In designing multivariate experiments, it will often be the case that different responses have different design matrices. This most often occurs when certain responses are not influenced by various factors. If not all responses are measured on each observational unit, this gives rise to the More General Linear Multiresponse (MGLM) design. For a factorial experiment, denote by V a variance-covariance matrix of estimable functions of the parameters of any effects one wishes to study. Optimal designs are found that minimize the trace of V when the size of the design is restricted by a total cost constraint, thus minimizing the average variance of each estimable function. It is shown that Hierarchical MGLM designs (HMGLM), a subset of the MGLM designs, need only be considered. In a HMGLM design a hierarchy of the responses V₁,...,V_p’ can be found such that if i < j (i,j = 1,...,p'), V_i should be measured on at least as many experimental units as V_j is measured and V_j is measured only on units where V_i is also measured. Given the costs and 'a priori' variance estimates, optimal designs for 2^v factorial experiments are found where k_i effects are under study for V_i. The procedure is then extended to include p^v factorial experiments. We consider next the minimization of the determinant of V as a criterion for optimality. This criterion results in the confidence ellipsoids for the estimable functions to be of minimum volume. Due to the difficulty in defining the off-diagonal covariance matrices of V for the general class of MGLM designs, certain well-defined subclasses were considered where the covariance matrices of these designs could be found in general. First a rather natural subclass of MGLM designs, called Restricted MGLM designs (RMGLM), was investigated. HMGLM designs are ·a subclass of RMGLM designs, as are Multiple Design Multiresponse (MDM) designs which assume that all responses are measured on each experimental unit. The general situation, assuming p' responses, was investigated first. Due to difficulty in finding the determinant of the matrix V, a general solution for the optimal RMGLM design was found for only certain specific situations. In an attempt to ease the difficulty in determining the general form of det(V), the two-response case was considered. The optimal RMGLM design was then determined for more general situations. Finally, the complement subclass (CRMGLM designs) of the RMGLM designs in the class of MGLM designs was investigated for two-response situations. The optimal MGLM design can then be determined by comparing the optimal RMGLM and CRMGLM designs. For most situations, the optimal CRMGLM design can be found, but for those situations where it cannot be found, the optimal RMGLM design (a HMGLM design) can still be determined, giving a design at least as.good as, and often better than, the generally accepted MDM design. / Ph. D.

Community colleges because of their philosophy have developed programs which are sensitive to the communities and students they serve. Thus, it would appear that data are needed in order to better prepare students for the writing they will do at a transfer institution. The present study was designed to provide such information for students transferring to one institution. A survey was made with two questionnaires, one sent to professors teaching 300-level courses at Virginia Polytechnic Institute and State University, and the second questionnaire sent to the Virginia public community college transfer students who were taking at least one 300-level course in the fall of 1972. Eighty percent return from professors and seventy-six percent from students was obtained. The results were tabulated by colleges: Agriculture, Architecture, Arts and Sciences (divided into two divisions), Business, Education, Engineering, and Home Economics. Tabulation of writing required by professors ranked from most frequently assigned to least frequently assigned were: Report on Material Read, Laboratory Report, Expository Essay, Case Study, Thematic Essay, Project Report, Argumentative Essay, Descriptive Ess , Term Paper, Independent Study, Narrative Essay, Journalistic Writing, and Creative Writing. A wide variance in the amount of writing assigned per class was observed; however, professors seemed to prefer the assignment of frequent short papers of about three pages in length. Writing assigned for test requirements accounted for about fifty percent of the writing assignments made by prossors. A combination of essay and objective examination appeared to be the most popular overall; however, the objective examination was used more often than the essay type except for the Arts division. In general, students reported doing the same types of writing which professors assigned, but when compared to the responses given by professors, students reported completing fewer writing assignments, doing shorter papers (except for the courses taken in the College of Engineering), and completing examinations which were more objective in character. Implications for the community colleges and suggestions for additional research were made. / Ed. D.

