Discrete Vector Models for Catalysis and Autocatalysis

Table 7

Notations for pairs of partition vectors and majorisation.

𝑛∈ℕ	Positive integer (natural number) to be partitioned.
𝑢𝐮=^1𝑢2⋮𝑢𝑑𝐮𝑣,𝐯=^1𝑣2⋮𝑣𝑑𝐯	Discrete partition vectors (column vectors) with nonnegative integers ui and vi as components (𝑖,𝑢𝑖,𝑣𝑖∈ℕ). They are used as pair of partition vectors to be compared with respect to majorisation and discrete vector catalysis. The following conventions are assumed. If a pair of different vectors u and v (𝐮≠𝐯) is compared with respect to majorisation, u is chosen as the vector with the first occurrence of a component 𝐮𝑖<𝐯𝑖.
𝑑𝐮,𝑑𝐯∈ℕ	Dimensions, that is, number of nonzero components, of u and v, respectively. In general, 𝑑𝐮 and 𝑑𝐯 are different.
𝑑𝐮,𝐯=sup(𝑑𝐮,𝑑𝐯)	Supremum of dimensions 𝑑𝐮 and 𝑑𝐯. The vector with the lower dimension is filled up with zeros to 𝑑𝐮,𝐯 components.
𝑛𝐮=tr(𝐮)=∑^𝑑𝐮,𝐯𝑗=1𝑢𝑗,𝑛𝐯=tr(𝐯)=∑^𝑑𝐮,𝐯𝑗=1𝑣𝑗,𝑛𝐮,𝑛𝐯∈ℕ	Trace (sum of components) of u and v, respectively; (classical) majorisation, 𝑛𝐮=𝑛𝐯=𝑛.
𝐨𝐮=𝐨𝐮,1𝐨𝐮,2⋮𝐨𝐮,𝑑𝐮,𝐯,𝐨𝐯=𝐨𝐯,1𝐨𝐯,2⋮𝐨𝐯,𝑑𝐮,𝐯𝐨𝐮,𝑖=∑^𝑖𝑗=1𝑢𝑗,𝐨𝐯,𝑖=∑^𝑖𝑗=1𝑣𝑗,𝑖=1,2,…,𝑑𝐮,𝐯	Vectors of partial sums of partition vectors u and v, respectively.
≺,≻∶𝐮≺𝐯≡𝐯≻𝐮∶=𝐨𝐮,𝑖≤𝐨𝐯,𝑖,𝑖=1,2,…,𝑑𝐮,𝐯	(Classical) majorisation relation. Comparable pair, where u is being majorised by v and v is majorising. u In quantum informatics the corresponding transition of entangled states is possible with certainty by LOCC (local operation and classical communication).
𝐮≈𝐯:=(𝐮≺𝐯)∨(𝐮≻𝐯)	Pair of partition vectors which is comparable with respect to majorisation.
𝐮≉𝐯=¬(𝐮≈𝐯)	Pair of partition vectors which is incomparable with respect to majorisation.
𝐮≺≻𝐯≡𝐯≻≺𝐮	Incomparable pair, where both u and v, respectively, are divided in two subsections of contiguous components, such that in the first section all partial sums 𝐨𝐯 of v are greater or equal to 𝐨𝐮 of 𝐨𝐮 and in the second section vice versa. (For this and the following definitions see also Section 3.2).
𝐮(≺≻)𝑘𝐯≡𝐯(≻≺)𝑘𝐮,𝑘∈ℕ	Same as 𝐮≺≻𝐯 but changing subsections of components are repeated k times; in case of 𝑘=1 one obtains the notation above.
𝐮≺≻≺𝐯≡𝐯≻≺≻𝐮	Incomparable pair, where both u and v, respectively, are divided in three subsections of contiguous components, such that in the first and last (third) section all partial sums 𝐨𝐯 of v are greater or equal to 𝐨𝐮 of u and in the second section vice versa.
𝐮≺(≻≺)𝑘𝐯≡𝐯≻(≺≻)𝑘𝐮,𝑘∈ℕ^0	Same as 𝐮≺≻≺𝐯 but changing subsections of components are repeated k times; in case of 𝑘=1 one obtains the notation above, in case of 𝑘=0 one gets the majorisation relation ≻(≺).