We consider a multivariate situation with means µ₁,...,µ_p and covariance matrix Σ. We wish to derive sequential procedures for testing the hypothesis: H₀: (µ̲ - µ̲ₒ)Σ⁻¹(µ̲ - μ̲₀)’= λ₀²( usually zero) against the alternative: μ̲₀ H₁: (µ̲ - µ̲ₒ)Σ⁻¹µ̲ - μ̲₀)’=λ₁² both for the case where Σ is known (the sequential X²-test) and where Σ is unknown and must be estimated from the sample (the sequential T²-test). These sequential procedures should guarantee that the probability of accepting H₁ when H₀ is true is equal to a and the probability of accepting H₀ when H₁ is true is equal to β. For the case where Σ is known, λ₀² = 0 and λ₁² = λ², the test procedure is as follows: for a sample of n observations form the probability ratio: P_1n/P_0n = e^-nλ²₀F₁(p/2;nλ²X_n²/4) where p denotes the number of variables, _nx[with horizontal bars above and below the x] denotes the vector of the sample means based on n observations, X²_n = n(_nx[with horizontal bars above and below the x] - μ̲₀) Σ⁻¹(_nx[with horizontal bars above and below the x] - μ̲₀)’ and ₀F₁ (c;x) is a type of generalized hypergeometric function. a. If P_1n/P_0n ≤ β/(1-α), accept H₀; b. If P_1n/P_0n ≥ (1- β)/α, accept H₁; c. If β/(1-α) < P_1n/P_0n < (1-β)/α, continue sampling. For the case where Σ is unknown, the procedure is exactly the same except that the probability ratio is now: P_1n/P_0n = e⁻^-nλ²/2 ₁F₁[n/2,p/2;nλ²T_n²/2(n-1+T_n²)] where T_n² = n(_nx[with horizontal bars above and below the x] - μ̲₀)S<sub>n</n>⁻¹(_nx[with horizontal bars above and below the x] - μ̲₀)' , S_ndenotes the sample covariance matrix based on n observations and ₁F₁(a,c;x) is a confluent hypergeometric function. Procedures are also given for the case λ²₀ ≠ 0. Similar procedures are given to test the hypothesis: H₀ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₀² (usually zero) against the alternative: H₁ = (μ̲₁ - μ̲₂- δ ̲)Σ⁻¹(μ̲₁ - μ̲₂ - δ̲)’ = λ₁² It is shown that these sequential procedures all exist in the sense that the risks of accepting H₀ when H₁ is true and of accepting H₁ when H₀ is true are approximately α and β respectively and that these sequential procedures terminate with probability unity. Some of these situation have been generalized to give simultaneous tests and the means and covariance matrix of a sample. No expressions yet exist for the OC or ASN functions although some conjectured values have been determined for the latter and suggest, in comparison with their corresponding fixed-sample tests, substantial reductions in the sample sizes required when either H₀ or H₁ is true. The general problem of tolerances is discussed and then some of these procedures are demonstrated with a numerical example drawn from the field of ballistic missiles. The determination of P_1n/P_0n is quite laborious for both the sequential X² - and T²-tests since it requires the evaluation of a hypergeometric function each time an observation is made. It would be better for each value of n, given p, α, β and λ² under H₁, to compute the values of X²_n or T²_n which would correspond to the boundaries of the tests indicated by β/(1-α) and (1-β)/α. Tables to facilitate both the sequential X²- and T²-tests are given for p = 2,3,...,9; λ² = 0.5, l.0, 2.0; α = β = 0.05 for n ranging from the minimum value necessary to reach a decision to 30, 45 and 60 for λ² = 0.5, 1.0, 2.0 respectively. These tables were prepared on the IBM 650 computer using the Newton-Raphson iterative procedure. Finally, a discussion is given for the hypergeometric function ₀F₁ (c;x) and a table given of this function for c = .5(.5)5.0 and x = .1(.1)1(1) 10(10)100(50)1000. / Doctor of Philosophy

