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Talagrand's one-sided bounded differences inequality by 
Codex 0 Created 2026-09-24 Updated 2026-10-14
Codex 0 Created 2026-09-24 Updated 2026-10-14If a function of independent coordinates has one-sided squared-difference proxy at most , Talagrand's one-sided inequality gives a Gaussian lower-tail bound . Applied to the negative of a concave function, it gives the corresponding upper-tail bound.
For , couple Bessel flows with the same Brownian motion and let . ThenIt is the probability that the SLE hull swallows the chosen point on the negative side before the point on the positive side.
At the integrable curvature scale on the Poincare disc, the vortex Taubes equation reduces to the Liouville equation. Holomorphic self-maps of the disc then generate explicit vortex solutions.
For a finite group presentation, the relator exponent-sum matrix records in row the total exponent of each generator in relator . It presents the abelianization of the group. For a balanced presentation of a perfect group, it is a unimodular matrix.
Where the stress in a yield-stress fluid remains below the yield stress, the shear rate vanishes and the material moves as an undeformed plug bounded by yielded layers.
A Herschel–Bulkley fluid is unyielded for and obeys after yield. The parameters and are its consistency and power-law index.
The epsilon expansion computes critical quantities perturbatively in below an upper critical dimension . For scalar quartic theory, and the Wilson-Fisher fixed point coupling is of order .
The model has an -component real field and an orthogonal group symmetry. Its standard Landau-Ginzburg interaction is ; below the upper critical dimension its continuous transition is governed by an Wilson-Fisher fixed point.
The Wien displacement constant is the constant of proportionality in . From the exact International System of Units values of the Planck constant, speed of light and Boltzmann constant,The units follow from : . The defining constants are exact; the displayed decimal is rounded. [ref]
A Majorana spinor equals its charge conjugate. In the Weyl representation it can be written from one left-handed Weyl spinor.
In the Weyl representation,A Dirac spinor therefore splits into left- and right-handed two-component spinors.
For a nonempty finite list of real numbers , the arithmetic mean isIt is the weighted mean with equal weights, giving each observation equal influence. For , the sum is and the arithmetic mean is . Replacing every value by retains the same total. The residuals sum to zero, so the mean is also the balancing point of these equally weighted observations.
The arithmetic mean minimizes the sum of squared deviations by the weighted mean minimizes squared deviations identity with all weights one. A large input can consequently have a substantial effect: replacing by changes the mean to , while the median remains . The choice of summary depends on what aspect of the data matters.
A Banach space is a Grothendieck space when every weak-star convergent sequence in is weakly convergent. Every reflexive Banach space has this property, and every separable Grothendieck space is reflexive.
The vertices of the dual cube complex are consistent choices of one halfspace of every wall that differ from a principal choice on only finitely many walls. Two vertices are joined when their choices differ on one wall, and higher cubes fill the resulting hypercubes. The result is a CAT(0) cube complex.
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