Applied Mathematics Commons^™

Measure Theory And Lebesgue Integration, Evan S. Brossard

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The Lebesgue integral is a generalization of the Riemann integral which extends the collection of functions which are integrable.

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 …

Tests Of Methods That Control Round-Off Error, Dale M. Rasmuson

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Methods of controlling round-off error in one-step methods in the numerical solution of ordinary differential equations are compared. A new Algorithm called theoretical cumulative rounding is formulated. Round-off error bounds are obtained for single precision, and theoretical cumulative rounding. Limits of these bounds are obtained as the step length approaches zero. It is shown that the limit of the bound on the round-off error is unbounded for single precision and double precision, is constant for theoretical partial double precision, and is zero for theoretical cumulative rounding.

The limits of round-off bounds are not obtainable in actual practice. The round-off error …

Algebraic Properties Of Endomorphisms Of Abelian Groups And Rings, Johnnie George Slagle

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The main objective of the thesis was to extend the definition of an M-Group to what is called an M-Ring. From this extension a system called an expanded ring follows naturally. To facilitate the development of the expanded ring, chapter I is devoted to developing properties on systems that are not quite rings where many interesting examples are constructed. In chapter II the definition of an M-Ring is given and some of its properties are derived. In chapter III some of the properties of expanded rings are proved, and examples of expanded rings are given to show their existence.

Principal Component Factor Analysis, Herbert H. Jolliff

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Factor analysis came into being around 1900 in the field f psychology to explain theories of human ability. Several methods of factor analysis exist; but according to Harman (1967) principal component factor analysis is unique in the mathematical sense, therefore, quite often the preferred method. The centroid method is computationally easier, and it gives close approximations to the principal component method on some data sets. An example of this is shown in Appendixes B and F by comparison.

Factor analysis is being used in many fields. A few of the fields are sociology, meteorology, political science, medicine, geography, business, economics, …