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Groups

In this subdiscipline: 269 papers

Discipline summary

Start with a set and an operation acting on pairs formed from that set.
0) Assume every possible pair yields something from that operation.
1) Assume you can chain that operation without taking care in which order that operation is performed.
2) Assume there exists an element which leaves any other element invariant under that operation.
3) Assume for each element there exists an inverse: Another unique element, such that the operation applied to them in any order will result in the element leaving others invariant.

you get a structure composed of the set and a 2-parameter function representing the operation(0). That function or operation has a neutral element(2) and is associative(1). Additionally the function can be inverted in each parameter: Applying the operation to that parameter's inverse object(3) and the given output. Of course, applied in the same order as was the order of the parameters.

That's a group! It's called commutative when the order of the operator doesn't change the result. Groups can be thought of as sets of matrices, along with matrix multiplication. Another interesting point of view is bijective functions on some set of numbers...

Popular papers

  1. An early version of these notes was prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated and expanded…
  2. Investigators have modeled oceanic and atmospheric vortices in the laboratory in a number of different ways, employing background rotation, density effects, and geometrical confinement. In this article, we address barotropic vortices in a rotating…
  3. Let $\Gamma$ be a finitely generated group, and let S be a fi- nite, non-necessarily symmetric, generating subset of $\Gamma$. Let h be the transition operator of the directed Cayley graph \g(F,S), acting on l^2(F). Staring with Kesten's seminal…
  4. The heat shock promoter is useful for regulating transgene expression in small water-living organisms. In zebrafish embryos, downstream gene expression can be greatly induced throughout the body by raising the temperature from 28.5 degrees C to 38.0…
  5. Increasingly, for many application areas, it is becoming important to include elements of nonlinearity and non-Gaussianity in order to model accurately the underlying dynamics of a physical system. Moreover, it is typically crucial to process data…
  6. Introduction This is a concise summary of recommended features in LATEX and a couple of extension packages for writing math formulas. Readers needing greater depth of detail are referred to the sources listed in the bibliography, especially…
  7. We describe latent Dirichlet allocation (LDA), a generative probabilistic model for collections of discrete data such as text corpora. LDA is a three-level hierarchical Bayesian model, in which each item of a collection is modeled as a finite…
  8. Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to…
  9. This document lists 4947 symbols and the corresponding LATEX commands that produce them. Some of these symbols are guaranteed to be available in every LATEX2e system; others require fonts and packages that may not accompany a given distribution and…
  10. Cytoscape is an open source software project for integrating biomolecular interaction networks with high-throughput expression data and other molecular states into a unified conceptual framework. Although applicable to any system of molecular…