Conditions under which constant block size implies constant replication number and vice versa are established for balanced ternary designs. This note is motivated by a note of almost the identical title of W. D. Wallis [2]. In a balanced binary design with constant block size, the replication number must be constant, but the converse is not true. Wallis [2] proved the following partial converse...

An-balanced pair in a partially ordered set P = (X; <) is a pair (x; y) of elements of X such that the proportion of linear extensions of P with x below y lies between and 1 ?. The 1=3 ? 2=3 Conjecture states that, in every nite partial order P , not a chain, there is a 1 3-balanced pair. This was rst conjectured in a 1968 paper of Kislitsyn, and remains unsolved. We survey progress towards a r...

The research presented in this thesis is on the application of the injection-locked oscillator technique to microwave balanced oscillators. The balanced oscillator design is primarily analysed using the extended resonance technique. A transmission line is connected between the two active devices, so that the active device resonate each other. The electrical length of the transmission line is al...

The Andrews-Curtis conjecture states that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of the elementary Nielsen transformations and conjugations. In this paper we describe all balanced presentations of the trivial group on two generators and with the total length of relators ≤ 12. We show that all these presentations satisfy the Andrews-Curt...

A balanced scorecard (BSC) is a management decision tool intended to be the corporate performance measurement. It also can play an important role in transforming an organization’s mission and strategy into a balanced set of integrated performance measures. Assigning suitable weight to each level of balanced scorecard is crucial to conduct performance evaluation effectively. In this research a c...

A balanced part ternary design (BPTD) is a balanced ternary design (BTD) with a specified number of blocks, say b2, each having repeated elements. We prove some necessary conditions on b2 and show the existence of some particular BPTDs. We also give constructions of infinite families of BPTDs with b1 = 0, including families of ternary t-designs with t ≥ 3.