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The problem discussed in this paper is that of finding a best approximation to a given real-valued function f(x) over a continuum by means of finding a best approximation over a discrete set of points. We are also seeking to find a numerical method of finding a best approximation over our discrete set of points. A best approximation is one which minimizes the maximum deviation of our approximation from our given function f(x). We first discuss the concept of linear programming. In this paper we are not so much concerned with the theory behind linear programming as we are with the method to solve a linear programming problem, namely the simplex method. We discuss the simplex method from the point of view of a programmer, noting how results are continually updated until an optimal solution to t he problem is found. The only theoretical aspect of linear programming which we discuss is the notion of duals and the relationship between the solution of a primal and a dual problem. This becomes very important later in the paper when we try to formulate a best a proximat ion problem as a linear programming problem. Next we discuss the theoretical aspects of best approximation over a continuum. We prove existence, uniqueness, and, most important for our purposes, characterization. Our approximating functions are assumed to form a Chebyschev set throughout this paper. Finally we discuss best approximation over a discrete set of points. We first prove that the characterization theorem holds for problems of this type. Now that we have a way to tell whether our approximation is the best that can be obtained, we turn our attention to the relationship between the best approximation problem over a continuum and a discrete set of points. We prove in a quite general context that the best approximation over a discrete set of points converges uniformly to the solution to the problem over the continuum. We then retrace our steps and establish similar results for the particular case of polynomial approximation. After this we try to find out about the rate at which this convergence takes place. In general this question has no answer for its depends on the smoothness of the functions involved; if, however, we assume the fun ctions satisfy a Holder condition we may obtain some bounds on the rate of convergence. Finally, we reformulate the best approximation problem, showing how it can be considered as a linear programming problem which we already have a means of solving.

The thesis deals with some of the mathematical techniques that are used in solving queueing theory problems. The organization of the paper is basically the same as that used by Goddard [3] in his treatment of queueing theory. The first method investigated is the analysis of queueing problems as Markovian processes. This analysis is due to D. G. Kendall [4,5] and is primarily for the case M/G/1. Formulas are found for E(n) and E(w). The limitations of this method in dealing with more general queues are mentioned. For the case M/M/s, the differential-difference equations are developed with examples of their use in machine breakdown problems and telephone trunk line congestion. The treatment is primarily for the case in which the system is in statistical equilibrium. The uses of Laplace transforms and probability generating functions are illustrated by Pollazoek's method of finding the moments of the waiting time distribution for the queue M/G/1. They are also shown in finding Pn(t) and Pn in an example of welders using a power supply. A method of expressing the distribution of the waiting time for the queue G/G/1, due to Lindley [7], is outlined at the conclusion of the paper. The result is an expression for the waiting time distribution in terms of the density function of the difference between the service time and inter-arrival time, rather than either density function separately.

In asserting that people in other fields often tend to be afraid to use mathematics, F.K. Mechta, an economist, expressed an uncertainty which contributes to the hesitancy to use mathematics. “Mathematics is tricky, it maintains silence, does its work quietly; and when we do not understand its ways and misinterpret its message, it just smiles. It never loses its temper, never laughs; we can observe a suppressed smile on its lips. Such is mathematics.” It is the purpose of this paper to conduct a brief survey of the applications of difference equations. The use of these equations is often rather elementary, frequently involving little material more technical than college algebra. An individual does not necessarily need a calculus background to be able to apply them. In the research for this work, applications of difference equations were looked for in the social and physical sciences. This paper covers the types of applications found, and the way they were developed. There is particular emphasis on what is left for the mathematician to do; what he can do to help these other fields benefit from mathematics. It will be sometimes noted throughout the paper whither or not the uniqueness of the solution has been checked for by each author. The first section in this paper is an introduction to difference equations. This section includes a brief summary of some of the theory of difference equations with emphasis on methods of solution, as well as a study of difference equations in probability. In the remaining section, the types of applications found in different fields are reviewed.

