This study is primarily an attempt to discover the trends to be found in professional magazines published during the period from 1950 to 1956, inclusive, relating to classroom control and having particular reference to high school situations. It emerged from an awareness of the importance of maintaining conditions conducive to teaching and learning in the individual teacher’s classroom; as well as in other instructional centers such as the laboratory, shop, or workroom, all of Which are actually types of classrooms. It is in such situations that assignments are made, instructions given, examinations taken, and working relationships between teacher and students developed. / M.S.

General extensions of the probability model for paired comparisons, which was developed by R. A. Bradley and M. E. Terry, are considered. Four generalizations to triple comparisons are discussed. One of these models is used to develop methods of analysis of data obtained from the ranks of items compared in groups of size three. / Ph. D.

Master of Science

In the investigation the effects of excess maleic anhydride, hydrazine hydrate-solvent reagent addition time, volume of reaction mass after concentration by heating at 100 °C, mixing of reactants, and the solvent selected were studied for the heterogeneous reaction of maleic anhydride and hydrazine hydrate. A reaction time of 12 minutes was employed with the ratio of solvent to reactants held constant at 75 weight per cent throughout the tests. Solvents employed in the investigation included ethanol, methanol, isopropanol, glacial acetic acid, water, hydrachloric acid, and benzene. Atmospheric drying tests at 25 to 88 °C were employed on hydrazide slurries of free moisture content from 1.258 to 1.515 pounds of water per pound of hydrazide. Centrifuge tests at 2000 to 4700 revolutions per minute and a rotary filtration test under a 10 inch vacuum were employed on 13 weight per cent hydrazide slurries. Hydrazide filtrate evaporation tests at 100 °C were performed on samples of 18 to 1715 milliliters to determine the approximate hydrazide content in the slurries. Field applications of 0.10 to 0.22 weight per cent hydrazide solutions in water were made on "wild" varieties of briers, bermuda grass, johnson grass, milkweed, red pine, ragweed, and honey locust in the Blacksburg, Virginia, area from May to August, 1956. The yield of maleic acid hydrazide was increased from 42.6 to 67.0 per cent when the maleic anhydride excess was increased to 20 per cent in the reaction. The optimum addition time for the ethanol-hydrazine hydrate reagent to the maleic anhydride was found to be 3.8 seconds, while the optimum volume of reaction mass after concentration by beating at 100 °C was 10 to 15 milliliters for the non-agitated reactions. Agitation of the reaction mass and the solvent chosen were determined to increase the yield of the hydrazide. The optimum drying temperature and time for the drying of the hydrazide slurries were determined to be 88 °C and 75 minutes, respectively. Rotary vacuum filtration of the hydrazide slurries was determined to produce a cake free moisture of 1.33 pounds of water per pound of hydrazide as compared with 1.38 for the centrifuge test at 4700 revolutions per minute. The hydrazide content of the filtrate samples was determined to be approximately 10 to 15 per cent. Field applications on "wild" plots indicated that 40 to 80 per cent control of briers, bermuda grass, ragweed, johnson grass, and red pine could be achieved from one application of 0.10 to 0.22 weight per cent hydrazide solutions in early spring. On milkweed and honey locust growth, the spraying solution would not adhere to the leaf. A total fixed plus working capital of $1,151,740 was determined to be necessary to build a plant for the production of 242 tons of 95.5 percent pure maleic acid hydrazide per year. On this basis, a selling price of $3.00 per pound ($0.05 per gallon) would yield a 13.7 percent return as new earnings on total fixed plus working capital. / Ph. D.

no abstract provided by author / Ph. D.

