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The word problem for free monoidal categories, i.e. deciding whether two given diagrams are equal, can be solved in polynomial time. The interchanger is a confluent rewriting system on the subset of boundary connected diagrams, i.e. whenever the plane graphs have no more than one connected component which is not connected to the domain or ...
A biological network is a method of representing systems as complex sets of binary interactions or relations between various biological entities. [1] In general, networks or graphs are used to capture relationships between entities or objects. [1]
This is one of the diagrams used in the definition of a monoidal cateogory. It takes care of the case for when there is an instance of an identity between two objects. commutes. A strict monoidal category is one for which the natural isomorphisms α, λ and ρ are identities. Every monoidal category is monoidally equivalent to a strict monoidal ...
Such categories where the multiplicative monoidal structure is the categorical product and the additive monoidal structure is the coproduct are called distributive categories. Vect , the category of vector spaces over a field, with the direct sum as ⊕ {\displaystyle \oplus } and the tensor product as ⊗ {\displaystyle \otimes } .
such that the pentagon diagram. and the unitor diagram commute. In the above notation, 1 is the identity morphism of M, I is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C. Dually, a comonoid in a monoidal category C is a monoid in the dual category C op.
Network motif: Motifs are unique sub-graphs composed of n-nodes found embedded in a network. For instance there exist thirteen unique motif structures containing three species, some of these correspond to familiar interaction modules studied by population ecologists such as food chains , apparent competition , or intraguild predation .
Figure 1. Example of a biological network between genes and proteins that controls entry into S phase. However, with knowledge of network interactions and a set of parameters for the proteins and protein interactions (usually obtained through empirical research), it is often possible to construct a model of the network as a dynamical system.
For example, the sample diagram does not indicate the physical type of connection between the PCs and the switch, but since a modern LAN is depicted, Ethernet may be assumed. If the same style of line was used in a WAN (wide area network) diagram, however, it may indicate a different type of connection.