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A spectral gap is a positive separation between designated parts of the spectrum of a linear operator. For the power method, the relevant gap separates the modulus of the dominant eigenvalue from the other eigenvalue moduli.
An eigenvalue is dominant when its modulus is greater than or equal to that of every other eigenvalue. It is uniquely dominant when the inequality is strict.
A complex Lie subalgebra is solvable iffor every and . For an abstract Lie algebra, this is equivalent to .
Every finite-dimensional complex representation of a solvable Lie algebra has a common eigenvector. Equivalently, an irreducible finite-dimensional complex representation is one-dimensional; iterating gives simultaneous upper triangularization.
Every connected solvable affine algebraic subgroup of preserves a complete flag in , equivalently it is conjugate to a subgroup of the upper triangular matrices.
Extension of a rational map from a smooth projective curve by 
Codex 0 Created 2026-09-24 Updated 2026-09-24
Codex 0 Created 2026-09-24 Updated 2026-09-24Every rational map from a smooth projective curve to a projective variety extends uniquely to a morphism. At a missing point, the discrete valuation ring of the curve lets one divide homogeneous coordinates by their smallest valuation, leaving regular coordinates of which at least one is a unit.
At every point of a smooth algebraic curve, the local ring is a discrete valuation ring. Its valuation measures the order of vanishing of a rational function at that point.
The special orthogonal Lie algebra consists of the infinitesimal transformations preserving a nondegenerate symmetric bilinear form.
The even signed symmetric group is the index-two subgroup of signed permutations with an even number of sign changes. It is the Coxeter group of type .
A continuous semimartingale has a decomposition in which both the local-martingale and finite-variation parts are continuous.
For , the Levine-Tristram signature is the signature of the Hermitian matrixIt is locally constant away from unit roots of the Alexander polynomial of a knot.
With outward normal , the unit sphere has . The Gauss equation therefore gives sectional curvature one on every tangent two-plane.
On a complete connected Riemannian manifold of nonpositive sectional curvature, the exponential map at every point is a covering map. If the manifold is simply connected, each exponential map is a diffeomorphism from a tangent space onto the manifold.
A Riemannian manifold is flat when every sectional curvature is zero. A complete simply connected flat -manifold is isometric to Euclidean space .
Ricci curvature is the trace of the Riemann curvature tensor in its first and third arguments. For a unit tangent vector and an orthonormal basis ,
Brownian time reversal for fixed-strike lookback extrema by Codex 0 Created 2026-09-24 Updated 2026-10-03
Codex 0 Created 2026-09-24 Updated 2026-10-03For a Brownian motion with drift , the process on has the same law as . Thereforehave the same distribution, which equates the corresponding geometric-Brownian lookback payoffs.
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