This thesis presents two alternative procedures for comparing standard and quick tests of a null hypothesis H_o. This comparison is usually made by plotting the power curves of each test for a fixed Type I error. However, the power curves give only an indication of the individual performances of each test and not of the extent to which they agree when applied to the same problem. The procedures discussed in this paper deal with determining this degree of agreement. The first method determines the probability, P, that the quick test leads to a significant result at a level α given that the standard test is just significant at level α. If the standard and quick tests are based on the statistics u₁ and u₂ , respectively, the second approach determines the level of significance corresponding to the expected value of u₂ given that u₁ is just significant at level α. This level of significance is termed the "equivalent Type I error" of the quick test and denoted by γ. Both methods are applied to compare tests of location, dispersion, and the paired t-test with the sign test, all in samples taken from a normal population. In the first two cases, values of P and γ are given for different sample sizes, and in the third case only the “equivalent Type I error" of the sign test is given, P being rather difficult to evaluate. / Master of Science

The Organization tor European Economic Co-operation(O.E.E.C.) was a result of Marshall Plan pressure for co-operation among the states receiving American aid. The cooperative spirit engendered by O.E.E.C. prompted the formation of the Council of Europe, where the international problems of the members were debated. Conclusions of the Council were addressed to the states, but were not binding. A. closer association was sought by men such as Paul-Henri Spaak, Robert Schuman, and Jean Monnet. The Schuman Plan launched the European Coal and Steel Community, whereby six states - France, Germany, Italy, Belgium, Luxembourg, and the Netherlands - pooled their coal and steel industries, and relinquished national control over them. This was followed by an abortive attempt to form a common European army and a common political community. The effort failed in the French National Assembly in 1954. The following year brought the “new drive for European union” led by MM. Schuman and Monnet. The result was the establishment of the European Economic Community (Common Market) and the European Atomic Energy Community (Euratom) in January of 1958. Complications in the relations between the Community and the other western European states - particularly the United Kingdom - have yet to be resolved, but negotiations are currently being conducted. The effectiveness, even the very existence, of the Common Market and the unity movement may well depend on the result of these negotiations. / Master of Science

A one-tenth scale model of a skewed reinforced concrete rigid frame bridge was constructed and tested both within the elastic range and to ultimate load. The model was fabricated of reinforced concrete using a 1: 3.8 : 0 mix and 1/8 inch deformed reinforcing bars. A total of 57 SR-4 type AR-7 rosette gages were used to measure the strains on the surface of the concrete at selected positions. An additional 108 SR-4 type AR-7 and AR-7-4 strain gages were fastened to the steel reinforcing. Deflections were measured at 11 points of the deck and supporting reactions determined by suitable dynamometers. The vast amount of strain data was presented in tables. Part of it was converted into stresses and these integrated to find the internal reactive forces and moments at selected sections. However, the unknown behavior of concrete in biaxial tension and difficulty in establishing the precise position of the neutral surface prevented as close a check of static equilibrium as would have been desirable. / Master of Science

In order to solve a highly indeterminate rigid frame structure by the method of slope deflection a high order stiffness matrix needs to be inverted. Existing computer programs for the IBM Type 650 Computer are entirely too time consuming when the matrix coefficients and program instructions exceed the storage capacity of the computer drum. The method described in this paper involves the solution of the problem by the inversion of a partitioned matrix originally described by Duncan, Prazer, and Collar. Machine time was reduced by 91% using the new method. The program retains accuracy of six significant figures for a matrix normally twice as large as a machine can handle at one time, and four plus significant figures if the matrix is four times as large as the computer can handle in one storage operation. The method can be applied to digital computers of any size, thus increasing their apparent storage capacity or decreasing their operating time for a matrix inversion. The method is applicable to symmetrical as well as unsymmetrical matrices. / Master of Science

Master of Science

The magnetic amplifier is a device employing saturable reactors as its basic element. The theory or magnetic amplifiers is developed from the standpoint of rectangular hysteresis loops that are simplified in such a manner as to eliminate the problems of nonlinearity. The saturable reactor may be connected in either series or parallel combinations and their operation analyzed on the basis of voltage, current, and flux waveforms. The characteristics or the saturable reactor circuits may be enhanced by the addition of feedback. Feedback is accomplished either by magnetic means using additional windings, or by self-saturation in which rectifiers are used to secure feedback. The result is a device in which a small amount of power can effectively be used to control larger amounts of power. / Master of Science

