CONTINUOUS ON RAYS SOLUTIONS OF A GOŁA̧B–SCHINZEL TYPE EQUATION

We show that if the pair $(f,g)$ of functions mapping a linear space $X$ over the field $\mathbb{K}=\mathbb{R}\text{ or }\mathbb{C}$ into $\mathbb{K}$ satisfies the composite equation $$\begin{eqnarray}f(x+g(x)y)=f(x)f(y)\quad \text{for }x,y\in X\end{eqnarray}$$ and $f$ is nonconstant, then the continuity on rays of $f$ implies the same property for $g$. Applying this result, we determine the solutions of the equation.

Kozak's Variable-Exponent Taper Equation Regionalized for White Spruce in Alberta

Kozak's (1988) variable-exponent taper equation was estimated for white spruce by ecoregion of Alberta to reflect stem form variability among different ecoregions. Inspection of fit statistics and residual plots showed that the taper equation fitted the data quite well. Since the relative height constraint p had little impact on the overall performance of the taper equation, the optimum p value was estimated as a part of the nonlinear least squares procedures. Regional differences of the taper equation were examined and tested using the nonlinear extra sum of squares method. Ecoregions of similar taper relationships were combined to provide a composite equation. Performance of the taper equation in predicting diameter inside bark, total volume, and merchantable height was evaluated. Results indicated that the biases in predictions were small, both across different parts of the stem and for various tree sizes. West. J. Appl. For. 15(2):75-85.

Hydraulic Resistance of Plants. I. Constant or Variable?

Two popular equations for describing the flow of water through plants are discussed and criticized. They are the Ohm's Law analogue, in which the flow rate is assumed to be proportional to the difference in water potential across the plant, between leaf and soil; and an equation, used widely to describe transmembrane fluxes of single cells, in which the flow rate is assumed to be the sum of two components, one proportional to the difference in hydrostatic pressure, the other proportional to the difference in osmotic pressure, between specified points in the plant. The well-established theoretical inadequacies of the Ohm's Law analogue are reviewed, especially the poorly recognized fact that it is of little use if substantial amounts of solutes are flowing with the water. The other, composite, equation does not have the same grave theoretical inadequacies, but its failure to agree with published data is highlighted.

Development of a Reduced Set of Composite Equations for Three Predictors

5 primary and 3 reserve equations were developed to span the quadrant containing all possible equations for 3 predictors, with the restriction that no weights be negative. Any empirical equation ever likely to be encountered operationally should be represented adequately by at least one of the primary theoretical composite equations with no more than .01 loss in predictive validity. The theoretical equations were positioned to achieve this result. It is not likely that additional equations will be needed; if so, the equations should be selected from the set of reserve equations to provide the most efficient coverage. The theoretical system was applied to 47 prediction equations for different colleges using SAT-V, SAT-M, and High School Rank-in-class as predictors and college performance as a criterion. The primary composite equations were selected as the appropriate composite equations for all of the original prediction equations except one. For that prediction equation, the appropriate composite equation is one of the reserve theoretical composite equations.

A PRELIMINARY STUDY OF AERIAL VOLUME TABLE CONSTRUCTION FOR LODGEPOLE PINE IN WEST-CENTRAL ALBERTA

Some aspects of photo-measurement and aerial volume table construction for the pure lodgepole pine type are dealt with. Tables giving gross stand volume in cubic feet are shown with stand height as the independent variable. Crown closure was not significantly correlated with gross stand volume in this study; probably because only fully-stocked stands were surveyed.The 70-plot sample was divided into three strata composed of distinct parent materials. The data from stands on one stratum, the lacustrine deposits which are mapped as the Caroline Silt Loam soil series, were developed into a regression equation which differed from the composite equation (for all parent materials).This study indicated that trained photogrammetrists can consistently measure tree heights to 8-foot height classes with a uniformly small standard error. It also indicates that a larger sample may permit the development of different aerial volume tables for different parent materials or soil types.