Discrete Vector Models for Catalysis and Autocatalysis

Table 8

Notations for tensor operations and discrete vector catalysis.

𝑐𝐜=^1𝑐2⋮𝑐𝑑𝐜𝐜^𝑇=(𝑐1𝑐2…𝑐𝑑𝐜)	Catalyst partition vector, represented as column vector c or as row vector 𝐜^𝑇; it will be used for tensor multiplication with u or v, respectively, in vector catalysis.
𝑑𝐜∈ℕ	Dimension (number of nonzero components) of catalyst partition vector c.
𝑚=tr(𝐜)=∑^𝑑𝐜𝑗=1𝑐𝑗,𝑚∈ℕ	Trace (sum of components) of catalyst partition vector c.
⊗	Symbol for tensor product relation.
𝐮⊗𝐜^𝑇,𝐯⊗𝐜^𝑇 (vector product notation)	Tensor products of u with c(𝐜^𝑇) and v with c(𝐜^𝑇), respectively, in both cases represented as a (𝑑𝐮,𝐯×𝑑𝐜)-matrix.
𝐮⊗𝐜,𝐯⊗𝐜 (tensor product notation)
𝜽𝐮⊗𝐜,𝜽𝐯⊗𝐜	Vectors of tensor products 𝐮⊗𝐜 and 𝐯⊗𝐜, respectively, where all components (tensor product elements) are arranged in decreasing (nonincreasing) order.
𝑑𝐮⊗𝐜=𝑑𝐯⊗𝐜=𝑑𝐮,𝐯⋅𝑑𝐜	Dimension (number of components) of tensor product vectors, that is, dimension 𝑑𝐮,𝐯 of partition pair vectors multiplied with dimension 𝑑𝐜 of catalyst vector.
𝑛𝐮⊗𝐜=𝑛𝐯⊗𝐜=𝑛⋅𝑚	Trace (sum of components) of tensor product vectors, that is, trace of partition vectors multiplied with trace of catalyst vector.
𝐨𝐮⊗𝐜,𝐨𝐯⊗𝐜	Vectors of partial sums of tensor product vectors 𝜽𝐮⊗𝐜 and 𝜽𝐯⊗𝐜, respectively.
𝐮^⊗𝑝=𝐮⊗𝐮⊗…⊗𝐮𝑝-times,𝐯^⊗𝑞=𝐯⊗𝐯⊗…⊗𝐯𝑞-times,𝑝,𝑞∈ℕ^0	Tensor exponentiation of partition vectors u and v with exponents p and q, respectively. Special cases, 𝐮^⊗1=𝐮,𝐯^⊗1𝐮=𝐯,⊗0=𝐯^⊗0=(1).