Divisibility graph for symmetric and alternating groups

Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides $a$. Let $X=cs~{G}$ be the set of conjugacy class sizes of a group $G$. In this case, we denote $D(cs~{G})$ by $D(G)$. In this paper we will find the number of connected components of $D(G)$ where $G$ is the symmetric group $S_n$ or is the alternating group $A_n$

On divisibility graph for simple Zassenhaus groups

The divisibility graph $D(G)$ for a finite group $G$ is a graph with vertex set $cs~(G)\setminus\{1\}$ where $cs~(G)$ is the set of conjugacy class sizes of $G$. Two vertices $a$ and $b$ are adjacent whenever $a$ divides $b$ or $b$ divides $a$. In this paper we will find $D(G)$ where $G$ is a simple Zassenhaus group

The Effects Of Marketing Factors On Consumers’ Intention To Purchase Products Under Volume Discount Through Perceived Benefits

Kajian tentang kaedah promosi jualan skim pengurangan harga mengikut jumlah kurang mendapat perhatian. Objektif utama tesis ini untuk mengenalpasti faktor- faktor pemasaran yang mempengaruhi keinginan pengguna untuk membeli barangan skim pengurangan harga mengikut jumlah. Jangkaan manfaat turut diambil kira sebagai perantara di antara hubungan faktor-faktor pemasaran dan keinginan membeli barangan skim pengurangan harga mengikut jumlah. Volume discount is a common type of sales promotion that has received less attention in the literature. The primary objective of this thesis is to find marketing factors influencing the intention to purchase products under volume discount. It is also of interest to determine the mediating effects of the perceived benefits on the relationship between marketing factors and intention to purchase products under volume discount

Risk Analysis - An Economic Comparison of Oil and Coal Power Plants

The demand for electric energy increases every year. However, due to recent changes in the U.S. energy supplies, a growing gas shortage forced suppliers to curtail deliveries of natural gas for power generation. Many utilities anticipating supply problems switched to burning more costly light distillate oil. Unfortunately the Arab boycott of 1973 and the following price increases for oil forced again utilities to seek a cheaper source of fuel, namely coal, as a substitute for oil. Even though the U.S. has abundant supply in coal, the use of coal in power generation was limited in the past because of a higher capital cost associated with installing air pollution control devices. Therefore, current utilities primary concerns are does the lower fuel price of the coal power plant really outweigh its disadvantage of higher construction costs as compared to the oil-burning power plant? . Thus, the purpose of this paper is to evaluate the economic preference of the coal burning power plant compared to the oil-burning power plant in suppling base load power. An extensive analytical model accounting for the effects of escalating fuel prices was examined and a computer simulation model was developed to handle risk associated with various input parameters using the SLAM as a simulation language

Assessing the Economic Feasibility of Synthetic Natural Gas Under Conditions of Uncertainty

The science of synthetic fuel production began in the seventeenth century. However, large-scale production of synthetic fuels started in the early 1900\u27s and, for several decades, gas manufactured from coal significantly contributed to the U.S. economy. The production of synthetic fuels declined due to increases in the price of coal and discoveries of predominantly methane natural gas. Today, an extensive network of pipelines is used to transmit and distribute natural gas for industrial and residential applications. The decline of natural gas reserves in the United States, in conjunction with the availability of very large coal reserves, has provided the incentive for development of coal gasification plants. Synthetic fuels are expected to contribute significantly to the supply of energy before the end of this century, and coal will be the primary source for production of these fuels. By many accounts, difficulties in raising the high amount of initial capital and future uncertainties with regard to fuel and operating costs have made development of synthetic fuels economically infeasible. However, as the prices of oil and natural gas increase, synthetic fuels production becomes a more attractive alternative. The purpose of this study is to evaluate the economics of synthetic natural gas with the current state of technology and to determine its future role as prices of oil and gas increase. In this report, a general methodology of production of synthetic natural gas is explained. For the economic analysis, the Lurgi Model was selected because it has been the most common model used for commercial production of high BTU gases. An extensive analytical model is described in which inflated capital, fuel, and operating and maintenance costs were accounted for and the equivalent annual cost of cash flows over the project life was calculated. The risk analysis was accomplished by applying Monte Carlo techniques through a simulation model which handles risks associated with various input parameters. SLAM, a FORTRAN-based language, was selected as the simulation language. Based on the results, all the cost elements were evaluated and the sensitivity of the total cost to each element was examined. This study was extended to the calculation of costs associated with he generation of electricity by burning synthetic natural gas. The results were then compared to the respective costs related to oil-burning power plants. The results show that high cost of synthetic high BTU gas makes it difficult to compete with natural gas at current prices. Coal feed stocks represent a major portion of the total cost of synthetic gases. The cost of capital, which is a critical factor at the developing stage, constitutes a relatively small portion of the total cost over the plant life. A similar observation was made for operating and maintenance costs. However, the future regulations regarding pollution control could have a strong impact on this portion of the cost. For power generations, oil was found to be far more economical than using synthetic natural gas. The computer simulation also revealed that the total cost of each alternative is very sensitive to this fuel cost. The conclusion of this study points to the fuel costs as the dominant factor in the choice of fuel alternatives in the future

The Divisibility Graph of finite groups of Lie Type

The Divisibility Graph of a finite group $G$ has vertex set the set of conjugacy class lengths of non-central elements in $G$ and two vertices are connected by an edge if one divides the other. We determine the connected components of the Divisibility Graph of the finite groups of Lie type in odd characteristic

Quotient graphs for power graphs

In a previous paper of the first author a procedure was developed for counting the components of a graph through the knowledge of the components of its quotient graphs. We apply here that procedure to the proper power graph $\mathcal{P}_0(G)$ of a finite group $G$, finding a formula for the number $c(\mathcal{P}_0(G))$ of its components which is particularly illuminative when $G\leq S_n$ is a fusion controlled permutation group. We make use of the proper quotient power graph $\widetilde{\mathcal{P}}_0(G)$, the proper order graph $\mathcal{O}_0(G)$ and the proper type graph $\mathcal{T}_0(G)$. We show that all those graphs are quotient of $\mathcal{P}_0(G)$ and demonstrate a strong link between them dealing with $G=S_n$. We find simultaneously $c(\mathcal{P}_0(S_n))$ as well as the number of components of $\widetilde{\mathcal{P}}_0(S_n)$, $\mathcal{O}_0(S_n)$ and $\mathcal{T}_0(S_n)$

ON SOME EQUIVALENCE RELATION ON NON-ABELIAN \CA-GROUPS

A non-abelian group $G$ is called a \CA-group (\CC-group) if $C_G(x)$ is abelian(cyclic) for all $x\in G\setminus Z(G)$. We say $x\sim y$ if and only if $C_G(x)=C_G(y)$.We denote the equivalence class including $x$ by$[x]_{\sim}$. In this paper, we prove thatif $G$ is a \CA-group and $[x]_{\sim}=xZ(G)$, for all $x\in G$, then $2^{r-1}\leq|G'|\leq 2^{r\choose 2}$.where $\frac {|G|}{|Z(G)|}=2^{r}, 2\leq r$ and characterize all groups whose $[x]_{\sim}=xZ(G)$for all $x\in G$ and $|G|\leq 100$. Also, we will show that if $G$ is a \CC-group and $[x]_{\sim}=xZ(G)$,for all $x \in G$, then $G\cong C_m\times Q_8$ where $C_m$ is a cyclic group of odd order $m$ andif $G$ is a \CC-group and $[x]_{\sim}=x^G$, for all $x\in G\setminus Z(G)$, then $G\cong Q_8$