In mathematics, particularly in algebraic geometry and differential geometry, a **moduli space** is a geometric space that parametrizes a family of algebraic structures, such as curves, vector bundles, or more generally, geometrical objects. The idea is to organize the objects of a particular type into a space, where each point in this space corresponds to a distinct structure (often up to some kind of equivalence).
A **semialgebraic space** is a concept primarily arising in the field of real algebraic geometry and relates to the study of sets defined by polynomial inequalities.
Semidefinite programming (SDP) is a subfield of convex optimization that deals with the minimization of a linear objective function subject to semidefinite constraints.
In mathematics, particularly in set theory and algebra, the concept of a quotient by an equivalence relation is an important one. When you have a set \( S \) and an equivalence relation \( \sim \) defined on that set, you can partition the set into disjoint subsets, known as equivalence classes.
In mathematics, particularly in the field of algebraic geometry, a **Scheme** is a fundamental concept that generalizes the notion of algebraic varieties. Introduced by Alexander Grothendieck in the 1960s, schemes provide a framework for working with solutions to polynomial equations, accommodating both classical geometric objects and more abstract algebraic structures.
The Bott residue formula is a result in the field of differential topology and algebraic topology that relates the topology of smooth manifolds with specific types of mappings. It is particularly associated with the study of smooth maps between manifolds and the properties of their critical points. The formula is named after Raoul Bott, and it generalizes the classical concept of residues from complex analysis into the realm of differential forms and manifolds.
A CR manifold (Cauchy-Riemann manifold) is a differential manifold equipped with a special kind of geometric structure that generalizes the concept of complex structures to a more general setting. Specifically, a CR manifold has a certain kind of foliation by complex subspaces that leads to the definition of CR structures. To define a CR manifold, consider the following components: 1. **Smooth Manifold**: Start with a smooth manifold \( M \) of real dimension \( 2n \).
A complex differential form is a mathematical object used in the field of complex analysis and differential geometry. It extends the notion of differential forms to the context of complex manifolds, allowing for the study of functions, integrals, and geometry in complex spaces. ### Key Concepts: 1. **Differential Forms**: In real analysis, differential forms are antisymmetric tensor fields that can be integrated over manifolds.
In the context of Jordan algebras, a **mutation** refers to a particular process or operation that alters the elements of the algebra in a structured way. Jordan algebras are a class of non-associative algebras that arise in various areas of mathematics, particularly in the study of symmetries, physics, and quantum mechanics.
Foliation is a term that can refer to different concepts depending on the context, primarily in geology and botany. Here are the key meanings: 1. **Geology**: In geology, foliation refers to the parallel layering that can occur in metamorphic rocks due to the alignment of mineral grains under directional pressure. This structure is typically produced by processes such as metamorphism, where heat and pressure cause the minerals in the rock to recrystallize and realign.
A Hermitian connection is a specific type of affine connection defined on a complex Hermitian manifold that preserves the Hermitian metric when parallel transporting vectors along curves. This concept arises in differential geometry and is particularly significant in the study of complex manifolds and the geometry of Hilbert spaces in quantum mechanics.
The Kosmann lift is an example of a construction in the realm of mathematics, specifically in the field of topology and homotopy theory. It's related to the study of vector spaces and can be viewed as a method to construct new spaces from existing ones. Named after the mathematician K. Kosmann, the Kosmann lift is often discussed in the context of differential geometry or in the analysis of various types of fiber bundles.
A Poisson-Lie group is a mathematical structure that arises in the study of both differential geometry and theoretical physics, particularly in the context of integrable systems and quantum groups. It combines the ideas of Poisson geometry and Lie group theory. ### Definitions 1. **Lie Group**: A Lie group is a group that is also a smooth manifold, where the group operations (multiplication and inversion) are smooth maps.
Diaconescu's theorem is a result in the field of mathematical logic, particularly in the area of set theory and the foundations of mathematics. It is concerned with the characterizations of certain types of spaces in topology, specifically regarding the utility of countable bases. The theorem states that in the context of a particular type of topological space, the existence of a certain type of convergence implies the existence of a countable base.
Finitism is a philosophical and mathematical position that emphasizes the importance of finitism in the foundations of mathematics. It is characterized by the rejection of the actual existence of infinite entities or concepts, instead focusing exclusively on finite quantities and operations. This means that finitists do not accept infinitely large numbers, infinite sets, or processes that involve infinite steps as part of their foundational framework.
Ramified forcing is a method in set theory, particularly in the context of forcing and cardinals, used to create new sets with specific properties. It is a more intricate form of traditional forcing, designed to handle certain situations where standard forcing techniques may not suffice, especially in the context of constructing models of set theory or analyzing the properties of large cardinals. The concept of ramified forcing often involves a hierarchical approach to the forcing construction, where one levels the conditions and the models involved.
The Jech–Kunen tree is a specific type of tree used in set theory, particularly in the context of analyzing and demonstrating properties of model theory, forcing, and the structure of certain sets. Named after the mathematicians Thomas Jech and Kenneth Kunen, the tree is often discussed within the framework of large cardinals, set-theoretic forcing, and consistency results in mathematics. A Jech–Kunen tree is defined as an infinite tree that possesses specific properties.
Pinned article: ourbigbook/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