Finite p-Groups in Which the Number of Subgroups of Possible Order Is Less Than or Equal to p^3

In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order p^n with a cyclic subgroup of index p2 are the same.

Error term concerning number of subgroups of group Z_(m)×Z_(n) with m^(2)+n^(2)≤x

Let Zm be the additive group of residue classes modulo m.Let s(m,n)denote the number of subgroups of the group Z_(m)×Z_(n),where m and n are arbitrary positive integers.For any x≥1,we consider the asymptotic behavior of D_(s)(x):=∑m^(2)+n^(2)≤xS(M,n)and obtain an asymptotic formula by using the elementary method.

Monitoring Saccharification Process in Brewery Industry Using Quality Control Charts

The aim of this study was to establish a control system for saccharification process using quality control charts. To achieve this goal, temperature, pH and brix were measured at 12 minutes intervals for 15 consecutive batches which took 2 hours each. The time variations for three process parameters were assessed to establish a good understanding of the saccharification process. The temperature varied between 58℃ and 62℃ while the pH decreased slowly due to oxidation, values of which varied between 5.7 and 5.0. Brix values increased linearly with time. The initial and final values of the three parameters varied from one batch to another. Of the three parameters, brix was not well represented on the quality control charts due to wide difference between initial and final values during saccharification. The final brix values varied between batches, from 10.6% to 11.6%. The control charts used in this study were X-bar and Range charts. The rules for interpreting control charts were implemented for both X-bar and R charts, results of which showed that the process was out of control, although some rules were not violated due to little number of batches studied. The values of for temperature and pH data (2.27℃ and 0.35, respectively) were lower compared to brix data (11.2%). The corresponding values of span between control limits, SPx and SPR for temperature and pH were also comparatively lower than those established from brix data. Due to larger values of for brix measurements, the corresponding control charts for brix were insensitive in identifying out-of-control points during saccharification process.