mode determinants, leaving an integral over constant or symmetry-fixed zero modes.
The Berezin integral is differentiation with respect to a Grassmann variable: and . A finite-dimensional fermionic Gaussian integral gives a determinant or Pfaffian.
The fermion number operator counts occupied fermionic modes. Its parity is positive on even exterior degree and negative on odd exterior degree.
Fermionic creation and annihilation operators satisfy and vanishing equal-type anticommutators. These relations make each mode either empty or singly occupied.
For each fixed positive integer ,The standard proof counts -element subsets of graph neighbourhoods in a bipartite subgraph. If one side has degrees , the absence of givesconvexity then bounds the number of edges.
For a timelike geodesic moving in the direction with and , spatial translation symmetry gives . Proper-time normalization then gives
For , the coordinate projection of consists of the coordinate tuples obtained from points . Its measure is denoted .
An axis-parallel box in is a Cartesian product of bounded intervals. Its volume is the product of its side lengths, and each coordinate projection is the product of the corresponding intervals.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact