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Congeneric Intervals Distributions

Congeneric Intervals Distributions is a stack of Partials Intervals Distribution that represents Intervals Distribution.

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Mathematical Definition

Let \(ID_p\) is Partial Interval Distribution

Define a Congeric intervals distributions

as m-tuple of congeneric intervals distributions

\(CID\) is m-tuple of \(ID_c\) intervals distributions

\[CID = <cid_1, cid_2, ..., cid_m>,\]
\[\forall j \in \{1, ..., m\} \exists cid_j \in \{ID_c\}, \]

where:

  • \(cid_j\)​ is called the \(j\)-th congeneric intervals distribution.
  • \(l := |CID(1)|\) is length, \(l \in N\)
  • \(m := |CID|\) is power, \(m \in N\)

as matrix (m,l)

\(CID\) is (m,l) matrix of \(\{0,..,, l\}\)

\[ CID = \begin{pmatrix} cid_{1,1} & cid_{1,2} & \cdots & cid_{1,l} \\ cid_{2,1} & cid_{2,2} & \cdots & cid_{2,l} \\ \vdots & \vdots & \ddots & \vdots \\ cid_{m,1} & cid_{m,2} & \cdots & cid_{m,l} \end{pmatrix} \]
\[\forall j \in \{1, ..., m\}, i \in \{1, ..., l\} \exists cid_{j,i} \in \{0,..,, l\}, \]

where:

  • \(cid_j\)​ is called the \(j\)-th congeneric intervals distribution or \(j\)-th row.
  • \(cid_{j,i}\)​ is called the \(i\)-th element of \(j\)-th congeneric intervals distribution.
  • \(l\) is length, \(l \in N\)
  • \(m\) is power, \(m \in N\)

from Congeneric Intervals Chains

Let \(IC_{p}\) is Partial Interval Chain

Let \(ID_p\) is Partial Intervals distribution described as function

\[ID_p : \big\{ IC_p \big\} \longrightarrow \big\{ \{1,...,l\} \longrightarrow N_0 \big\},\]

Let \(CIC\) is Congeneric Intervals Chains - (m,l) matrix of \(\{1,...,l\} \cup \{-\}\)

\[ CIC = \begin{pmatrix} \Delta_{1,1} & \Delta_{1,2} & \cdots & \Delta_{1,l} \\ \Delta_{2,1} & \Delta_{2,2} & \cdots & \Delta_{2,l} \\ \vdots & \vdots & \ddots & \vdots \\ \Delta_{m,1} & \Delta_{m,2} & \cdots & \Delta_{m,l} \end{pmatrix} \]

Define

\[congenerics\_intervals\_distributions : \{CIC\} \longrightarrow \{CID\}\]
\[congenerics\_intervals\_distributions(CIC)(j)(i) = ID_p(CIC(j))(i) \]