Ideal ring bundle

Ideal ring bundle or gold ring bundle is a mathematical term which means an n-stage cyclic sequence of semi-measured terms, e.g. integers for which the set of all circular sums enumerates row of natural numbers by fixed times. The circular sum is called a sum of consecutive terms in the n-sequence of any number of terms (from 1 to n − 1).

Examples

For example, the cyclic sequence (1, 3, 2, 7) is an Ideal Ring Bundle because four (n = 4) its terms enumerate of all natural numbers from 1 to n(n − 1) = 12 as its starting term, and can be of any number of summing terms by exactly one (R = 1) way:

1,

2,

3,

4 = 1 + 3,

5 = 3 + 2,

6 = 1 + 3 + 2,

7,

8 = 7 + 1,

9 = 2 + 7,

10 = 2 + 7 + 1,

11 = 7 + 1 + 3,

12 = 3 + 2 + 7,

13 = 1 + 3 + 2 + 7.

The cyclic sequence (1, 1, 2, 3) is an ideal ring bundle also, because four (n = 4) its terms enumerate all numbers of the natural row from 1 to n(n − 1)/R = 6 as its starting term, and can be of any number of summing terms by exactly two (R = 2) ways:

1, 1

2, 2 = 1 + 1

3, 3 = 2 + 1

4 = 3 + 1, 4 = 1 + 1 + 2

5 = 2 + 3, 5 = 3 + 1 + 1

6 = 1 + 2 + 3, 6 = 2 + 3 + 1

References

• Синтез оптимальних комбінаторних систем. - Львів: Вища школа, 1989.- 168 с.
• «Multi-dimensional Systems Based on Perfect Combinatorial Models», IEEE, Multidimensional Systems: Problems and Solutions, #225, London, 1998. doi:10.1049/ic:19980164
• RIZNYK, V.; BANDYRSKA, O.; SKRYBAYLO-LESKIV, D.. Application of the gold ring bundles for innovative non-redundant sonar systems. Archives of Acoustics, [S.l.], v. 31, n. 4(S), p. 379-384, jan. 2014. ISSN 2300-262X.