The present investigation was undertaken to determine experimentally whether or not instability of the laminar boundary layer on blunt convex bodies exists when the wall-to-stagnation temperature is lowered. It was found that instability existed and theoretical considerations are given to the transition being associated with the formation of ice on the model surface and with an increase in roughness Reynolds numbers due to thinning of the laminar boundary layer at low wall-to-stagnation temperature ratios. The experimental tests were conducted on two-inch diameter spheres at M = 4.95 and free-stream Reynolds numbers per foot of approximately 72. 5 x 10⁶ or 12.1 x 10⁶ based on the model base diameter. Data were obtained tor both the hot wall and cold wall case. The stagnation temperature was approximately 400° F. Initial model wall temperatures were 97°F, for the hot wall test and -320° F for the cold wall tests. / M.S.

The velocities which are associated with the phenomenon of flame propagation can be studied by diverse experimental techniques. From the information gained through an extensive survey of literature a versatile apparatus which enables the measurement of uniform name and detonation velocities has been constructed. This was achieved by constructing a name tube made up of three sections of 1 ½” extra-heavy piping. Velocity measurements were made by sensing the name front at two locations along the axis or the flame tube through the use of ionization probes operated in conjunction with two amplification circuits. An oscilloscope and an electronic counter, operated in parallel with the amplifiers, provided the measurement of the time of flame travel. A series of tests were run in order to justify the capability of the apparatus to measure uniform name and detonation velocities. Stoichiometric mixtures of acetylene and air were used throughout the investigation. Uniform name velocities were determined to be between 339 and 469 cm/sec. Detonation velocities were observed to be around 1800 m/sec. / M.S.

A soluble acid-heat extracted Brucella vaccine: Immunological and physiological studies in guinea pigs, rabbits and calves. 143 p. Dissertation. 1959. -- A soluble-type vaccine was prepared by the acid-heat extraction of Brucella abortus Strain 2 308 and its metabolic by-products in Stuart• s medium. A comparison was made, in guinea pigs, of 2-day-old and 13-day-old cultures for the preparation of the immunogenic agent. Further comparisons were made in guinea pigs and calves of the 13-day-old culture and cell-free 13-day-old culture vaccine. The agent made from the 2-day-old culture produced no significant protection against various challenge levels of virulent Br. abortus Strain 2308. The agent prepared from 13-day-old cultures not only produced significant protection against homologous strain challenge but, produced insignificant serum-agglutination titers at effective dosage levels. The 13-day-old whole and cell-free culture vaccines gave similar results in guinea pigs. In calves the 13-day-old whole-culture vaccine produced higher transient serum-agglutination titers than the 13-day-old cell-free culture vaccine. Protective studies in calves were inconclusive due to inadequate infection in the control animals. The vaccine was shown to be essentially protein in nature and contained two distinct fractions on paper electrophoretic examination. The less mobile fraction apparently contained the agglutinogenic material and the more mobile fraction apparently contained the immunogenic material. Removal of the Brucella cells, prior to acid-heat extraction, decreased the less mobile fraction by more than one-half. The degree of serum agglutinin titer response was apparently contingent on this fraction, which indicated the agglutinogens were an index of metabolic byproducts, but this did not imply that they were an index of protection. Paper electrophoretic serum-protein patterns of 15 male and 15 female rabbits were studied. It was determined that: the serum fraction percentages of normal rabbits showed little variation, with no differences between sex or breed. The various pathologic conditions were indicated first by serum-protein patterns, and later diagnosed by histopathological examination of necropsy material. An additional 14 rabbits with a natural Eimeria stiedae infection were also investigated. The use of paper electrophoresis as an aid in the selection of normal animals for experimental investigation was demonstrated. The 13-day-old whole culture vaccine was employed in rabbits to study serum-agglutinin titer response. The results indicated that the maximum gamma globulin response followed the peak agglutinin response by seven to twenty-one days, this indicated a secondary response the nature of which was not determined. Female rabbits responded to the vaccine with higher initial titers than males and the titer decline was one and one-half times more rapid in the males than. in the females. Sex was shown to be the most significant factor in this finding. Castration and ovario-hysterectomy indicated that the above results could be reversed when an estrogenic hormone was given to the castrates and when testosterone was given to the ovario-hysterectomized animals. Insufficient data was available to elucidate the role of steroid hormones in the serum-agglutinin titer response. / Ph. D.