The purpose of this dissertation is to examine the problem of estimation of a univariate probability density function. Let Y₁, Y₂, …, Y_n be a sample of n independent observations, each distributed according to an unknown continuous density function f(y). Given this sequence of observations, how can one estimate f(y)? Chapter I presents the historical background and a literature review.of existing methods for the estimation of a probability. density function. In addition, a section is devoted to the application of density estimation. In particular, we consider the practical application to reliability analysis. The estimator of the unknown density function developed in Chapter II is similar to one proposed by Rosenblatt (1956) and by Parzen (1962). However, the kernel we consider is a function of the rank of each observation. We use, as our kernel, the asymptotic distribution of the order statistics of a sample. We refer to this estimator as the normal rank kernel estimator. In order to test the performance of our estimator, we have performed an experimental analysis by monte carlo studies. Chapter III uses the estimator of Chapter II as the foundation for the development of a. recursive estimation procedure. The technique employed is the method of successive substitution in which the solution at each iteration is used to generate the next solution until convergence is achieved. Consequently, we call this estimator the iterative estimator. Again, we have performed a simulation to compare the estimator of Chapter III with that of Chapter II. In Chapter IV, a sequential procedure is developed for estimation of a probability density function. Initially, a normal distribution with mean and variance y̅and s², respectively, is fitted to the data and a goodness of fit test is performed. This hypothesis rejected, a sequential procedure employing the concept of spline functions is used. Several examples are given in Chapter V which illustrate the various methods of density estimation introduced.in the preceding chapters. The examples use data that are both simulated and real. Also, an example estimates both the reliability and hazard functions. Finally, relevant computer programs (Fortran) and descriptions of their utilization appear in the appendices. The program contained in Appendix A has a dual purpose in that by proper choice of an input parameter, the program will be executed for either the rank kernel estimator or for the iterative estimator, while Appendix B contains the computer program for Chapter IV. / Ph. D.

A new approach to Kneser's Theorem, which achieves a simplification of the analysis through the introduction of maximal sets, the basic sequence of maximal e-transformations, and the limit set, B*, is presented. For two sets of non-negative integers, A and B, with C∈A⋂B, the maximal sets, Aᴹ and Bᴹ, are the largest supersets of A and B, respectively, such that Aᴹ + Bᴹ = A + B. By shifting from A and B to Aᴹ and Bᴹ to initiate the analysis, the maximal properties of Aᴹ and Bᴹ are exploited to simplify the analysis. A maximal e-transformation is a Kneser e-transformation in which the image sets are maximized in order to preserve the properties of maximal sets. The basic sequence of maximal e-transformation is a specific sequence of maximal a-transformations which is exclusively used throughout the analysis. B* is the set of all non-negative elements of sM which are not deleted by any transformation in the basic sequence of maximal e-transformations. Whether or not B* = {O} divides the analysis into two cases. One significant result is that B* = {O} implies δ (A + B) = δ (A, B) where δ(A + B) is asymptotic density of A + B and δ (A, B) is the two-fold asymptotic density of A and B. The second major result describes the structure of A + B when δ(A + B) < δ(A, B). With B* ≠ {0} it is shown, using only elementary properties of greatest common divisor and residue classes, that there exists C⊆ A+ B, 0εC, such that δ(C) ≥ δ(A, B) -1/g where g is the greatest common divisor of B* and C is asymptotically equal to C^(g), the union of all residue classes, mod g, which have a representative in C. The existence of C provides the crucial step in obtaining an equivalent form of Kneser’s Theorem: If A and B are two subsets of non-negative integer, 0εA⋂B, and δ(A + B) < δ(A, B), then there exists a positive integer g such that A + B is asymptotically equal to (A + B)^(g) and δ(A + B) = δ ((A + B)^(g)) ≥ δ (A^(g) , B^(g)) - 1/g ≥ δ(A, B) -1/g. / Ph. D.

The Jost function method is extended to study scattering phenomena produced by potentials having spheroidal symmetry. The spheroidal radial functions are constructed by taking the spherical wave functions as bases. The role of the Jost function in spherical potential scattering is reviewed. The relation between zeros of the Jost function and the formation of bound and resonant states is then established for spheroidal potentials. The domain of analyticity of the spheroidal Jost function is studied for three classes of basis functions: those belonging to spherical potentials having only a bounded first and second moments, those belonging to a Yukawa potential, and those belonging to truncated potentials. / Ph. D.

