By truncation, or censoring, the information can be obtained in a shorter period of time, since fewer items are tested. These statistical situations have frequently been encountered in what are called life testing, dosage response studies, target analysis, biological analysis, biological assays, and in other related investigations.

The methods applicable to the study of truncation may be classified roughly as follows: 1. Method of maximum likelihood estimator. This method is to be recommended when sample sizes are at least moderately large.The estimators for truncated and censored samples are consistent and asymptotically efficient. Solutions are always approximated by straightforward iterative procedures; hence, the calculations often become tedious and laborious. 2. Method of least squares or order statistics. The method should be employed when estimators must be based on samples of size 20 or less. The approach to the general case in truncation is of value not only for its numerical results but also for the drawing of inferences concerning interesting and important patterns for the coefficients, variances, and the relative efficiencies of the estimates. 3. Computer method of maximum likelihood estimation. Recently with the development and availability of electronic computers, the exhausting calculations involved in the maximum likelihood estimations have been greatly alleviated. One program furnished by Hurst (1966) has been appended for reference.

]]>The first portion of the paper discusses transformations and subgroups. However, many basic definitions and theorems will not be stated for example definition of a group, subgroup, normal subgroup, factor group, etc. Topics to be emphasized include centralizer, center, and normalizer of a group, characteristic subgroups, conjugacy and commutator.

In the second part of the paper direct sums are discussed with the ultimate proof of the famous Remak-Krull-Schmidt Theorem.

]]>An additional basic objective is to obtain explicit algebraic expressions for different types of linear trans- formations.

The first concepts to be covered are arbitrary linear transformations, various ways of looking at linear transformations, and the effects of a linear transformation on a vector of normally distributed random variables.

Next orthogonal transformations to independence, and then oblique transformations to independence will be developed in turn.

]]>It is the purpose of this study to present associated numerical methods for digital computer which are satisfactorily accurate and which are reasonably economical in both time and machine memory capacity. To carry out this objective the following procedures were used:

1. A review of literature on numerical approximations-both texts and articles from statistical journals and computer science publications.

2. .Writing test programs in Fortran for all the associated methods which can be obtained.

3. Checking the answers obtained by numerical approximation with the known answers in the table in order to determine usefulness of the numerical method.

4. Writing Fortran subprograms to evaluate those integrals by using the most accurate methods according to the experimental results.

]]>For example we show that the plane (E^{2}) does not contain uncountably many pairwise disjoint contina each of which contains a simple triod (Corollary 4. 1 ). We prove that in an uncountable collection G of pairwise disjoint simple closed curves in E^{2} "almost all" elements of G must be converged to homeomorphically "from both sides" by sequences of elements of G (see Theorem 4. 3 ). The same technique allows us to prove the nonexistence of uncountably many pairwise dis -joint wild 2 -spheres in E^{3}.

Another interesting consequence of Borsuk's Theorem is Theorem 3. 4 which shows that in each set G consisting of uncountably many compact subsets of a metric space, some element of G is an element of convergence. Proofs for this theorem do not often appear in the literature, and, as far as the author knows, the proof given here does not appear in the literature.

We wish to emphasize that all the proofs given in this report were constructed by the author without reference to the literature, in fact the author was unaware of the references until after the proofs were given. We given reference at the end of the paper where proofs in the literature can be compared with the proofs given here.

We wish to emphasize that all the proofs given in this report were constructed by the author without reference to the literature, in fact the author was unaware of the references until after the proofs were given. We given reference at the end of the paper where proofs in the literature can be compared with the proofs given here.

]]>Consequently, there have been methods developed to approximate chi-square, t and F value, when degrees of freedom and probability are known. It is the purpose of this study to present the methods of each individual distribution and evaluate its accuracy. Thus, the scope of this paper includes the following:

1. The definition and inverse function of each distribution.

2. The numerical approximate methods and examples.

3. A computer Fortran IV program to maximize the accuracy of calculation.

4. A comparison of the results obtained by numerical approximation with the known tabular value.

5. An evaluation of the capacity of these numerical methods.

]]>The first part of this report will be devoted to the general development of such functions by means of definitions and theorems. The second part will consist of generalizations of a particular function, the -r-function.

