Development and Evaluation of an Audio System Trainer

Guided by the desire to contribute a little in the development of an innovative instructional technology for the teaching of electronics, this researcher constructed a prototype audio system trainer. Electronics teachers will find the instructional tool convenient and practical to use for visual instruction, laboratory activities and troubleshooting works in electronics. To support the functionality of the audio system trainer, the researcher developed a supplementary laboratory manual. The project whose production costs totaled fourteen thousand two hundred seventy two pesos and fifty centavos (Php 14,272.50) was finished in two weeks’ time. However, this amount is small if compared to the prices of commercially available instructional device developed for the same purpose. The completed project and the supplementary laboratory exercises were evaluated by selected technical faculty members of the College of Industrial Technology of the Bulacan State University, Bulacan Polytechnic College, University of Rizal System, and Lyceum of the Philippines University. Sampling technique as used in the study is both purposive and incidental. The prototype audio system trainer received an over-all mean rating of 4.62 which means that the project is highly acceptable on a set of criteria which includes – physical features, cost, function / operations, and durability. Furthermore, the supplementary laboratory exercises obtained a mean of 4.70 which could be interpreted that the respondents strongly agree on the validity of the manual.

On the Metric Dimension and Minimal Doubly Resolving Sets of Families of Graphs

Let G = (V (G);E(G)) be a connected graph. The distance between two vertices u; v 2 V (G) is the length of shortest path between them and is denoted by d(u; v). A vertex x is said to resolve a pair of vertices u; v 2 V (G) if d(u; x) 6= d(v; x). For an ordered subset, B = fb1; b2; : : : ; bng of vertices of G, the n-tuple r(vjB) = (d(v; b1); d(v; b2); : : : ; d(v; bn)) is called representation of vertex v with respect to B or vector of metric coordinates of v with respect to B. The set B is called a resolving set of G if r(ujB) 6= r(vjB) for every pair of vertices u; v 2 V (G), i.e., the representation of each vertex with respect to B is unique. The resolving set with minimum cardinality is called metric basis of G. This minimum cardinality is called metric dimension and is denoted by _(G). Notice that the i-th coordinate in r(vjB) is 0 if and only if v = bi. Thus in order to show that B is a resolving set of G, it su_ces to verify that r(ujB) 6= r(vjB) for every pair of distinct vertices u; v 2 V (G) n B. Let G be a graph of order at least 2. Two vertices x; y 2 V (G) are said to doubly resolve the vertices u; v of G if d(u; x) ? d(u; y) 6= d(v; x) ? d(v; y): A subset D _ V (G) is called a doubly resolving set of G if every two distinct vertices of G are doubly resolved by some two vertices in D, i.e., all coordinates of the vector r(ujD)?r(vjD) can not be same for every pair of distinct vertices u; v 2 V (G). The minimal doubly resolving set problem is to _nd a doubly resolving set of G with the minimum cardinality. The cardinality of minimal doubly resolving set of G is denoted by(G). We have _(G) _(G) always. Therefore these sets can contribute in finding upper bounds on the metric dimension of graphs. In this thesis, we have investigated the minimal doubly resolving set problem for necklace graph, circulant graph, antiprism graph and M obius ladders. Also, in last part of thesis, the metric dimension problem has been investigated for kayak paddle graph and cycles with chord.