The purpose of this investigation was to study the factors affecting the low-temperature carbonization of seam number one, Penn-Lee coal from southwestern Virginia, by determining the effect of pretreatment of the charge, temperature of the retorting gas, and time of retorting on the yields and properties of the products of carbonization. After an examination of the literature pertinent to the investigation, a pilot-scale carbonization unit was constructed. The construction of the unit consisted partly of the modification of an existing fluidization unit. Various auxiliary apparatus were also added to the modified unit for more satisfactory operation. A series of bench-scale tests were performed to study the effect of pretreatment conditions on the free-swelling index, the weight loss, and the volatile matter in Penn-Lee coal. Time and temperature had a significant effect upon the free-swelling index. Temperature of pretreatment had a significant effect on the weight loss during pretreatment. Time and the time by temperature effect were found to be significant with regard to the volatile matter remaining in the coal after pretreatment. Also, the decomposition point of Penn-Lee coal was found to lie between 427 and 454 °C (800 and 850 °F). Twelve pilot-scale tests were performed to study the effect of carbonizing conditions on the products of carbonization. The pretreatment time had a significant effect upon the dry tar yield, the tar viscosity, the tar specific gravity, and the twenty per cent tar distillation temperature. The time of carbonization had a significant effect upon the gas yield. None of the variables studied showed a significant effect within the range studied, upon the volatile matter in the char product, the bulk density of the char, or the calorific value of the char. / Ph. D.

This study has been concerned specifically with the problem of estimating the power spectrum associated with a random process. It has shown how the power spectral density function φ(ω) can be used to specify completely a stationary Gaussian process. Estimation of this function is therefore one of the fundamental problems in random time-series. The power spectral density function is given by φ(ω) = [2/π] ∫₀^+∞ ρ(τ)cos ωτ dτ And must be estimated from a partial realization of the process. To accomplish this, the usual procedure is to use estimated auto-covariance functions ρ̂(τ), computed from a set of observations X(t_i) from which φ(ω) is approximated by numerical integration. This gives φ̂(ω) = 1/W [ρ̂(0)+2∑_j=1^m-1 ρ̂[jπ/W]cos(ωjπ/W) + ρ̂[mπ/W]cos(ωmπ/W)] where the ρ[jπ/W]’s are estimated from ρ̂[jπ/W] = (1/n)∑_i=1ⁿX(t_i)X(t_i + jπ/W) j = 0, 1, …, m. φ̂(ω_α) is widely used in power spectral analysis and although it can be shown to be biased, the side lobes of its spectral window can be soothed in such a way that the bias is greatly reduced. The difficulty with the estimator is not so much with its bias, but rather with the considerable numerical task it creates even when digital computing equipment is available. The primary objective of this research was to devise an estimator which would simulate the bias of the classical estimator φ̂(ω_α) but which would require much less work to compute. To this end the randomizes estimator φ*(ω_α) = (1/n)_i=1ⁿ X(t_i)X(t_i + k_iΔt)G_α(k_i) was considered. Unlike φ̂(ω_α) which was constructed by systematically forming all possible lagged products X(t_i)X(t_i + k_iΔt), i=1, 2, …, n and k=0, 1, …, m , the new estimator utilizes a random subsample of lagged products. This is made possible by choosing the k_i at random. The weighting function G_α(k_i) is determined in such a way that the bias of φ*(ω_α) is the same as the bias of φ̂(ω_α). As would be expected, the sampling variance of φ*(ω_α) is larger than the variance of φ̂(ω_α), since φ*(ω_a) is based on considerably fewer points. It was discovered, however, that the variance of φ*(ω_α) was affected by the probabilities used in the selection of the k_i. Thus, the difference between the variances of the two estimators can be minimized by an appropriate choice of the probabilities P(j). It was shown also that by selecting the integers j = 0, 1, …, m with probabilities P(j) = (√(f(j))) / ((∑_j=1^m)(√(f(j)), where f(j) = (1/n)[ρ²(0) + ρ²(jΔt)]ρ²(j)G²_α²(j), that the variance of φ*(ω_α) is minimized. For the special case, φ(ω) = λ and the point W/2, it was shown that sampling with equal probabilities is about half as efficient as with probabilities. Finally, some of the areas in which research has been carried out using power spectral analysis were considered. In particular, a problem from the field of aeronautical engineering research was used to demonstrate how the randomized estimator φ*(ω_α) would be calculated from real data. Using 900 observations on the pitching velocity of an aircraft, the power spectrum was estimated at ten points. The new estimator proved very tractable and it is felt that the loss of precision due to sampling will be more than offset by the economy and ease with which it produces estimates. This will be especially true when the need is for quick pilot estimates of spectra to be used in preliminary studies, as guidance for future research. / Ph. D.