INTRODUCTION This paper will amount to an inquiry into the mathematical structure and properties of a matrix of approximate integration formulas due to Werner Romberg 1 which have found considerable favor in numerical analysis during the past decade. Like other devices for approximate integration the Romberg method has its origin in the exhaustion techniques developed by the Greeks for determining the areas of figures enclosed by lines in the plane. It is particularly closely associated with the scheme employed by Archimedes2 (circa 250 B.C.) for approximating the value of [3.14]. This involves a doubling process which is essentially the basis for the Romberg method. Archimedes began with a circle of radius one within which was inscribed some regular polygon whose area A00 he could readily compute. After doubling the number of sides he was able to compute a new area A01, then a third area A02 after a second doubling and so on to A03, .... ,A0n, ect. It is geometrically clear that such a sequence should converge to [3.14] from below. As an example, if Aon corresponds to a 24-sided polygon it will he found that Aon= 3.(32,(,,Zq) Ao, the latter correct for [3.14] to 3 figures. Much later, in about 1654 in fact, Christian Huygens rather ingeneously deduced an improved version of the Archimedes sequence. This involves combing two successive Archimedes terms with the weiphts,-1/3 and 4/3, to produce a new sequence Ain = 4 Ao,(n+1) -Aon 3 which Huygens showed would converge to [3.14] much more rapidly. Thus for term A1n, corresponding to a 48-sided polygon it will be found that A in = 3. 141590 a value correct for [3.14] to 6 figures! Huygens' numerical innovation lay practically unnoticed for almost three centuries (most likely because of the emphasis on methods of infinitesimal calculus during this period) when in 1936 K. Komqerellˆ3 undertook to extend it by constructing additional sequences, the first of which, A2n, combines two successive weighted Huygens' terms, the second of which, A2n, combines two successive weighted terms of A2n, etc. The general prescription Kommerellused for the kth sequence starting Kˆth, sequence, starting with k=1 for the case of Huygens, was ARN= 4ˆR (R-1), (n+1)- A (R-1), n 4ˆR-1 This scheme, of course, produces as many sequences as desired each one of which, as it turns out, converges to [3.14] faster than any of its predecessors. 4 As evidence of the speed of convergence, the first term in the highest sequence corresponding to a 48-sided polygon gives [3.14] accurate to 9 figures. It remained for Romberg in 1955 to obtain the notion of Applying the gist of Kommerell’s results to the problem of finding the area under the graphs of function in the problem of approximate integration of such functions. In so doing he provided a rationale for the choice of the weighting factors and developed the matrix of integration formulas that are to be the subject of this paper. It was while studying a discussion of the Romberg method in a textbook by E. Stiefelˆ5 that the writer’s interest was aroused. In particular there occurred in Stiefel’s development some unusual "bonus" properties in which formulas constructed to be exact for polynomials of some degree n continually turned out to hold as well for degree (n+1). These results, which were merely pointed out by Stiefel with little comment, motivated the writer to find their explanations. Doing so led to a fairly complete study of the Romberg method as a whole which culminated in the writing of this paper. The primary purpose of the paper will be simply to elaborate on the structure of the Romberg method. This will be accomplished by developing the matrix of formulas in considerable detail, while emphasizing completeness in the presentation and analysis of all basic ideas essential to understand the structure. In the process, explanations will be provided for the aforementioned "bonus" properties as well as others. A secondary purpose, which in many ways overlaps the primary one, will he to evaluate the effectiveness of the Romberg matrix as a numerical integration tool. To this end some theorems on convergence obtained from the liter­ature will be presented and discussed (without proof), and a generalization will be used in the procedure for formula construction which, together with some computations, will produce graphical demonstrations of the capabilities of the Romberg method.

