The Cook–Levin theorem, established by Stephen Cook in 1971 and independently by Leonid Levin, is a fundamental result in computational complexity theory. It states that the Boolean satisfiability problem (SAT) is NP-complete. This means that SAT is at least as hard as any problem in the complexity class NP (nondeterministic polynomial time), and any problem in NP can be reduced to SAT in polynomial time.
The Karp–Lipton theorem is an important result in computational complexity theory that connects the complexity classes \(P\), \(NP\), and \(PSPACE\). It was established by Richard Karp and Richard J. Lipton in the early 1980s. The theorem states that if \(NP\) problems can be solved in polynomial time by a non-deterministic Turing machine using polynomial space (i.e.
The PCP (Probabilistically Checkable Proofs) theorem is a significant result in computational complexity theory that characterizes the class of decision problems that can be efficiently verified by a probabilistic verifier using a limited amount of randomness and reading only a small portion of the proof.
De Bruijn's theorem, named after the Dutch mathematician Nicolaas Govert de Bruijn, is primarily known in the context of combinatorics and graph theory. It refers to several important results, but the most widely recognized version is in relation to the properties of sequences and combinatorial structures.
The Four-Vertex Theorem is a result in differential geometry and the study of curves. It states that for a simple, closed, smooth curve in the plane (which means a curve that does not intersect itself and is continuously differentiable), there are at least four distinct points at which the curvature of the curve attains a local maximum or minimum. To elaborate, curvature is a measure of how sharply a curve bends at a given point.
Kirchberger's theorem pertains to the field of mathematics, specifically in the area of graph theory and combinatorial optimization. The theorem is often involved with properties of vertices and edges in graphs, particularly in relation to specific configurations or arrangements. However, it’s important to note that Kirchberger's theorem is not as widely known as some other mathematical theorems, so detailed and widely recognized references might be limited.
Tverberg's theorem is a result in combinatorial geometry that concerns the division of points in Euclidean space. It states that for any set of \( (r-1)(d+1) + 1 \) points in \( \mathbb{R}^d \), it is possible to partition these points into \( r \) groups such that the \( r \) groups share a common point in their convex hulls.
The Erdős–Stone theorem is a fundamental result in extremal graph theory, which deals with understanding the maximum number of edges in a graph that does not contain a particular subgraph. Specifically, the theorem provides a way to determine the asymptotic behavior of the maximum number of edges in a graph on \( n \) vertices that does not contain a complete subgraph \( K_r \) (the complete graph on \( r \) vertices) as a subgraph.
Theorems about polygons constitute a significant part of geometry, focusing on the properties, relationships, and characteristics of various types of polygons.
Barbier's theorem is a result in geometry concerning the relationship between the perimeter of a plane figure and the circumference of a circle that has the same area as that figure. Specifically, Barbier's theorem states that for any plane figure, the perimeter of the figure is greater than or equal to the circumference of the circle that has the same area. The equality holds if and only if the figure is a circle.
Holditch's theorem is a result in the field of geometry, specifically in topology related to convex polyhedra. It states that any two convex polyhedra with the same number of vertices, edges, and faces are combinatorially equivalent, meaning they can be transformed into one another through a series of edge-edge and face-face correspondences while preserving the connectivity structure.
The "Tribune of Galileo" refers to an important technological and scientific development associated with Galileo Galilei, the renowned Italian astronomer, physicist, and mathematician of the Renaissance period. The term may be used in several contexts, but it primarily relates to Galileo's contributions to the fields of astronomy and observational science. One of the most notable aspects of Galileo's work involved the use of the telescope, which he improved and used to make groundbreaking astronomical observations.
"Hypatia" is a historical novel written by Charles Kingsley, published in 1853. The story is set in Alexandria, Egypt, during the late Roman Empire and revolves around the life of Hypatia, a renowned female philosopher, mathematician, and astronomer of the time. Kingsley's portrayal of Hypatia highlights her intellectual pursuits and her struggles against the rising tide of religious conflict, particularly the tensions between paganism and Christianity.
"The Physicists" is a play written by Swiss dramatist Friedrich Dürrenmatt, first performed in 1962. The play is a dark comedy that explores themes of science, ethics, responsibility, and the consequences of knowledge. The story is set in a sanatorium for the mentally ill, where three physicists—each pretending to be insane for various reasons—find themselves in a complex situation that reflects on human nature and the potential dangers of scientific discoveries.
"Kepler" is an opera composed by the German composer and conductor, Philip Glass. It premiered in 2009 and is based on the life and work of the renowned astronomer Johannes Kepler, who is known for his laws of planetary motion and contributions to the scientific revolution during the 17th century. The opera explores themes of science, metaphysics, and the human experience, focusing on Kepler's quest for knowledge and his struggles against the societal and intellectual challenges of his time.
The Nicolaus Copernicus Monument in Chicago is a statue commemorating the famous Renaissance astronomer Nicolaus Copernicus, who is best known for his heliocentric theory, which posited that the Earth and other planets revolve around the Sun. Located in a park dedicated to his memory, the monument celebrates Copernicus's contributions to science and astronomy.
Bruno Buchberger is an Austrian mathematician and computer scientist, known primarily for his contributions to the field of computational mathematics and computer algebra. He is particularly recognized for developing the Gröbner basis theory, which is fundamental in solving systems of polynomial equations and has applications in various areas such as algebraic geometry, robotics, and coding theory.
Conrad Habicht can refer to different things, but it’s often associated with historical figures or specific ventures. One notable reference is to **Conrad Habicht (1860–1933)**, who was a physicist primarily remembered for his contributions to the early development of quantum mechanics. He was associated with various scientific endeavors and collaborations during his time.
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 3. Visual Studio Code extension installation.Figure 4. Visual Studio Code extension tree navigation.Figure 5. Web editor. 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.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. - 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





