The Hausdorff dimension of the visible sets of planar continua

For a compact set $\Gamma\subset\Bbb{R}^2$ and a point $x$, we define the visible part of $\Gamma$ from $x$ to be the set $\Gamma_x = \{u \in\Gamma : [x, u] \cap\Gamma = \{u\}\}.$ (Here $[x, u]$ denotes the closed line segment joining $x$ to $u$.) In this paper, we use energies to show that if $\Gamma$ is a compact connected set of Hausdorff dimension larger than one, then for (Lebesgue) almost every point $x\in\Bbb{R}^2$, the Hausdorff dimension of $\Gamma_x$ is strictly less than the Hausdorff dimension of $\Gamma$. In fact, for almost every $x$, $\dim_H(\Gamma_x)\leq \frac{1}{2}+\sqrt{\dim_H(\Gamma){-}\frac{3}{4}}.$ We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension larger than $\sigma+\frac{1}{2}+\sqrt{{\dim_H}{(\Gamma)}{-}{\frac{3}{4}}}$ for $\sigma > 0$

New Reference Galaxy Standards for HI Emission Observations

We have taken advantage of the improved baselines and higher sensitivity available with the upgraded Arecibo 305m telescope to create a new HI spectral line catalog of disk galaxies which can be used as a reference catalog for anyone interested in 21-cm spectral line work. In all 108 galaxies were observed, covering 24h of the sky at declinations between 0 < delta < 36 degrees and velocities between 0 - 25,000 km s-1. The majority of the galaxies were observed at least two times on different nights to avoid problems with RFI, baselines fluctuations, etc. Comparing our measured values with all those available in the literature show that while large individual variations may exist, the average differences between the measurements to be zero. In all we have considerable confidence in our measurements, and the resultant catalog should be extremely useful as a well defined reference catalog for anyone interested in 21-cm spectral line work.Comment: Accepted for publication by AJ. 23 pages, with 10 figures and 3 tables. Data tables, paper, etc. available online at http://www.gb.nrao.edu/~koneil/HIsurvey Replacement paper corrects one error in Table 1 and two errors in Table

Measuring Anisotropy in Planar Sets

We define and discuss a pure mathematics formulation of an approach proposed in the physics literature to analysing anisotropy of fractal sets

Factors influencing net investment decision making for a group of lower North Island sheep and beef farmers : a thesis presented in partial fulfilment of the requirements for the degree of Master of Agricultural Economics at Massey University

This study investigated the process of net investment decision-making on a group of New Zealand sheep and beef farmers. A review of previous theoretical and empirical research led to the study's objectives, namely to test that investment decision making on New Zealand farms could be incorporated in two dimensions: the determination of a desired level of capital stock and a description of the rate of adjustment of actual capital stock to the desired level. A study of net investment decision-making was chosen because net investment was seen by policy-makers in the 1970's to be an ingredient in planned growth in output. Information on net investment at the individual farmer level was not, however, available to policy-makers at the time. The study was at the individual farmer level to complement previous reserarch at the macro-level on investment in the New Zealand pastoral sector. An investment model was tested using ordinary least squares combining time-series and cross-section data. The initial specification included individual farm dummy variables to account for cross-sectional differences in net investment decision-making. Later, candidate variables hypothesised as explaining cross-section differences were included in the model. The regression results led support to the study's objective. Demand for desired capital stock was viewed as determined by Government policy measures, farm size, farmer age and the initial development state of the farm. Adjustment of actual capital stock to the desired level was viewed as determined by the level of cash at the beginning of each period and windfall gains or losses in net income in the current period. The results provide some basis for the better targeting of future policy measures to the farm sector. The study was limited by lack of a priori knowledge of inter-farm differences in the desire for capital, by the lack of a precise measurement of actual capital stock and the failure to account for interdependencies in the consumption-investment decisions that take place on farms. These limitations could provide avenues for future research

The Hausdorff dimension of the visible sets of connected compact sets

For a compact subset K of the plane and a point x, we define the visible part of K from x to be the set K_x={u\in K : [x,u]\cap K={u}}. (Here [x,u] denotes the closed line segment joining x to u.) In this paper, we use energies to show that if K is a compact connected set of Hausdorff dimension larger than one, then for (Lebesgue) almost every point x in the plane, the Hausdorff dimension of K_x is strictly less than the Hausdorff dimension of K. In fact, for almost every x, dim(K_x)\leq {1/2}+\sqrt{dim(K)-{3/4}}. We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension larger than s+{1/2}+\sqrt{dim(K)-{3/4}}, for s>0.Comment: Approximately 40 pages with 6 figure

The development of a handbook for Nashua Junior High School students

Thesis (Ed.M.)--Boston Universit