Isomorphism Task
An Isomorphism Task is an Exact Matching Task that requires the production of an Isomorphism Function between two items.
- AKA: Homeomorphism Task, Diffeomorphism Task.
- Example(s):
- Counter-Example(s):
- See: Category Theory, Group (Mathematics), Roots of Unity, Mathematics, Homomorphism, Morphism, Map (Mathematics), Inverse Function, Mathematical Object.
References
2018
- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Isomorphism Retrieved:2018-7-21.
- In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.
For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective.
In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called diffeomorphisms.
A canonical isomorphism is a canonical map that is an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space V to its second dual space is a canonical isomorphism; on the other hand, V is isomorphic to its dual space but not canonically in general.
Isomorphisms are formalized using category theory. A morphism in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism in that category such that and , where 1X and 1Y are the identity morphisms of X and Y, respectively.
- In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.
2016
- (Redmond & Cunningham, 2016) ⇒ Ursula Redmond, and Padraig Cunninghama (2016). [hhttps://arxiv.org/pdf/1605.02174.pdf Subgraph isomorphism in temporal networks]. arXiv preprint arXiv:1605.02174.
- QUOTE: The subgraph isomorphism problem determines whether a given graph contains a subgraph which has the same topological structure as another given graph. Subgraph isomorphism is an NP-complete problem (Michael & David, 1979). Thus, the time complexity of brute force matching algorithms increases exponentially with the size of the graphs and query graphs to be matched. This makes the problem prohibitively expensive to solve for graphs of large scale.