Subsets of finite groups exhibiting additive regularity

In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In particular, we show that any sum set must exhibit higher-order regularity and that an abelian sum set is necessarily a reversible difference set. We next develop several general construction techniques under the hypothesis that the over-riding group contains a normal subgroup of order 2. Finally, by exploiting properties of dihedral groups and Frobenius groups, several infinite classes of sum sets and partial sum sets are introduced

The Implications of the Piagetian Stages to Readiness for Baptism

The Problem. This study was designed to discover how children from Seventh-day Adventist families react to their baptism after the event. Do they feel that they were ready at the time of baptism? Do they think they understood the Bible doctrines when they were baptized? Which people and factors influenced them in their decision? Have these children reached cognitive maturity according to the Formal Operations stage in Piagetian terms, the stage at which children are accountable for their decisions. Method. A questionnaire of thirty-three questions was given to children in selected Adventist schools who had been baptized between June 1972 and June 1982. Five hundred and eighty two answered these questions. Chi Square tests were employed to analyze the data. All the answers reported on the questionnaires were related to the Piagetian stages of cognitive development and especially to the level of Formal Operations beginning at around 10-11 years according to Jean Piaget. Results. The research indicated that the peak age for baptism was 12 years in the Seventh-day Adventist church and that most of the children felt ready when they were baptized. However, as they grew older more felt that they did not understand Bible doctrines as well as when they decided to be baptized, and considered they were too young at the time of baptism. Concerning the degrees of influence in their decision for baptism the children as a whole revealed that the parents had the greatest influence, followed by their minister, adult relatives and friends, peers, and a week of spiritual emphasis. In the research a progression was found toward maturation from age 6 to 14 and above, supported in large measure by Piagetian stages of cognitive development. Conclusion. The findings of the study conducted among Seventh-day Adventist children about readiness for baptism support in part Piaget\u27s theory of cognitive development. When they wait until the Formal Operations stage, young people are aware of the importance of the decision for baptism. It is their commitment, a step forward in their spiritual life, encouraged by their environment, their family, their church and their school

Logarithmic Representability of Integers as k-Sums

A set A=A_{k,n} in [n]\cup{0} is said to be an additive k-basis if each element in {0,1,...,kn} can be written as a k-sum of elements of A in at least one way. Seeking multiple representations as k-sums, and given any function phi(n), with lim(phi(n))=infinity, we say that A is a truncated phi(n)-representative k-basis for [n] if for each j in [alpha n, (k-alpha)n] the number of ways that j can be represented as a k-sum of elements of A_{k,n} is Theta(phi(n)). In this paper, we follow tradition and focus on the case phi(n)=log n, and show that a randomly selected set in an appropriate probability space is a truncated log-representative basis with probability that tends to one as n tends to infinity. This result is a finite version of a result proved by Erdos (1956) and extended by Erdos and Tetali (1990).Comment: 18 page

Feedback Control for Average Output Systems

In this work we propose new methods for the design of economic Nonlinear Model Predictive Control (NMPC) feedback schemes for Average Output Optimal Control Problems (AOCPs). AOCPs are Optimal Control Problems (OCPs) defined on infinite time horizons with averaging performance critera as objective functionals. Such problems arise frequently for continuously operating systems such as for example power plants. Due to the infinite time horizon and the resulting intrinsic nonuniqueness of solutions, the design of appropriate NMPC schemes for AOCPs is challenging. Often, the analysis of the closed-loop behavior of economic NMPC schemes depends on dissipativity conditions on the dynamical system and the associated performance criterion, which sometimes can be hard to check. The methods we develop are based on the observation that periodic solutions exhibit excellent approximation properties for AOCPs, which is exploited by splitting the time horizon and the objective functional of the NMPC subproblems into a transient and a periodic part. For the analysis of the closed-loop behavior of the resulting controller we develop new methods that essentially work by showing that the (appropriately defined) difference of two subsequent NMPC subproblem solutions vanishes asymptotically. Complementary to many other economic NMPC schemes, this approach is not based on dissipativity assumptions on the dynamical system and the associated performance criterion but rather on assumptions on existence of periodic orbits, controllability of the dynamical system, and uniqueness of the NMPC subproblem solutions itself. As a result, we can show that the economic performance of the closed-loop system is equal to the economic performance of the optimal periodic solutions. Furthermore, the approach is extended in two directions. First, we consider the general setting of a parameter dependent dynamical system where the parameter can be subject to change during operation. This parameter change can lead to a change in the optimal periodic behavior, in particular also to a change of the optimal period, which we take into account by including the period as an optimization variable in the NMPC subproblem. Second, we show that the approach can also be applied to systems with time-dependent periodic performance criteria. All the described methods are implemented within the MATLAB NMPC toolkit MLI and are applied to a number of demanding applications. The simulation results confirm that the generated closed-loop trajectories perform economically equally well as the optimal periodic trajectories