This paper explores two kinds of graphs, isopathic graphs and air­ port graphs. A distance property of graphs in general is also examined. Isopathic graphs are graphs in which every maximal path has the same length. The major theorem of this section characterizes isopathic graphs as extended stars, bipartite or hamiltonian. There is then a discussion of the latter two classes of isopathic graphs. At the end of Section I, there is an introduction to isopathic di­graphs, a natural concern after an exposure to isopathic graphs. Airport graphs, more appropriately snob graphs, can be thought of in the following way. It might be advantageous in some milieu that before a person would become friends with another person, A, he would first de­velop friendships with all people who are stronger than A. If in a graph strength corresponds to degree, and friendship to adjacency, an airport graph is formed when all points have this property. The resulting graph has a special structure which is examined in Section II. The concept of the "distance of a graph" is presented in Section III. Here the degree of a point is seen to be of greater importance than is the condition of being a special point. (Special points correspond to the snobs of Section II.) Notation used in this paper is that given in Frank Harary's book, Graph Theory, unless it is explicitly defined as the need arises.

This thesis purposes to study a certain group of movements which can be expressed as substitutions. The groups of movements which send a square into itself is to be studied as a group of eight substitutions on the vertices for the purpose of leading up to the real problem of this paper. From the octic group, it is natural to proceed to a study of the movements which send a cube into itself. In particular, it is the aim of this thesis to discover the group of the cube and to analyze some of its properties. There are twenty-eight rotations and reflections with respect to diagonals and central axes of the cube which possess special geometrical properties. One of the problems of this thesis is to determine whether or not these twenty-eight elements constitute a group. Once the group of the cube has been determined, other problems are those of finding subgroups within the original group and of enumerating their properties. This paper is to be concerned chiefly with subgroups composed entirely of elements from twenty-eight rotations and reflections with the special geometrical properties. Also a few theorems relative to groups in general will be demonstrated and application will be made to the group of the cube.

This paper was written in response to the following question: what conditions are sufficient to guarantee that if a compact subset A of a topological linear space L^3 is not convex, then for every point x belonging to the complement of A relative to the convex hull of A there exists a line segment yz such that x belongs to yz and y belongs to A and z belongs to A? Restated in the terminology of this paper the question bay be given as follow: what conditions may be imposed upon a compact subset A of L^3 to insure that A is braced?

"Each of the ancient nations had a number system of their own, however, only a few of these will be mentioned in this paper, those being the ones which probably contributed to the development of our own number system. Therefore, the primary purpose of this paper is to give a gist of these ancient number systems, and to give in more details the history of our present system, namely, the Hindu-Arabic number system. Although this is not a complete history of our system of numbers, it is hoped that the reader will see some advantages the system has over the number systems of the ancient world"--Introduction.

The main purpose of this thesis is to collect and coordinate some known results in the study of characteristic roots of certain real symmetric matrices. Specifically, most of the matrices considered have their elements aij = 0 except for I i - J I < 1 and are known as Jacobi matrices. The thesis fills in the details of two papers by D. E. Rutherford [12; 13]1 , with some translation of his results for determinants to the problem of finding characteristic roots of matrices. Some of the results obtained are briefly discussed relative to certain theorems on bounds for characteristic roots. The first part of the paper is an attempt to show some of the physical sources of the problem and indicate some applications of the results. The aim here is to exhibit matrices of the type considered in the problem and little effort is made to give a detailed theory of the physical laws involved. A background for the study is the purpose of the next section. A rather general treatment of theorems concerning characteristic roots is given, especially with respect to real symmetric matrices. More recent discoveries concerning bounds of characteristic roots are given. Proofs are either sketched briefly or omitted entirely since most of the theorems are well known. The main problem is introduced by solving one of the physical problems discussed in the section on applications. The results obtained are used as a basis for a more generalized treatment With the aim of finding the characteristic roots of several classes of n by n matrices. The matrices to be considered are at best somewhat restricted in scope. This seems to limit the usefulness of the results. However, it is feasible from a practical viewpoint to reduce any real symmetric matrix to a Jacobi matrix in such a way as to preserve the characteristic roots. This fact, [6] , gives added value to the results obtained. Aside from the actual solution of certain physical problems, it is hoped that results here obtained might be of some importance for at least two reasons: first, it makes available several classes of matrices of order n for which the proper values can be obtained from tables of trigonometric functions and, secondly, the methods used here might suggest an approach to the solution of the problem for more general real symmetric matrices . Much remains to be done, even in the special case of the Jacobi matrix.

