Discrete Dynamics by Different Concepts of Majorization

Table 2

The order of the different concepts of majorization concerning bubble-size distributions is given. One sees the time steps in the first column. The second and third columns show the relation × of classical majorization with different time steps. The transitions of the weak submajorization can be examined in the fourth and fifth columns differing in the time steps. Incomparable distributions are denoted by 𝑘=𝑡[𝑠]/5𝑠.

ℎ(𝑘)𝑅ℎ(𝑘+1)	ℎ(2𝑘)𝑅ℎ(2𝑘+2)	𝑓(𝑘)𝑅𝑓(𝑘+1)	𝑓(2𝑘)𝑅𝑓(2𝑘+2)	≻
0	≻	≻𝑤	≻𝑤	≻
1	≻𝑤		≻
2	≻	≻𝑤	≻𝑤	≻
3	≻𝑤		≻
4	≻	≻𝑤	≻𝑤	×
5	≻𝑤		×
6	×	≻𝑤	≻𝑤	×
7	≻𝑤		×
8	≻	×	≻𝑤	≻
9	≻𝑤		≻
10	≻	≻𝑤	≻𝑤	×
11	≻𝑤		×
12	×	≻𝑤	≻𝑤	≻
13	≻𝑤		×
14	×	×	≻𝑤	×
15	≻𝑤		×
16	×	×	×	×
17	×		×
18	×	≻𝑤	≻𝑤	×
19	×		×
20	×	≻𝑤	≻𝑤	×
21	≻𝑤		≻
22	×	≻𝑤	≻𝑤	×
23	×		×
24	≻	≻𝑤	≻𝑤	≻
25	≻𝑤		×
26	×	≻𝑤	≻𝑤	×
27	≻𝑤		≻
28	×	≻𝑤	≻𝑤	×
29	×		≻
30	≻	≻𝑤	≻𝑤	×
31	≻𝑤		×
32	≻	≻𝑤	≻𝑤	×
33	≻𝑤		×
34	×	×	≻𝑤	×
35	≻𝑤		×
36	≻	≻𝑤	≻𝑤	×
37	≻𝑤		×
38	×	≻𝑤	≻𝑤	≻
39	≻𝑤		×
40	≻	≻𝑤	≻𝑤	≻
41	≻𝑤		×
42	×	×	≻𝑤	×
43	≻𝑤		×
44	×	≻𝑤	≻𝑤	×
45	≻𝑤		×
46	×	≻𝑤	≻𝑤	×
47	≻𝑤		≻
48	≻	≻𝑤	≻𝑤	×
49	≻𝑤		×
50	×		𝑅