Class Oriented Ring Associated Language - ορισμός. Τι είναι το Class Oriented Ring Associated Language
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Τι (ποιος) είναι Class Oriented Ring Associated Language - ορισμός

(N+1)-MANIFOLD W LINKING TWO N-MANIFOLDS M AND N, WITH BOUNDARY OF W CONSISTING OF M AND N
Bordism; Cobordism theory; Cobordant; Cobordism class; Oriented cobordism ring; Oriented cobordism; Oriented cobordant
  • Fig. 1
  • A cobordism (''W''; ''M'', ''N'').
  • The 3-dimensional cobordism <math>W = \mathbb{S}^1 \times \mathbb{D}^2 - \mathbb{D}^3</math> between the 2-[[sphere]] <math>M = \mathbb{S}^2</math>  and the 2-[[torus]] <math>N = \mathbb{S}^1 \times \mathbb{S}^1,</math> with ''N'' obtained from ''M'' by surgery on <math>\mathbb{S}^0 \times \mathbb{D}^2 \subset M,</math>and ''W'' obtained from ''M'' × ''I'' by attaching a 1-handle <math>\mathbb{D}^1 \times \mathbb{D}^2.</math>
  •  A cobordism between a single circle (at the top) and a pair of disjoint circles (at the bottom).
  • Fig. 2a
  • Fig. 2b
  • Fig. 2c. This shape cannot be embedded in 3-space.

Class Oriented Ring Associated Language      
<language> (CORAL) A language developed by L.G. Roberts at MIT in 1964 for graphical display and systems programming on the TX-2. It used "rings" (circular lists) from Sketchpad. ["Graphical Communication and Control Languages", L.B. Roberts, Information System Sciences: Proc Second Congress, 1965]. [Sammet 1969, p.462]. (1994-11-30)
Quotient ring         
CONSTRUCTION IN ABSTRACT ALGEBRA
Factor ring; Residue class ring; Residue ring; Quotient Ring; Factor Ring; Quotient associative algebra
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra.
Ring languages         
BRANCH OF THE NARROW GRASSFIELDS LANGUAGES; BEST KNOWN MEMBER IS KOM; NAMED AFTER THE OLD RING ROAD OF CENTRAL CAMEROON
Ring language
The Ring or Ring Road languages, spoken in the Western Grassfields of Cameroon, form a branch of the Narrow Grassfields languages. The best-known Ring language is Kom.

Βικιπαίδεια

Cobordism

In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.

The boundary of an (n + 1)-dimensional manifold W is an n-dimensional manifold ∂W that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds.

A cobordism between manifolds M and N is a compact manifold W whose boundary is the disjoint union of M and N, W = M N {\displaystyle \partial W=M\sqcup N} .

Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4 – because the word problem for groups cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in geometric topology and algebraic topology. In geometric topology, cobordisms are intimately connected with Morse theory, and h-cobordisms are fundamental in the study of high-dimensional manifolds, namely surgery theory. In algebraic topology, cobordism theories are fundamental extraordinary cohomology theories, and categories of cobordisms are the domains of topological quantum field theories.