In the literature of ideals a left ideal L of a system S has usually been defined by the inclusion of SL ⊆ L and a right ideal R by the inclusion RS ⊆ R. On the other hand a left zero element z is usually defined by the equation za = z and a right zero element by the equation az = z ; similarly a left or right identity element e is defined by the equation ea = a or ae = a respectively. The author has in this paper made the terms left and right when applied to ideals consistent with the use of the terms left and right when applied to zeros or identities; thus L is a left ideal of a multiplicative system S if LS ⊆ L ; and R is a right ideal of S if SR ⊆ R. The reader is therefore cautioned to interchange the words left and right throughout this paper when referring to other literature on the subject. If such an interchange of words is effected, no change in the symbolism is required.

There have been many contributions of work in different fields of convexity giving various metrical properties of convex sets. In this thesis we shall consider some further ideas which seem interesting to study. A standard way of tackling certain types of problems is to prove the existence of an 'extremal' convex set with respect to the property in consideration and by a series of arguments determine its construction. Generally speaking the extremal set turns out to be regular in some sense with a correspondingly easy geometry. In Chapters 1 and 2 we shall concern ourselves entirely with polytopes and we shall give some results on the metric properties of their faces. Following these results, we shall in Chapter 3 consider some continuity properties of the more general class of cell-complexes. In Chapters 4, 5 and 6, we shall confine ourselves to the plane. In Chapter 4, we shall consider sets which incertain senses correspond to the sets of constant width. This leads us in Chapter 5 to give some results concerning the minimal widths of triangles circumscribing convex sets. Finally, in Chapter 6 we consider the areas of certain subsets of a convex set which are determined by partitions of that set by three concurrent lines. Papers which are relevant to the field of study in a particular chapter are mentioned briefly in an introduction to that chapter.

In this dissertation we consider the minimax approximation of functions f(x) E"C[O, l] rotated about the origin, and the characterization of the optimal rotation, a*, of f in the sense of least minimax error over all possible rotations. The paper divides naturally into two sections: a) Existence, uniqueness, and characterization for unisolvent minimax approximation for each rotation a of f. These results are applications of Dunham (1967). b) Existence, non-uniqueness, and com.putation of a*; derivation of necessary conditions for the minimax [TRUNCATED]

The purpose of this paper is to discuss several tests of statistical normality. By normality it is meant that a simple random sample is drawn from a population with a normal or Gaussian distribution. There are a number of such tests in existence. For example, the chi-squared (CS), √b1 , and b2 are reasonably well known. Others, such as Geary's (a) are not as popular. The discussion of tests of normality has been quite thorough in the various journals. However, they have not brought together, with a complete discussion including exrunples and comparisons. The discussion will proceed in four parts. The effect of non-normality will be briefly reviewed. Several examples are indicated here, such as the t-statistic and prediction intervals where non-normality alters the significance level. Some of the tests where non-normality is not too harmful will also be indicated. [TRUNCATED]

In this paper we have determined the structure of divisible groups, certain primary groups, and countable torsion groups. Chapter 1 introduces two important infinite abelian groups, R and Z(p^∞). The structure of these groups is completely known and we have given most of the important properties of these groups in Chapter 1. Of special importance is the fact that a divisible group can be decomposed into a direct sum of groups each isomorphic to R or Z(p^∞). This is Theorem 2.12 and it classifies all divisible groups in terms of these two well-known groups. Theorem 1.6 is of great importance since it reduces the study of torsion groups to that of primary groups. We now have that Theorems 3.3 and 5.5 apply to countable torsion groups as well as primary groups. Theorem 3.3 gives a necessary and sufficient condition for an infinite torsion group to be a direct sum of cyclic groups. These conditions are more complicated than the finite case. From Theorem 3.3, we derived Corollary 3.5. This result is used later on to get that the Ulm factors of a group are direct sums of cyclic groups. In essence, Ulm's theorem says that a countable reduced primary group can be determined by knowing its Ulm type and its Ulm sequence. Now by Corollary 3.5, we have only to look at the number of cyclic direct summands of order p^n (for all integers n) for each Ulm factor. This gives us a system of invariants which we can assign to the group. Once again, these invariants are much harder to arrive at than in the finite case.

