Jensen's functional equation on the symmetric group Sn\bold{S_n}

Two natural extensions of Jensen's functional equation on the real line are the equations $f(xy)+f(xy^{-1}) = 2f(x)$ and $f(xy)+f(y^{-1}x) = 2f(x)$, where $f$ is a map from a multiplicative group $G$ into an abelian additive group $H$. In a series of papers \cite{Ng1}, \cite{Ng2}, \cite{Ng3}, C. T. Ng has solved these functional equations for the case where $G$ is a free group and the linear group $GL_n(R)$, R=\z,\r, a quadratically closed field or a finite field. He has also mentioned, without detailed proof, in the above papers and in \cite{Ng4} that when $G$ is the symmetric group $S_n$ the group of all solutions of these functional equations coincides with the group of all homomorphisms from $(S_n,\cdot)$ to $(H,+)$. The aim of this paper is to give an elementary and direct proof of this fact.Comment: 8 pages, Abstract changed, the proof of Proposition 2.1 and Lemma 2.4 changed (minor), one reference added, final version, to be published in Aequationes Mathematicae (2011

Substitution of labour and energy in agriculture and options for growth

Viewed from the technical angle, labour and energy may be considered the basic resources for production. Methods to compute the requirement of energy and labour for the production processes from raw material to the product at the farm gate are discussed. It is shown that limited amounts of added energy and labour are required to achieve the maximum possible production per hectare. The form of the functions that relate the requirement for these resources with the level of production is considered in two limiting situations. These concern the minimum amount of added labour that is required when energy is abundantly available and the minimum amount of added energy that is required when labour is abundantly available. By means of these functions and the added energy and added labour used in the actual farm situation, iso-yield functions are constructed. These functions identify the possibilities for substitution of energy and labour during the production process at various yield levels. The shape of the expansion paths indicate that there may be a considerable range of production situations where the yield per unit of added energy, the yield per unit of added labour and the yield per unit of surface increase with increasing use of added labour and energy per hectare, whereas the added energy use per unit of added labour decreases at the same time. Some resulting options for growth, with special reference to the situation in the Netherlands, are discussed

A preliminary investigation into the effects of a sex-reversal androgen, methyltestosterone, on food utilization and growth of Sarotherodon niloticus (L) fry

Three groups of Sarotherodon niloticus fry were fed for 8 weeks on diets either treated with 17- & methyltestosterone (MT), alcohol (CA), or untreated (CO). Growth rate and food utilization in the different groups were compared. Results indicate that the best growth, Feed Conversion Ratio (FCR), Protein Efficiency Ratio (PER) and Mean Growth Rate (MGR) were obtained with the MT diet. There was no significant difference (P 0.05) in growth and food utilization of the CA and CO fry, nor in the mortality rate of the 3 treatments. The androgen, methyltestosterone promotes growth and protein anabolism without producing toxic effects in S. niloticu

The effects of brood stock size on the economy of catfish (Clarias anguillaris) fry production using the hormone induced natural breeding technique

89 ripe female brooders of the catfish, Clarias anguillaris (Body wt. Range 150g-1, 200g) were induced to spawn by hormone (Ovaprim) induced natural spawning technique over a period of 10 weeks. Matching ripe males were used for pairing the females at the ratio of two males to a female. Six ranges of brood stock body weights were considered as follows; 1000g and the number of fry produced by each female brooder was scored/recorded against the corresponding body weight range. The number of fry per unit quantity of hormone and the cost of production a fry based on the current price of Ovaprim (hormon) were determined so as to ascertain most economic size range. The best and most economic size range was between 400g-599g body weight with about 20,000 fry per ml of hormone and N0.028 per fry, while the females above 1000g gave the poorest results of 9,519 fry per ml of hormone and N0.059 per fry. For optimum production of Clarias anguillaris fry and maximum return on investment female brooders of body weights ranging between 400g-599g are recommended for hormone induced natural breeding exercise

Argumentation of the River Derwent using Thirlmere

This is the Argumentation of the River Derwent using Thirlmere produced by the Cumberland River Authority in 1972. This report focuses on the augmentation of the River Derwent by a limited discharge from the Thirlmere Reservoir in short terms. A Working Party of officers from both the Authority and the Corporation was established and experimental releases of water from Thirlmere were made in dry periods in order to assess more clearly their effect upon river flows, and on the level of Bassenthwaite Lake. This report contains the Working Party's evaluations, conclusions and recommendations