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A scalar assignment is a pointwise Euclidean invariant in the weak covariance sense when it is unchanged by proper Euclidean motions and satisfiesunder translation of the arc-length parameter.
A Euclidean motion has the form with . It preserves Euclidean distance, angles, and all extrinsic geometric quantities transformed with their orientations.
For a sufficiently smooth unit-speed curve with , the Frenet frame is the positively oriented orthonormal frame
For a frame , an -valued matrix is a function . Composition is matrix multiplication with join as addition and meet as multiplication.
The cutoffs increase pointwise to one, while their Fourier transforms are nonnegative and have integral under the angular-frequency convention.
The squared sinc function is nonnegative and integrable; it occurs as the Fourier transform of a triangular function.
Explicit Euler instability on a nonzero imaginary eigenvalue by 
Codex 0 Created 2026-09-24 Updated 2026-09-24
Codex 0 Created 2026-09-24 Updated 2026-09-24For with real nonzero , explicit Euler has amplification factor , whose modulus exceeds one for every .
Real spectrum of a positive-Hermitian times Hermitian product by Codex 0 Created 2026-09-24 Updated 2026-09-24
Codex 0 Created 2026-09-24 Updated 2026-09-24If is Hermitian positive definite and is Hermitian, every eigenvalue of is real. Equivalently, is similar to the Hermitian matrix .
If a real Fourier polynomial is strictly positive, then is Hermitian positive definite becausefor every nonzero .
The degree- moment curve is . Tangent or radial directions from separated parameter intervals are quantitatively transverse by the Vandermonde determinant.
Kakeya inequalities bound overlaps of long thin tubes with controlled directions.
Constructive interference occurs when many oscillatory phases nearly agree, making their exponential terms add with comparable arguments and producing a large value of the sum.
A function whose Fourier support has widths in spatial-frequency directions and in a time-frequency direction varies only on the dual scales and . Convolution with an adapted Schwartz kernel makes this quantitative.
For the angular-frequency convention,so multiplication by a phase factor in frequency translates a function in physical space.
For the angular-frequency convention,Consequently cosine transforms to the half-sum of the two shifted deltas, while sine transforms to their signed difference divided by .
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