Throughout this paper, lower case Greek letters will represent real numbers and lower case English letters will represent integers. Also the basic ideas of summation and product will be assumed as already familiar to the reader.

]]>Lebesgue integration differs from Riemann integration in the way the approximations to the integral are taken. Riemann approximations use step functions which have a constant value on any given interval of the domain corresponding to some partition. Lebesgue approximations use what are called simple functions which, like the step functions, take on only a finite number of values. However, these values are not necessarily taken on by the function on intervals of the domain, but rather on arbitrary subsets of the domain. The integration of simple functions under the most general circumstances possible necessitates a generalization of our notion of length of a set when the set is more complicated than a simple interval. We define the Iebesgue measure "m" of a set E Є M, where M is some collection of sets of real numbers, to be a certain set function which assigns to Ea nonnegative extended real number 'mE '.

This report consists of the solutions of exercises found in 'Real Analysis", by H. L. Hoyden. Quotations from the book are all accompanied by the title "Definition" or "Theorem". The exercises are all entitled "Proposition" and all proofs in this report are my own. All theorems are quoted without proof The theorems and definitions occur as they are needed throughout the paper1 but some of the most basic definitions and theorems are lumped together in section II.

It is assumed in this paper that the reader is familiar with the basic concepts of advanced calculus and set theory,

]]>Several different types of problems are solved in this report. Among these are Bessel's classical differential equation of index n, two electrical circuit problems, a beam problem, a vibrating string problem, a heat flow problem, and a temperature gradient problem.

One of the objectives of this report is to illustrate several operation properties of the Laplace and finite Fourier sine transforms. Therefore, various methods of inverting transforms are employed to provide diversification.

]]>It is the purpose of this paper to provide a bird's-eye view of Bayesian Inference with emphasis on the comparison of Bayesian approaches and conventional approaches. This paper is written for those who have had about one year's background in mathematical statistics.

Some conclusions from Bayesian Inferences and classic inferences are essentially the same. However, it is informative to know how a problem can be handled by two completely different procedures, even though the conclusions are the same. What is more, the Bayesian approach does have some advantages over the classic approach. Sometimes the Bayesian approach can handle a problem which has not been solved by the classic approach. For example. an exact test of the differences between two normal means with unequal variances can be developed by introducing Behrens' distribution; a solution has not been obtained by classical methods.

]]>Since 1940 the number of papers on semigroups has increased to approximately 30 per year. Thus we find ourselves in the midst of a new and expanding area of mathematics. Our interest centers around the structure of semigroups and their representation by mappings. Therefore it is necessary that the reader be familiar with sets, mappings, groups, and lattices.

]]>be given explicitely, whereas the metalanguage will consist of, only that portion of the English language needed to clearly describe the formal system. In some instances, since no confusion will result, certain symbols may appear not only in the object language but also in the metalanguage. This will be evident by the two-fold use of the symbols: ~, >,, (, ), and '. Should specific reference be made to some one of the symbols of the theory, this symbol will be enclosed in single quotes. Furthermore, the reasoning employed in establishing results about the formal systems will consist only of those notions which have great intuitive appeal. Included among these will be mathematical induction.

]]>Various methods of examining phenomena in the Snake River Plain were employed. Unsupervised classification of Landsat ETM data indicates that spectra can be used to differentiate between flows. However, photographs of flows at Craters of the Moon National Monument show wide variations in texture and color within a single flow. Vegetation grows preferentially on the smoother pahoehoe slabs and significantly impacting the spectra observed through remote sensing. Additional vegetation would prohibit using Landsat ETM to map lava flows reliably. Landsat ETM is not a viable method to map the lava flows in quadrangles Twin Falls NE, Hunt, and Eden NE, since they have significantly more loess and vegetative cover.

Elevation data reveal the locations of the ramparts at the source vent, but do not define the edge of a flow except when an inflated lobe terminates the flow. Field observations of terrain and geochemical analysis of samples defines the flows and their source vent.

]]>may be a potentially effective strategy to help decrease the ACL injury risk.

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