This paper discusses the following problem of approximation theory: a continuous function f(x) is to approximated over an interval alpha

This paper deals with methods of Chebyshev approximation. In particular polynomial approximation of continuous functions on a finite interval are discussed. Chapter I deals with the existence and uniqueness of Chebyshev or C-polynomials. In addition, some properties of the extremal points of the error function are derived, where the error fUnction E(x) = f(x) - p(x), p(x) being the C-polynomial. Chapter II discusses a method for finding the C-polynomial of degree n--the exchange method. After choosing a set of n+2 distinct abscissas, or a reference set, the so-called levelled reference polynomial is computed by the method of divided differences or by using the approximation errors of this polynomial. A point xj of maximal error is obtained and introduced into a new reference. A new levelled reference polynomial is then computed. This process continues until a reference is gotten, whose reference deviation equals the maximal approximating error of the levelled reference polynomial. The reference deviation is the common absolute value of the levelled reference polynomial at each of the reference points. The levelled reference polynomial for this reference is then shown to be the desired C-polynomial. Chapter III deals with phase methods for constructing the a-polynomial. It is shown that under suitable restrictions, if a Pn, A and €(phi) can be found such that the basic relation f(cos phi) = Pn(cos phi) + A cos[(n+1)phi + E(phi)] is satisfied on the approximation interval, then Pn is the a-polynomial. Two methods for finding the amplitude A and the phase function €(phi) are discussed. The complex method assumes f to be analytic on a domain and uses Cauchy's integral formula to obtain new values of €(phi), starting with a set of initial values. These values in turn generate new values of Pn and A. The values of Pn as well as values of A and €(phi) at certain points are gotten through convergence of this iterative scheme. Then an interpolation formula is used to obtain Pn from its values at these points. The second method attempts to find A, €(phi) and Pn so as to satisfy the basic relation only on a discrete set of points. First, assuming €(phi) so small that cos €(phi) may be replaced by 1, an expression is obtained for Pn(cos phi). In the general case, a system of phase equations is given, from which €(phi), A and hence Pn may be obtained. Although these results are valid only on a discrete set of points in the approximation interval, the polynomial derived in this way represents a good approximation to f(x).

The purpose of this thesis was to examine the Hilbert Nullstellensatz and some of its more important corollaries. We approached the problem, working primarily with varieties and ideals. The major result is that the ideal associated with the variety of any ideal is the radical of that ideal. Definitions of the terms may be found in the text. It was necessary to develop considerable machinery in order to accomplish the proof. The preliminary details are done at a leisurely pace, with some propositions brought in for their own interest, rather than any application to our future proof. Propositions, which we consider only tools, will be labelled lemmas. Propositions, which are particularly important as independent results, are labelled theorems. Every effort was made to have the paper itself contained, although some experience with field extensions and polynomial rings is presupposed.

This paper is divided into five sections. It is concerned with the derivation and application of a formula known as Quadrature by Differentiation. Section One derives the basic formula by applying integration by parts to a suitably chosen 2n^th. degree polynomial. By applying this method to a polynomial of degree m + n, Hummel and Seebeck's Generalized Taylor Expansion is obtained and shown identical with the Quadrature Formula when m is set equal to n. Finally the quadrature approximation is proved convergent if f(x) is analytic in a certain domain of the complex plane. Section Two deals with the representation of certain elementary functions using quadrature methods. These expansions, because they have integer coefficients and appear in a rational form, are far easier to compute than the corresponding Maclaurin Series with the same degree of accuracy. Section Three uses quadrature methods to solve ordinary differential equations whose boundary data are given at a single point. The method that is devoloped is a variation of the predictor corrector type. It is very accurate and is easily extended to solve almost every type of initial value problem. Section Four treats the linear "Two Point" and eigenvalue problem. This is accomplished by transforming the given differential equation into a system of linear algebraic relationships between the known and unknown boundary conditions. This section also deals briefly with the non linear "Two Point Problem" suggesting a iterative method, based on the results of Section Three, to obtain the missing boundary data. Section Five improves on something that Quadrature by Differentiation already is; an accurate integration formula. This is achieved by replacing derivatives with central differences. The final result is three integration formulas based only on the tabular values of the function being integrated. Since these formulas are derived using the basic interval, xg< x < xg + h, integration can be extended into s successive intervals using the same or different values of h.

