Search results
Results from the WOW.Com Content Network
The following is known about retracts: A subgroup is a retract if and only if it has a normal complement. [4] The normal complement, specifically, is the kernel of the retraction. Every direct factor is a retract. [1] Conversely, any retract which is a normal subgroup is a direct factor. [5] Every retract has the congruence extension property.
Karol Borsuk (8 May 1905 – 24 January 1982) was a Polish mathematician. His main area of interest was topology . He made significant contributions to shape theory , a term which he coined.
[7] For all n for fields of revolution — shown by Boris Dekster (1995). [8] The problem was finally solved in 1993 by Jeff Kahn and Gil Kalai, who showed that the general answer to Borsuk's question is no. [9] They claim that their construction shows that n + 1 pieces do not suffice for n = 1325 and for each n > 2014.
The concept of a retraction in category theory comes from the essentially similar notion of a retraction in topology: : where is a subspace of is a retraction in the topological sense, if it's a retraction of the inclusion map : in the category theory sense.
Structural balance theory posits that some types of triads are forbidden and others are permitted on the basis of four rules. [4]Using the term “friend” to designate a positive sentiment and the term “enemy” to designate a negative sentiment, the classic balance model defines a sentiment network as balanced if it contains no violations of four assumptions:
A space is an absolute neighborhood retract for the class , written (), if is in and whenever is a closed subset of a space in , is a neighborhood retract of . Various classes C {\displaystyle {\mathcal {C}}} such as normal spaces have been considered in this definition, but the class M {\displaystyle {\mathcal {M}}} of metrizable spaces ...
Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets .
[8] [9] The resulting equations are somewhere between detailed balance and global balance equations. Any solution to the local balance equations is always a solution to the global balance equations (we can recover the global balance equations by summing the relevant local balance equations), but the converse is not always true. [2] Often ...