No description available.

No description available.

M.S.

M.S.

The problem considered in this dissertation is the following: let π₁ and π₂ be two bivariate populations having unknown cumulative distribution functions F₁(x₁, x₂) and F₂(x₁, x₂), respectively. Assume that F₁ and F₂ are continuous and identical except possibly in location parameters. It is desired to test the null hypothesis H₀: F₁(x₁, x₂) ≡ F₂(x₁, x₂) against the alternative H₀: F₁(x₁, x₂) ≠ F₂(x₁, x₂) It cannot be assumed that the variables x₁ and x₂ are statistically independent. Suppose there are n₁pairs of observations (x₁₁, x₂₁),..., (x_1n₁, x_2n₁) from the population π₁ and n₂ pairs of observations (x_ln+1. X_2n₁+1),..., (x_1N, x_2N) from population π₂, where N = n₁ + n₂. The x₁ᵢ (i = 1,2,..., N) are ranked according to magnitude, the largest being assigned rank 1 and the smallest assigned rank N. In a similar manner, ranks are assigned to the observations x₂ᵢ (i = 1, 2, …, N). It is assumed that there are no ties in ranks. Let u₁ᵢ and u₂ᵢ denote the ranks assigned to x₁ᵢ and x₂ᵢ if these observations belong to population π₁, and let u’₁ᵢ and u’₂ᵢ denote the ranks of the same observations if they belong to population π₂. Since the sum of the first N integers is (N(N+1))/2, it follows that Σ_k=1^n₁ u_ik + Σ_{k=n₁ + 1}^N u_ik’ = (N(N+1))/2 If the N pairs of ranks are plotted on a plane, it is likely that the n₁ points from population π₁ and the n₂ points from population π₂ will be interspersed forming a circular or elliptical pattern under the assumption that F₁(x₁, x₂) and F₂(x₁, x₂) are identical. Under the alternative hypothesis, it is likely that there will be a segregation of the points into two groups. A test statistic, S₁² is constructed to measure the extent of this segregation . The S₁²-statistic proposed here, is based on the Euclidean distance between the centroids of the ranks belonging to π₁ and π₂, in particular S₁²= (ū₁-ū₁')² + (ū₂-ū₂')² where ūᵢ = n₁⁻¹ Σ_k=1^n₁ u_ik , uᵢ’ = n₂⁻¹ Σ_{k=n₁ + 1}^N u_ik’ The first two moments of S₁² are derived under the following conditional randomization procedures keeping the ranks paired as given in the sample, n₁ pairs are selected at random (with equal probabilities) from among the N = n₁ + n₂ pairs and assigned to population π₁; the remaining n₂ pairs are assigned to population π₂. It is shown that E(S₁²) = (N²(N+1))/6n₁n₂ and σ²_S₁² = a₀₀ + a₁₁A₁₁+ a₁₂A₁₂ + a₂₁A₂₁ + a₂₂A₂₂ + a₁₁,₁₁A²₁₁ Where A_rs = Σ_k=1^N₁ u_1k^ru_2k^s are parameters depending on the sample, and the coefficients a₀₀, a₁₁, a₁₂, a₂₁, a₂₂ and a₁₁,₁₁ have been tabulated for values of n₁ and n₂ up to 20. The exact sampling distribution of S₁² is unknown However, it is sho•Nn that the distribution of (kE(S₁²))/ σ²_S₁² is approximately χ² with (2[E(S₁²)]²/ σ²_S₁² degrees of freedom. A rank analogue of Wald’s modification of Hotelling's T² is given and the first two moments obtained. Also, a multivariate extension is considered and a statistic, S₁²(k,2), constructed. The expectation and variance of S₁²(k,2) are derived. A multi-populatiun extension for the case of bivariate populations is given and the expectation is derived for a statistic, S₁²(2,p). A statistic, S₁²(k,p) is constructed for the most general case and its expectation is given. An alternative approach to the problem, also investigated, is by means of discriminant analysis. In this case simplified formulas are given for the calculation of the components of a vector which provides optimum discrimination. It is shown that this method is not a fruitful one for the construction of tests of significance pertaining to the original null hypothesis. / Doctor of Philosophy

A brief introduction to the concepts and terminology of spectral analysis and a review of the standard methods for cross-spectral estimation, based on discrete time history data, are incorporated in Chapter 1. Co-spectral and quadrature-spectral estimators which are characterized by non-negative spectral windows are developed in Chapter 2. While the spectral windows for the co-spectral estimators are non-negative for all relevant values of the assignable constants, certain restrictions on these constants are necessary to assure the non-negativity of the quadrature-spectral window. The properties of these estimators are considered in detail. In Chapter 3 randomized co-spectral and quadrature spectral estimators are presented. These estimators depend on the random selection of sets of time differences, as opposed to the systematic evaluation of all possible time differences for the standard estimators. By suitable choices of probability distributions for the time differences and of weight functions, the expectations of the randomized estimators can be made equivalent to the expectations of the standard estimators or the estimators of Chapter 2. Since the randomized estimator is much simpler to use than the standard estimator, these estimators are compared in terms of their variances, given that they have equal expectations. The choice of probability distributions to yield minimum variance, given that the expectation is specified, is considered. Extremely simple co-spectral and quadrature-spectral estimators, for the case where the coefficients of the Fourier series expansions of realizations of the processes over a finite time interval can be obtained by means of suitable analog equipment, are developed in Chapter 4. The expectations, variances and covariances of these estimators are derived. / Ph. D.