The bending of thin flat plates has occupied the interests of mechanicians and applied mathematicians since J. L. Lagrange discovered the differential equation characterizing the behavior of such structural members. One particular phase of investigation in this field concerns itself with the solution of the differential equation subject to given boundary conditions. Indeed, it may be safely stated that the bulk of the literature on the subject of flat plates is concerned with the solution of problems involving the specification of the transverse loading on the plate and the conditions at the boundary of the plate. Various mathematical techniques are available for the solution of such problems. Among these, the most prominent are, a.) the method of series, b) the method of singularities, and c) the complex variable techniques. A survey of the literature in this area has revealed a paucity of solutions of certain types of problems; notably, those problems in which boundary conditions are mixed a.long a portion of the edge of the plate which ha.s a continuously turning tangent. By mixed boundary conditions, we mean a. change in condition from prescription of bending moment and vertical shear to assignment of slope and deflection along a portion of the edge which has a continuously turning tangent. In the first section of this thesis, a number of problems are considered for the half-plane. The attendant boundary conditions considered a.re combinations of clamping and simple support. The second portion consists of a number of problems associated with the quarter-plane. Solutions for these problems are obtained by utilizing the method of images in conjunction with the solutions presented in the first section. After this, we examine some problems connected with the circular plate. In particular, a numerical solution is given for a uniformly loaded circular plate simply-supported over half of its boundary and clamped over the remaining portion. The last chapter is a brief discussion of plates in the form of rectangles. Here, a closed solution is presented for the bending moments in terms of Weierstrassian elliptic functions. Another numerical example is included for a uniformly loaded plate clamped over a portion of one edge and simply- supported over the remainder of its boundary. / Doctor of Philosophy

Linear discriminant analysis is the classification of an individual as having arisen from one or the other of two populations on the basis of a scalar linear function of measurements of the individual. This paper is a population and large sample study of linear discriminant analysis. The population study is carried out on three levels: (1.1) (a) with loss functions and prior probabilities, (b) without loss functions but with prior probabilities, (c) with neither. The first level leads to consideration of risks which may be split into two components, one for each type of misclassification, i.e. classification of an individual into population I given it arose from population II, and classification of it into II given it arose from I. Similarly, the second level leads to consideration of expected errors and the third level leads to consideration of conditional probabilities of misclassification, both again which may be divided into the same two components. At each level the "optimum" discriminator should jointly minimize the two probability components. These quantities are all positive for all hyperplanes. Either one or any pair may be made equal to zero by classifying all individuals of a sample into the appropriate population; but this maximizes the other one. Consequently, joint minmization must be obtained by some compromise, e.g. by selecting a single criterion to be minimized. Two types of criteria for judging discriminators are considered at each level: (1.4) (i) Total risk (a) (1.5) Total expected errors (b) (1.6) . Sum of conditional probabilities of misclassification (c) (1.7) (ii) Larger risk (a) (1.8) Larger expected error (b) (1.9) Larger conditional probability of misclassification (c). These criteria are not particularly new, but have not been applied to linear discrimination and not been all used jointly. If A is a k-dimensional row vector of direction numbers, X a k-dimensional row vector of variables, and a constant, a linear discriminator is (1.10) AX' = o, which also represents a hyperplane in k-space. An individual is classified as being from one or the other population on the basis of its position relative to the hyperplane. The parameters A and c ot (1.10) were investigated to find those sets of values which minimize each of the two criteria at various levels. Exact results were found for A under some circumstances and approximate results in others. At the levels (b) and (c), when exact results were obtained, they were the same for both criteria and were independent or c. Investigation of the c’s showed the c’s to be exact functions of A and the parameters and yielded one c for each criterion. At level (c), the c's for criteria (i) and (ii), c(min) and c(σ), respectively, were compared to c(m), a population analog of the c suggested by other authors, to discover the conditions under which it was better (i.e. having lesser criteria) than both c(min), c(σ) on criterion (ii), (i) respectively. In the large sample study, variances and covariances were found (in many cases approximately) for all estimates of the parameters entering into the conditional probabilities of misclassification (level (c)). Extension of results to level (b) and to special cases of level (a) were given. From these variances and covariances were derived the expectations of these probabilities for both criteria, at level (c), and comparisons were made where feasible. Results were tabulated. / Ph. D.

no abstract provided by author / M.S.