Boole's Method I. A Modern Version

A rigorous, modern version of Boole's algebra of logic is presented, based partly on the 1890s treatment of Ernst Schroder

A study of large, medium and small scale structures in the topside ionosphere

Alouette and ISIS data were studied for large, medium, and small scale structures in the ionosphere. Correlation was also sought with measurements by other satellites, such as the Atmosphere Explorer C and E and the Dynamic Explorer 2 satellites, of both neutrals and ionization, and with measurements by ground facilities, such as the incoherent scatter radars. Large scale coherent wavelike structures were found from ISIS 2 electron density contours from above the F peak to nearly the satellite altitude. Such structures were also found to correlate with the observation by AE-C below the F peak during a conjunction of the two satellites. Vertical wavefronts found in the upper F region suggest the dominance of diffusion along field lines as well. Also discovered were multiple, evenly spaced field-aligned ducts in the F region that, at low latitudes, extended to the other hemisphere and were in the form of field-aligned sheets in the east-west direction. Low latitude heating events were discovered that could serve as sources for waves in the ionosphere

Scaling behavior in economics: II. Modeling of company growth

In the preceding paper we presented empirical results describing the growth of publicly-traded United States manufacturing firms within the years 1974--1993. Our results suggest that the data can be described by a scaling approach. Here, we propose models that may lead to some insight into these phenomena. First, we study a model in which the growth rate of a company is affected by a tendency to retain an ``optimal'' size. That model leads to an exponential distribution of the logarithm of the growth rate in agreement with the empirical results. Then, we study a hierarchical tree-like model of a company that enables us to relate the two parameters of the model to the exponent $\beta$, which describes the dependence of the standard deviation of the distribution of growth rates on size. We find that $\beta = -\ln \Pi / \ln z$, where $z$ defines the mean branching ratio of the hierarchical tree and $\Pi$ is the probability that the lower levels follow the policy of higher levels in the hierarchy. We also study the distribution of growth rates of this hierarchical model. We find that the distribution is consistent with the exponential form found empirically.Comment: 19 pages LateX, RevTeX 3, 6 figures, to appear J. Phys. I France (April 1997

Scaling behavior in economics: I. Empirical results for company growth

We address the question of the growth of firm size. To this end, we analyze the Compustat data base comprising all publicly-traded United States manufacturing firms within the years 1974-1993. We find that the distribution of firm sizes remains stable for the 20 years we study, i.e., the mean value and standard deviation remain approximately constant. We study the distribution of sizes of the ``new'' companies in each year and find it to be well approximated by a log-normal. We find (i) the distribution of the logarithm of the growth rates, for a fixed growth period of one year, and for companies with approximately the same size $S$ displays an exponential form, and (ii) the fluctuations in the growth rates -- measured by the width of this distribution $\sigma_1$ -- scale as a power law with $S$, $\sigma_1\sim S^{-\beta}$. We find that the exponent $\beta$ takes the same value, within the error bars, for several measures of the size of a company. In particular, we obtain: $\beta=0.20\pm0.03$ for sales, $\beta=0.18\pm0.03$ for number of employees, $\beta=0.18\pm0.03$ for assets, $\beta=0.18\pm0.03$ for cost of goods sold, and $\beta=0.20\pm0.03$ for property, plant, & equipment.Comment: 16 pages LateX, RevTeX 3, 10 figures, to appear J. Phys. I France (April 1997

Effect of nonlinear filters on detrended fluctuation analysis

We investigate how various linear and nonlinear transformations affect the scaling properties of a signal, using the detrended fluctuation analysis (DFA). Specifically, we study the effect of three types of transforms: linear, nonlinear polynomial and logarithmic filters. We compare the scaling properties of signals before and after the transform. We find that linear filters do not change the correlation properties, while the effect of nonlinear polynomial and logarithmic filters strongly depends on (a) the strength of correlations in the original signal, (b) the power of the polynomial filter and (c) the offset in the logarithmic filter. We further investigate the correlation properties of three analytic functions: exponential, logarithmic, and power-law. While these three functions have in general different correlation properties, we find that there is a broad range of variable values, common for all three functions, where they exhibit identical scaling behavior. We further note that the scaling behavior of a class of other functions can be reduced to these three typical cases. We systematically test the performance of the DFA method in accurately estimating long-range power-law correlations in the output signals for different parameter values in the three types of filters, and the three analytic functions we consider.Comment: 12 pages, 7 figure

Food Demand Projections Using Full Demand Systems

Disaggregated food demand projections for developing countries, although essential for improved development planning and effective policy making, are rare. Moreover food demand projection models are usually based on aggregated, national-level data. In this article, under conditions of weak separability and multistage budgeting decisions, a structural model capable fo generating regional-level food demand projections for a disaggregated set of commodities is developed and estimated using data from an Indonesian expenditure survey. Regional food demand projections in Indonesia obtain under a scenario assuming constant real prices are then combines into national-level estimates.