An anaerobic, cellulose- decomposing bacteria was isolated in pure culture from the inghetata of a bovine rumen. The organism was a small, curved, Gram negative rod that occurred singly or in chains. The isolated bacterium was identified as a member of the genus Butyrivibrio because of its morphology and because of its production of certain volatile fatty acids in a rumen fluid-glucose medium fermentation. The characteristic fermentation waa the production of a large amount of butyric acid and 1ome lactic acid or succinic acid, and the lack of production of propionic acid. It was found that when rumen fluid was omitted from the fermentation medium the production of butyric acid decreased markedly. The organism was found to have a relatively narrow pH tolerance for the initiation of growth, pH 6.3 to 7.2, with an optimum of pH 6.9 to 7.0. Growth stopped at about pH 6.4, and continued fermentation decreased the pH to 6.0. The organism was cultured continuously on a medium which chemically defined except that vitamin-free casein hydrolysate was included, when the hydrolysate was replaced by a mixture of pure L-isomers of amino acids similar in composition to the hydrolysate, good growth occurred. However, when each amino acid was supplied at a uniform weight concentration, much weaker growth was obtained. Evidence was obtained indicating that amino acids were inhibitory, although the reason for the mild inhibition was not disclosed. The following amino acids were found to be critical nutrients, but in some cases their status, whether essential or stimulatory, was found to be dependent on the composition of the medium: - cysteine, histidine, isoleucine. methionine, lysine, tyrosine, asparagine and leucine. A medium containing only these amino acids supported only very weak growth. Aspartic acid was found to be mildly inhibitory to growth, and glutamic acid was found to reverse the inhibition. A mixture of the common purine and pyrimidine bases, and a mixture of volatile fatty acids, were each found to be inhibitory to growth. Biotin, folic acid, and pyridoxal were essential vitamins, but the other eight B-vitamins examined were found to be neither stimulatory nor essential. Carbon dioxide was found to be required in relatively high concentration in order for growth to start in a medium devoid of rumen fluid. Purines, pyrimidines, cobalamin, and Tween-80 in a mixture were found not to replace the carbon dioxide. Rumen fluid was found to contain material very stimulatory for growth of the Butyrivibrio. The stimulatory material was found to be possibly two compounds or types of compounds, one a peptide or peptide like substance, and the other an anionic substance which probably was polynucleotide-like. / Ph. D.

The estimation of several missing values in a randomized block design ls considered. The method used ls that of minimizing the error sum of squares, proposed originally by Yates (1933). Explicit equation for each absent value are derived for all cases in which not more than three values are missing. A general formula valid for any permissible number of missing observations ls given for the case in which no two values are missing in the same block or treatment, and also for the case in which all of the values missing are in a single block or treatment. A procedure for the completely general case is proposed. This, although requiring the inversion of s symmetric matrix of order equal to the number of missing observations, may prove to be less tedious in application than the iterative method proposed by Yates. / Master of Science

The purpose of this thesis has been achieved. Vector notation has been used in developing the equations necessary for the analysis of three dimensional elastic rings. An elastic centroid for the elastic ring has been located with a tabular procedure for exacting the solution to the problem. For a comparison of the solutions between the method based on three elastic centroids with the method based on one elastic centroid, one need refer to Figures 7 and 10. Comparing the two methods, it is the author's opinion that the method herein presented is superior if for no other reason than simplicity. The development of the six equations can be readily followed and the tabular form of Figure 7 presented as an orderly means for obtaining the coefficients of the six unknowns. Figure 10 presents a method involving a similar tabular form as in Figure 7 along with the additional calculations that must be made before the correction moments and shears can be obtained. The necessity of these additional terms serves to obscure the physical significance of the final equations. / Master of Science

A preliminary theoretical and experimental investigation of the effect of velocity diagram parameters on inlet total-pressure distortions through a single-stage subsonic axial-flow compressor for incompressible flow has been conducted. The wake of 1/4-inch diameter rod, measured both upstream and downstream of a rotor, has been compared for various velocity diagrams. The measured downstream wake was also compared with the downstream wake estimated by using a derived equation subject to the assumption that (a) the undistorted and distorted flows enter the rotor with the same absolute direction, (b) the static pressures of the undistorted and distorted flows are equal at the inlet and also at the exit of the rotor, (c) the increase of blade angle of attack in the distorted flow is not sufficient to cause blade stall, and (d) the distorted flow is turned in the rotor passage to the same exit direction, relative to the rotor, as the undistorted flow.> / Master of Science