Non-singular circulant graphs and digraphs

We give necessary and sufficient conditions for a few classes of known circulant graphs and/or digraphs to be singular. The above graph classes are generalized to $(r,s,t)$-digraphs for non-negative integers $r,s$ and $t$, and the digraph $C_n^{i,j,k,l}$, with certain restrictions. We also obtain a necessary and sufficient condition for the digraphs $C_n^{i,j,k,l}$ to be singular. Some necessary conditions are given under which the $(r,s,t)$-digraphs are singular.Comment: 12 page

Representation of Cyclotomic Fields and Their Subfields

Let \K be a finite extension of a characteristic zero field \F. We say that the pair of $n\times n$ matrices $(A,B)$ over \F represents \K if \K \cong \F[A]/ where \F[A] denotes the smallest subalgebra of M_n(\F) containing $A$ and $$ is an ideal in \F[A] generated by $B$. In particular, $A$ is said to represent the field \K if there exists an irreducible polynomial q(x)\in \F[x] which divides the minimal polynomial of $A$ and \K \cong \F[A]/. In this paper, we identify the smallest circulant-matrix representation for any subfield of a cyclotomic field. Furthermore, if $p$ is any prime and \K is a subfield of the $p$-th cyclotomic field, then we obtain a zero-one circulant matrix $A$ of size $p\times p$ such that (A,\J) represents \K, where \J is the matrix with all entries 1. In case, the integer $n$ has at most two distinct prime factors, we find the smallest 0-1 companion-matrix that represents the $n$-th cyclotomic field. We also find bounds on the size of such companion matrices when $n$ has more than two prime factors.Comment: 17 page

UTILIZING EXPECTED REVENUE IN SELECTING OPTIMAL MARKETING ALTERNATIVES FOR FIXED RESOURCE COW/CALF OPERATORS IN THE TEXAS PANHANDLE

Beef cattle enterprises for the 91% of Texas producers with less than 100 cows represent secondary or supplemental enterprises. In most cases the operator is depending on surplus or discretional labor and the alternate use of the physical resources is limited due to topography or environmental factors. These characteristics transform the decision making process from a variable resource allocation problem to a fixed resource problem. Monthly average prices for 300-900 lb stocker steers, divided into 50 lb intervals, are analyzed for three market locations for the period 1992-1999. Consistent price roll back is observed as the price per cwt decreases as the weight increases. However, as the reduction in price is more than offset by the increase in value due to additional weight, the expected revenue per animal increases as weight increases.Livestock Production/Industries,

Zn(II), Cu(II), Ni(II) & Co(II) Complexes of N-(Salicylidene)-N'-[2-(salicylaldimine)- benzoyl]hydrazine, the Schiff Base Derived from o-Aminobenzoylhydrazine & Salicylaldehyde

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Effects of Macronutrients on Seed Quality and Profitability Analysis of Sunflower Production in Northwest Pakistan

Potassium (K) and phosphorus (P) application to sunflower (Helianthus annus L.) reduced cost of production, increased seed oil and protein concentration, yield and net returns on the K and P deficient soils in Northwest Pakistan. Field experiment was conducted at the New Developmental Research Farm of NWFP (North West Frontier Province) Agricultural University Peshawar, Pakistan during summer 2006. The experiment was carried out in a randomized complete block design with split plot arrangements using three replications. Six levels of K (0, 25, 50, 75, 100 and 125 kg ha-1) were kept in main plots while four levels of P (0, 45, 90 and 135 kg ha-1) were kept in sub-plots. Increase in K and P levels enhanced seed oil concentration, on the other hand, increase in P level increased seed protein concentration but increase in K level decreased seed protein concentration. Both oil and protein yields per unit area increased significantly with increase in K and P levels. The increase in oil and protein yields of sunflower was mainly attributed to the improvement in yield components and significant increase in seed yield. The highest net returns of Rs. 25268 (grain yield basis) and Rs. 31274 (oil yield basis) was obtained from those plots that received a combination of 100 kg K + 45 kg P ha-1. Application of 100 kg K ha-1 + 45 kg P ha-1 was therefore, recommended for profitable sunflower production in the agro-climatic conditions in Northwest Pakistan.sunflower, potassium, phosphorus, oil, protein, economic analysis, Northwest Pakistan, Farm Management, International Development, International Relations/Trade, Production Economics, Productivity Analysis,