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Pappus's centroid theorem, named after the ancient Greek mathematician Pappus of Alexandria, is a principle concerning the geometry of figures in relation to their centroids (or centroids). It actually consists of two related theorems, often referred to as Pappus's centroid theorems.
The Non-Squeezing Theorem is a fundamental result in symplectic geometry, a branch of mathematics that studies structures and properties of spaces that are equipped with a symplectic form. Specifically, the theorem addresses the concept of symplectic embeddings, which are mappings between symplectic manifolds that preserve the symplectic structure. The Non-Squeezing Theorem asserts that there are limitations on how one can "squeeze" or transform symplectic spaces.
Liouville's theorem in the context of conformal mappings relates to the properties of holomorphic (or analytic) functions defined on the complex plane. Specifically, the theorem states that any entire (holomorphic everywhere in the complex plane) function that is bounded is constant.
The Lickorish–Wallace theorem is a result in the field of topology, specifically in the study of 3-manifolds. This theorem provides a criterion for when a connected sum of 3-manifolds can be represented as a connected sum of prime 3-manifolds.
Lexell's theorem, often associated with the field of celestial mechanics, pertains to the motion of celestial bodies in gravitational fields. Specifically, it describes the precession or gradual change in the orientation of the orbit of a celestial body due to perturbations from other bodies or non-uniformities in the gravitational field.
Jørgensen's inequality is a result in the field of functional analysis, particularly concerning the relationships between norms in Banach spaces. Specifically, Jørgensen's inequality pertains to the estimates of certain linear operators and is often discussed in the context of submartingales, Brownian motion, and processes in probability theory.
The Hyperbolization Theorem is a result in the field of topology and geometric group theory, specifically concerning the characteristics of 3-manifolds. It states that any compact, orientable 3-manifold that contains a certain type of submanifold (specifically, a “reducible” submanifold or one that can be "hyperbolized") can be decomposed into pieces that exhibit hyperbolic geometry.
The Fold-and-Cut theorem is a result in computational geometry and combinatorial geometry that deals with the problem of folding paper to achieve a desired cut. Specifically, it states that any shape that can be formed by a straight cut through a folded piece of paper can be realized by an appropriate folding of the paper beforehand.
Euler's rotation theorem states that any rotation of a rigid body in three-dimensional space can be represented as a single rotation about a specific axis. This means that for any arbitrary rotation, it is possible to find an axis in space such that the body can be considered to have rotated around this axis by a specific angle. More formally, the theorem states that given any rotation defined by a rigid body transformation, there exists a unique axis of rotation and a corresponding angle of rotation about that axis.
Dévissage is a French term that translates to "unscrewing" in English. In various contexts, it can refer to the act of removing screws or bolts from an object. However, the term can also have specialized meanings in different fields. In the context of watchmaking, for example, dévissage refers to the process of unscrewing the crown of a watch to adjust the time or date.
The Double Limit Theorem, often referred to in the context of limits in calculus, relates to the properties and behavior of limits involving functions of two variables.
The Collage Theorem, often referred to in the context of topology and geometry, is a concept related to the study of spaces and continuous functions. However, the term "Collage Theorem" may not be universally recognized under that name in all areas of mathematics, and its interpretation can vary depending on the context.
The Castelnuovo–de Franchis theorem is a result in algebraic geometry that deals with the embedding of algebraic curves in projective space. More specifically, it provides conditions under which a non-singular projective curve can be embedded into projective space of a certain dimension based on its genus and the degree of a line bundle.
Castelnuovo's contraction theorem is a result in algebraic geometry, specifically dealing with the properties of smooth projective varieties. The theorem is part of the study of the behavior of certain types of morphisms between algebraic varieties, particularly in the context of contraction maps in the context of the minimal model program (MMP).
Campbell's theorem is a result in differential geometry that pertains to the geometry of a Euclidean space and the properties of certain curves and surfaces within it. Specifically, it deals with the concept of the Frenet frame and the curvature of curves. In its essence, Campbell's theorem states that for a certain class of curves in Euclidean space, there exists a correspondence between curvature and torsion.
Blichfeldt's theorem is a result in the field of number theory, specifically in the study of lattice points and their distributions. Named after the mathematician A.B. Blichfeldt, the theorem deals with the packing of points in a convex geometry context.
Bang's theorem on tetrahedra is a result in geometry regarding the arrangement of points within a tetrahedron. Specifically, it concerns the maximal number of points that can be placed in the interior of a tetrahedron such that no three points are coplanar.
Anderson's theorem, formulated by P.W. Anderson in the context of condensed matter physics, primarily relates to the behavior of disordered systems, particularly in the study of superconductivity and localization effects. The theorem is often associated with the concept of Anderson localization, which describes how wavefunctions (such as those of electrons) can become localized in a disordered medium and thus inhibit electrical conductivity.
The Almgren regularity theorem is a result in the field of geometric measure theory, specifically concerning the regularity properties of minimizers of certain variational problems. Named after the mathematician Frederic J. Almgren Jr., the theorem addresses the behavior of minimizers of the area functional, which are often studied in the context of minimal surfaces.
The term "2π theorem" doesn't refer to a widely recognized theorem in mathematics or physics by that name. However, it might be associated with concepts involving the number \(2\pi\), which frequently appears in contexts related to circles, trigonometry, and periodic functions.
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
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