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AlphaGeometry is an artificial intelligence (AI) program that can solve hard problems in Euclidean geometry.It was developed by DeepMind, a subsidiary of Google.The program solved 25 geometry problems out of 30 from the International Mathematical Olympiad (IMO) under competition time limits—a performance almost as good as the average human gold medallist.
The projective dimension and the depth of a module over a commutative Noetherian local ring are complementary to each other. This is the content of the Auslander–Buchsbaum formula, which is not only of fundamental theoretical importance, but also provides an effective way to compute the depth of a module.
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.
Sections, particularly of principal bundles and vector bundles, are also very important tools in differential geometry. In this setting, the base space B {\displaystyle B} is a smooth manifold M {\displaystyle M} , and E {\displaystyle E} is assumed to be a smooth fiber bundle over M {\displaystyle M} (i.e., E {\displaystyle E} is a smooth ...
A graded vector space is an example of a graded module over a field (with the field having trivial grading). A graded ring is a graded module over itself. An ideal in a graded ring is homogeneous if and only if it is a graded submodule. The annihilator of a graded module is a homogeneous ideal.
Let R 1, R 2, R 3, R be rings, not necessarily commutative. For an R 1-R 2-bimodule M 12 and a left R 2-module M 20, is a left R 1-module. For a right R 2-module M 02 and an R 2-R 3-bimodule M 23, is a right R 3-module.
Some troops leave the battlefield injured. Others return from war with mental wounds. Yet many of the 2 million Iraq and Afghanistan veterans suffer from a condition the Defense Department refuses to acknowledge: Moral injury.
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution [1]) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to define invariants characterizing the structure of a specific module or object of this category.