In this paper we consider second order parabolic partial differential equations on the infinite strip. We confine our attention to the quasi-linear equations, and in particular, the problem (0-1) u͙(t,x) = f(u)uxx(t.x) , (t,x) ε(0,T)x(-∞,∞) ≡ DT, T>0 u(0,x) = u0(x) , x ε (-∞,∞). Existence, uniqueness and properties of the solution are discussed. These results can be immediately generalized to problems where the differential equation is of the form u͙(t,x) = f(t,x,u)u xx(t,x).

The purpose of this paper is to exhibit the main properties of Gamma and Beta distributions and show their relation to certain well known distributions. In chapter II the Gamma and Beta distributions are defined in terms of Gamma and Beta functions. The moments of these distributions are calculated, and the moment generating function and cumulant generating function for the Gamma distribution are obtained. The curves are classified with respect to parameter values and the curves are graphically illustrated in Figures 1, 2, and 3. The exponential distribution, as a special case of interest, is shown to be a Gamma distribution with parameter one.

In summary, it should be stressed that the numerous examples of this paper are important only in so much as they illustrate the use of the devices and concepts emphasized in the study. It is our premise that plane geometry, as usually taught, does not possess the value which it might have. Certain changes in content as well as in presentation would materially improve the course. This study, then, presents suggestions for improving the ordinary course in two directions. First, by presenting logical propositions from the students' experiences which are partially isomorphic to the theorems of plane geometry, it is believed that the student will appreciate and better be able to use the principles of deductive logic. Secondly, by placing considerable emphasis upon the concept of variation, the student will gain a greater knowledge of mathematical method and its relation to procedures of empirical science.

It is the purpose of this paper to show how vector methods can be applied to the geometry of the triangle. Because of the unlimited amount of material regarding the triangle, the discussion herein is restricted to those considerations involving only points or straight lines, or combinations of the two.

The field of knowledge known as mathematics is composed of the totality of mathematical systems. A mathematical system consists of a body of propositions known as the axioms. The concern of this paper is with the axioms; in fact, with the change which has taken place during the centuries in the ideas of mathematicians about the axioms.

The purpose of this paper is to study two particular entire functions which satisfy the conditions set up in a theorem due to Newsom. It is to be hoped that this report may be preliminary to the invention of a method which will lend itself toward the solution of certain general problems.

The essential purpose of this paper is to obtain further information in regard to the asymptotic representation of a power series.

The purpose of this paper is the investigation and classification of regular planar graphs. The motive behind this investigation was a desire to better understand those properties which allow a graph to be represented in the plane in such a manner that no two edges cross except perhaps at vertices.

This paper is a discussion of multifunctions, various types of continuity defined on multifunctions, and implications of continuity for the range and domain sets of the multifunctions.

The problem with which this paper is concerned is that of investigating a class of topological properties commonly called separation properties. A topological space which satisfies only the definition may be very limited in open sets. By use of the separation properties, specific families of open sets can be guaranteed.

The problem and purpose of this paper is to develop Lane's Integral in two-space, and then to expand these concepts into three-space and n-space. Lane's Integral can be used by both mathematicians and statisticians as one of the tools in the calculation of certain probabilities and expectations. The method of presentation is straightforward with the basic concepts of integration theory and Stieltjes Integral assumed.