Impact shear and static flexure tests were performed on pallet stringers notched with six different fillet radii. These notches were cut with a band-saw or a stringer-notcher. Attempts were made to reduce the incidence of stringer fracture between the notches (a) by nail-reinforcing the stringers adjacent to the inner notch ends and (b) by retarding longitudinal moisture movement through the notch ends by notch coating. The leading-edge design of the paIlet was studied by determining its impact resistance if fabricated with leadboards and stringers of different species and specific gravities. The impact shear resistance and the flexural strength of notched pallet stringers increased with increasing notch fillet radii. The optimum fillet radius was one inch. No performance difference was found between the band-sawn and machine-shaped notches. Nail-reinforcing slightly increased the immediate flexural strength at initial failure of green stringers; did not increase the delayed flexural strength of seasoned stringers; and increased their immediate and delayed ultimate flexural strength. Notch-coating progressively increased the delayed ultimate flexural strength during the seasoning period. Leadboard and stringer specific gravity effected the leading-edge impact resistance of wooden pallets. It increased more rapidly if the leadboard specific gravity was increased. An increase in the stringer specific gravity did not necessarily increase the leading-edge impact resistance. Optimum leading-edge impact resistance should be attained by using high-density lumber assembled with a supplementary nai I driven through the leadboard into the center stringer, provided lumber splitting is avoided. / Master of Architecture

Increased growth of Pinus taeda L. plantations is expected from genetic improvement and intensified cultural practice. It is necessary for forest managers to have a means for estimating the yield of forest stands after increased growth occurs. Several models were tested to determine which could best predict dry weight yield after certain constant growth increases were assumed. The data used consisted of 189, one-tenth acre sample plots taken in loblolly pine plantations located in piedmont and coastal plain Virginia, and coastal plain Delaware, Maryland, and North Carolina. Constant growth increases in diameter, height, and wood density were considered. The models considered were multiple regression and the diameter distribution approach using the beta distribution. Under the diameter distribution approach alternatives involving a piecewise integration over diameter class limits and a "unified" approach were considered. The models were compared to each other and to expected yield increases when determining their consistency, accuracy, flexibility, and simplicity. Recommendations concerning the different models were made. / Master of Science

The factors influencing shifts in the demand for professional baseball are investigated. Initially, the logical framework behind attendance patterns is presented as a prelude to the theoretical model. Graphical analysis and classical optimization techniques are then used to transform the model logic discussion into a theoretical model. Lastly, the validity of the theoretical postulates is tested through use of regression analysis. All the empirical findings are tabulated and analyzed in light of the theoretical model in order to determine the relative impact of each variable on attendance. Current winning percentage, population density, lagged winning percentage, and the percentage of workers employed in manufacturing prove to be of utmost importance. / Master of Arts

The mineral content of vegetables grown with organic and mineral fertilizers was investigated in order to determine the nutritional superiority or non-superiority of organically raised foods. Duplicate samples of split peas, pinto beans, kidney beans, blackeye peas and tomatoes of each treatment were wet washed with concentrated nitric and perchloric acid. Concentrations of Cu, Fe, Mn, Zn, Na, K, Ca and Mg were determined using the Perkin-Elmer Atomic Absorption Spectrophotometer. Unequal variation between treatments rendered the F test meaningless for many minerals. Where differences were found, they were in favor of chemically fertilized vegetables as often as in favor of those fertilized organically. It was concluded that methods of fertilization will not differently affect nutrient content of a crop and that organically grown foods are not worth the extra cost to the consumer in terms of nutritional quality. / Master of Science

The twofold purpose of this thesis included the operationalization of a measure of participatory style and a quest for correlates of said style in several selected areas. Using survey data based upon the responses of 827 delegates to the 1972 Republican National Convention, this study developed a stylistic index which was finally structured into four categories of Amateur-Professional orientation. The fourfold index was first cross-tabulated with several selected socio-economic variables and then compared to several variables directly related to party service longevity. The findings with respect to socio-economic variables largely confirmed the results of previous research which has failed to substantiate the existence of relationships between political style and such possible correlates. With respect to service longevity, findings indicated that the Amateur style is more prevalent among relative newcomers to political party activism. The Amateur-Professional index was compared to three indices computed for each delegate and designed to tap preferences for material, solidary, and purposive incentives for maintenance of political activism. It was found that while the majority of all delegates manifested a strong preference for purposive incentives, such a preference was greatest among individuals of the Amateur orientation. Alternatively, such persons indicated the strongest distaste for material incentives. The hypothesis, which was formulated in accordance with the findings of previous research, was confirmed. A comparison of the stylistic index with the intensity of views regarding current issues failed to confirm the hypothesis that while the content or direction of views or attitudes has not provided a basis of style differentiation, issue saliency well might. / Master of Arts