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In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems.As in other areas of mathematics, such problems are often made public at professional conferences and meetings.
Using the language of Lie algebra cohomology, the central extension can be described using a 2-cocycle on the loop algebra. This is the map φ : L g × L g → C {\displaystyle \varphi :L{\mathfrak {g}}\times L{\mathfrak {g}}\rightarrow \mathbb {C} } satisfying φ ( X ⊗ t m , Y ⊗ t n ) = m B ( X , Y ) δ m + n , 0 . {\displaystyle \varphi ...
In its most general form a loop group is a group of continuous mappings from a manifold M to a topological group G.. More specifically, [1] let M = S 1, the circle in the complex plane, and let LG denote the space of continuous maps S 1 → G, i.e.
The octonions are a special type of loop known as a Moufang loop. An associative quasigroup is either empty or is a group, since if there is at least one element, the invertibility of the quasigroup binary operation combined with associativity implies the existence of an identity element, which then implies the existence of inverse elements ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
The 100 prisoners problem is a mathematical problem in probability theory and combinatorics. In this problem, 100 numbered prisoners must find their own numbers in ...
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