Discrete Mathematics

Let L be a finite lattice. A sublattice K of a lattice L is said to be convex if a,b K; c L, a c b imply that c K. Let CS(L) be the set of convex sublattices of L including the empty set. Then CS(L) partially ordered by inclusion relation, forms an atomic algebric lattice. Let P be a finite graded poset. A finite graded poset P is said to be Eulerian if its M bius function assumes the value (x; y) = (-1)l(x;y) for all x y in P, where l(x, y) = (y)- (x) and is the rank function on P. In this paper, we prove that the lattice of convex sublattices of a Boolean algebra Bn, of rank n, CS(Bn) with respect to the set inclusion relation is a dual simplicial Eulerian lattice. But the structure of the lattice of convex sublattices of a non- Boolean Eulerian lattice with respect to the set inclusion relation is not yet clear.

The main objective of the present paper is to establish certain new formulae in the aspects of Hypergeometric function.

We prove that the upper domination number of a graph does not increase when a vertex is removed from the graph. Moreover, we consider well-dominated graphs and their upper domination number. We also define approximately well-dominated graphs.

This paper is aimed to discuss cordial graphs in the context of middle graph of a graph. We present here cordial labeling for the middle graphs of path, crown, star and tadpole.

The harmonic mean labeling is a variation of arithmetic mean labeling. We investigate harmonic mean labeling for various graphs resulted from the duplication of graph elements.

The P-Value models were designed by the researcher to teach Place Values in mathematics. The P-Value models are tools which is methodology in nature. It depicts yet another / alternative practical way of teaching Place Values in Mathematics to the usual / known method (Appendix A; Appendix B) of teaching Place Value (Kumassah, 2012; Price 2002; Price, 2001). The mathematical formulas of the P-Value models shows the step (s) / movement (s) involved in arriving at a specific number (s) / figure (s) and also the total / sum / overall number (s) / figure (s) in each P-Value models. The mathematical formulas of the P-Value models can be employed in teaching and used at the industry level. The researcher is of the view that the use of the P-Value models may help teachers teach Place Value effectively and students learn meaningfully of Place Values in mathematics. The P-Value model solution-boxes serve as a calculator to the actual calculators. It gives the same answers as that of the actual calculators and also shows and reveals the hidden mathematical operations of the actual calculator. Instead of a student manipulating the hidden operations of an actual calculator, with the P-Value model solution-box he/she practically sees and manipulates that hidden operations of a calculator.

Abstract. This paper is aimed at finding an optimal estimator of the normal population mean when the coefficient of variation. The paper proposes an “Efficient Minimum Mean Square Error (MMSE) Estimation Approach [Using Two ‘Sample Standard Deviation’, for improving the well-known Searls’ Usual MMSE Mean Estimator (1964)] for Normal Population with Known Coefficient of Variation”. The ‘Relative Efficiencies [as compared to the usual unbiased sample-mean estimator ] estimator, per the proposed approach, has no simple algebraic form, and hence is not amenable to an analytical study determining its relative gainfulness, as compared to the usual unbiased sample mean estimator. Nevertheless, we examine the relative efficiency of our estimator with respect to the usual unbiased estimator , using an illustrative simulation study with high replication. MATLAB R2013a is used in programming this illustrative “Simulated Empirical Numerical Study”.

In this paper we introduce the variant of domination called s- domination for semigraphs. The concept of s- domination is stronger than the concept of domination in semigraphs. We prove that the s- domination number of a semigraphs does not increase when a vertex is removed in a semigraphs. This is unlike in the case of domination. We also consider maximal independent sets in semigraphs and observe that they are exactly independent s- dominating sets in semigraphs. We prove some related results.

In recent years it has been acknowledged that to significantly improve biological system performance and productivity, control, including optimal control, should be implemented online. The dynamics of biological processes are highly non-linear and key variables are difficult to measure. As an aid to contributing solutions to the twin difficulties of complex dynamics and measurement difficulty a relatively simple mathematical model has been developed for the purpose of evaluating optimal controllers. In this paper a simulation of fed-batch fermentation is developed which includes the no-linear dynamics of the process using single-term Haar wavelet series (STHW) method.

In this paper we prove that the graphs obtained by duplication of an edge in , mutual duplication of pair of edges and mutual duplication of pair of vertices between two copies of cycle admit product cordial labeling. Moreover let and be two graphs such that their order and/or size differ by at most 1 then we prove that the new graph obtained by joining and by a path of arbitrary length is a product cordial graph.