This paper will be devoted to an exposition of some of the relationships existing between a field and certain of its extension fields. In particular, it will be shown that many fields may be characterized rather simply in terms of their subfields which, in turn, may be directly correlated with the subgroups of a finite group of automorphisms of the given field.

One of the classic theorems concerning the real numbers states that every open cover of a closed and bounded subset of the real line contains a finite subcover. Compactness is an abstraction of that notion, and there are several ideas concerning it which are equivalent and many which are similar. The purpose of this paper is to synthesize the more important of these ideas. This synthesis is accomplished by demonstrating either situations in which two ordinarily different conditions are equivalent or combinations of two or more properties which will guarantee a third.

This paper is concerned with equations in which all derivatives are ordinary rather than partial derivatives. The customary meanings of differential order and difference order of an equation are observed.

The purpose of this paper is to construct the real number system. The foundation upon which the real number system will be constructed will be the system of counting numbers.

The purpose of this paper is to study convergence of certain continued fractions.

The purpose of this paper is to make a detailed study of vector spaces and a certain vector-like system.

The purpose of this paper is to examine the properties of the envelope and the extraneous loci associated with the solution curves of ordinary differential equations of the first order and degree greater than one.

This paper is the result of a study of triangular matrices with particular emphasis on those which are convergence preserving transformations.

In the category of mathematics called partial differential equations there is a particular type of problem called the Dirichlet problem. Proof is given in many partial differential equation books that every Dirichlet problem has one and only one solution. The explicit solution is very often not easily determined, so that a method for approximating the solution at certain points becomes desirable. The purpose of this paper is to present and investigate one such method.

The purpose of this paper is to investigate Saunders MacLane's axioms for plane geometry. The wording of the axioms has been modified; however, the concept suggested by each axiom remains the same.

It may be said of certain pairs of elements of a set that one element precedes the other. If the collection of all such pairs of elements in a given set exhibits certain properties, the set and the collection of pairs is said to constitute a partially ordered set. The purpose of this paper is to explore some of the properties of partially ordered sets.

This paper consists of a study of the direct sum U of two rings S and T. Such a direct sum is defined as the set of all ordered pairs (s1, t1), where s1 is an arbitrary element in S and t1 is an arbitrary element in T.

This paper is primarily concerned with developing the theory of real-valued functions of bounded variation and those ideas which are closely related to this main topic. In addition to this, some emphasis has been placed on the relationship of the theory of functions of bounded variation to specific areas of analysis. In particular, integration theory has been chosen as the vehicle to demonstrate this connection.

The purpose of this paper is to present some fundamental properties of algebraic semigroups. The development of the theory of semigroups has appeared for the most part in the past few years of this century. A semigroup is the result of a weakening of the axioms for a group. Thus all groups are semigroups. That the study of semigroups is very closely related to the abstract study of general transformations is, perhaps, one of the reasons for the rapid development of semigroup theory.

This paper deals with the development of the convolution ring and the construction of a field from this ring.

The purpose of this paper is to study two differential equations. A method of approximation by iteration is used to define sequences of functions which converge to solutions of these equations. Some properties of the solutions are proved for general boundary conditions and certain special solutions are studied in detail.

The purpose of the paper is to prove that the Lp spaces, p ≥ 1, of equivalence classes of functions are Banach spaces.

The purpose of this paper is to develop some properties of uniformly locally compact spaces. The terminology and symbology used are the same as those used in General Topology, by J. L. Kelley.

The problem with which this paper is concerned is that of defining the Riemann-Complete Integral and comparing it with the Riemann and the Lebesgue Integrals.

This thesis is an investigation of some of the properties of polynomial rings, unique factorization domains, Euclidean domains, and principal ideal domains. The nature of some of the relationships between each of the above systems is also developed in this paper.

This paper establishes a basis for abelian categories, then gives the statement and proof of two equivalent definitions of an abelian category, the development of the basic theory of such categories, and the proof of some theorems involving this basic theory.