1. Introduction
Peng, J. introduced the definition of Shape of numbers in [1]:
there are M-1 intervals between adjacent numbers. Use A for continuity and B for discontinuity, record as a string of M-1 characters (e.g. AABB…) to represents a catalog.
Define collection of a catalog as Shape of numbers. Use the symbol PX to represent a catelog (if M = 1 then PX = 1).
The single
is an Item,
is the product of an item.
For example:
,
,
= Count of numbers of PX,
= Count of A,
= Count of B
= Count of items belonging to PX
= Minimum product of PX:
,
:
,
= Sum of all product of items belonging to PX in
For example:
= Set of items belonging to PX with the maximum factor = N-1
For example:
[1] obtain the conclusion:
(1)
(2)
Definition: Subdividethe
by interval of adjacent numbers. If the discontinuity interval is D > 1, the interval of adjacent numbers ≥ D is classified into a same catelog. Use the symbol PY represent a catelog and represented by [min item]
For example:
,
. Same as PX = B
,
,
,
,
Redefinition: PB(PY) = Count of discontinuity intervals in PY, compatible with PX
Redefinition: IDX(PY) = The maximum factor of MIN(PY) + 1, compatible with PX
Definition: BASE(PY) = PX, If PB(PX) = PB(PY), PM(PX) = PM(PY), PX has discontinuity intervals at the same positions of PY.
For example:
;
;
;
Definition: if
, then
Definition:
= Maximum factor of
,
(1.0)
[Proof]
(1.1)
[Proof]
By definition:
(1.2)
Derived from (1.2)
(1.3)
By definition:
(1.4)
(1.5)
According to the method in [1]:
(1.6)
compatible with (1).
2. Calculation Formula
If
, the calculation formula has been given by (2).
Otherwise, it can be deduced from (1.1)-(1.5).
(2.1)
[Proof]
(2.2)
[Proof]
(2.3)
[Proof]
(2.4)
[Proof]
2.1.
,
1*)
,
items in total.
2*)
3*)
Use the form
. The expansion function has 2M items in total.
4*)
5*)
,
[Proof]
,
Suppose
,
.
According to inductive hypothesis:
,
,
,
(2.1.1)
1*) is obvious.
Mi change form
to
2*) proved
3*) proved
4*) 5.1*)
4*) 5.2*)
4*) 5*) proved
,
(2.1.2)
1*) is obvious.
Mi change form
to
2*) proved
3*) proved
4*) 5.1*)
4*) 5.2*)
4*) 5*) proved
q.e.d.
Example 2.1:
2.2.
,
1*)
,
items in total.
2*)
3*)
Use the form
.
4*)
5*)
(Same as 2.1)
Example 2.2:
,
2.3. The Meaning of the Expansion of SUM (N, PY)
,
(means
) or
,
(2.3.1*)
Define 2.3.
in (2.3.1*) with the same PF
Example 2.3:
then can prove
,
(2.3.2*)
Theorem 2.3
in (2.3.2*) expand by {}, factors has the same
,
(2.3.3*)
,
[Proof]
Suppose
,
.
,
,
,
When
(2.3.3*)
(2.3.3*)
(2.3.3*)
When
(2.3.3*)
(2.3.3*)
(2.3.3*)
q.e.d.
2.3.1)
has the similar conclusions.
2.3.2)
,
3. A Theorem of Ring
,
,
When PY = BS,
This inspired us:
In the form
,
(Mi is same).
Definition 3.1:
;
,
,
are allowed.
Choice N from
,
Indicates that Ki was selected,
Indicates that Ki was unselected,
,
(*)
When S = 1, abbreviated as
Definition 3.2:
, Sum traverses all N1-Choice of K
Theorem 3.1:
,
[Proof]
When
or
, it is obvious.
Suppose
holds
Symmetry
holds
Suppose
holds
q.e.d.
Example 3.1:
Definition 3.3:
Ki come from q sources:
,
indicates Ki come from Sj
,
, (
);
;
then can change related parts of definition 3.1 to
and define
Theorem 3.2
,
[Proof]
Only need to prove S=1, and specify
, (
)
,
. An item has M factors,
Choice N1 factors, Ki in these factors,
, It is called invariant factor{F}.
Others are called variable factors {V}
, choice L2 factors in {V} to S2, L3 to S3
By definition,
and items gives all (N1, L2, L3)-Choice of K.
Prove
satisfies the definition of
Let
,
,
,
,
,
,
The invariant factors match the definition
Match the definition
Match the definition
Match the definition of
q.e.d.
Example 3.2
3.1)
is permutable in
3.2)
3.3) In the section 2,
is generated by the form
. It can be understood asgenerated by 2-SET:
,
.
If
, (
,
,
are allowed),
can be understood asgenerated by (X + 1)-SET:
3.4)
, Theorem 2.3 has the similar promotion.
4. Conclusion
Review the whole process,
Basic Shapes in [1]
More Shapes in this article.
Acknowledgements
The author is very grateful to Mr. HanSan Z. (Department of Electronic Information, Nanjing University) for his suggestions on this paper.