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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.

The one-dimensional time-dependent equation of heat conduction is solved analytically for an infinite two-layer circular cylinder whose core may be either hollow or solid. On the outer surface of the cylinder, which has no heat loss due to convection, a constant heat flux from an external source of heating is applied uniformly. The layers are in perfect thermal contact, and there is no heat loss at their interface. At the smaller radius of the inner layer is a perfect insulator so that the heat flux on the inner surface of the two-layer cylinder is zero. The temperature is uniform initially throughout the layers, and the thermal and physical properties of each layer remain constant. Equations for predicting the transient temperature distribution within each layer are determined from the heat-conduction equation by using a method of separation of variables, by homogenizing the boundary conditions, by applying the boundary conditions, and by determining eigenvalues and eigenfunctions. The constants or integration are determined by applying the boundary conditions to the general solutions. The solutions for the temperature distribution in layers 1 and 2, respectively, are and The various terms in these equations are defined in the text. These final solutions are shown analytically to satisfy the heat-conduction equation and the boundary conditions. Appendix A contains a discussion of Bessel functions of the first and second kinds of orders zero and one. Appendix B contains listings of computer programs to determine values of Beta which satisfy the “root equation” and to verify the initial conditions. Some values of Beta are given for a steel-phenolic cylinder. Subroutines are given for computing the Bessel functions of the first and second kinds of orders zero and one. The results may be of general academic interest to those working in heat conduction or studying thermal environments.

The subject of this thesis is topological but the approach is categorical. We consider the approach as important as the subject itself and so we try to indicate whenever we have a categorically defined concept. In fact, we are led by this approach to make definitions and constructions so that certain relationships are categorical in nature. Our coreflective completion functor of Chapter IV is one example of this approach. We wish here to give a brief outline of the historical develop­ment of convergence theory and how category theory has come to play a role in this development...

Let (Xi} be a sequence of independent, identically-distributed random variables with EX2i < ꝏ and E(Xi - EXi)2 = 1.

The problem is to find an efficient algorithm that, given the productions of a context-free grammar G, will discover whether G is LR(k) for given k and if it is build an efficient parser for G . The algorithm is given in Section 8. It is essentially a synthesis of the best parts of Knuth's and DeRemer's algorithms. On simple LR(k) grarranars it yields a result equivalent to DeRemer's algorithm, and like Knuth's algorithm it will work on all LR(k) grammars.

I. The existence of eigenvalues is shown for certain classes of integral equations with continuous kernels. A number of interesting and useful results are thereby treated in a unified and relatively elementary way. The simplicity of these new proofs make the results accessible to introductory courses on the theory of integral equations. II. Collocation with piecewise polynomial functions is developed as a method for solving two-point bour:rlary value problems. Convergence is shown for a general class of linear problems and a rather broad class of nonlinear problems. Some computational examples are presented to illustrate the wide applicability and efficiency of the procedure.

This study is divided into two seemingly disjoint parts -- one containing EMPIRICAL (Bayesian and Non-Bayesian) approach and the second containing INFORMATION-THEORETICAL techniques in problems of statistical estimation and tests of hypotheses. But in the end, both approaches have been brought together for solving ENCODING problems of COMMUNICATION THEORY to unify the whole dissertation.

A convergence function is a correspondence between the filters on a given set S and the subsets of S which specifies which filters converge to which points of S. This concept is defined to include types of convergence which are more general than that defined by specifying a topology on S. Thus a convergence function may be regarded as a generalization of a topology.

The set P(n) of all primes equal to or less than n has the obvious property that it contains exactly one multiple of each prime equal to or less than n. We use this partial description of P(n) as a basis for the following Definition 1.1. An increasing sequence {a1,...,ak} of integers greater than 1 is a coprime chain if it contains exactly one multiple of each prime equal to or less than ak.

The problem of multiple regression analysis where the dependent and independent variables are components of a two dimensional vector is discussed, and a complete statistical development of the solution of estimators for the parameters in the model given. The theory regarding predictions and confidence statements about such predictions is also developed. A computer code was written for the IBM 704 computer which solves the above problem and a description of the code appears in the appendix. The statistical model was applied to a meteorological problem in wind forecasting at the Eniwetok Proving Ground, and prediction equations were developed and evaluated.

The formulation and discussion of the simplest (fixed) end point direct problem of the calculus of Variation is a necessary preliminary to attack on the inverse problems considered in Chapters II and III of this thesis. Since the plane problem is already comprehensively treated in the literature, only enough of its theory is developed here to render intelligible to the reader the inverse problems studied in the sequel.

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.

A written contract to pay a certain amount of money on a specified redemption date, and to pay equal periodical dividends, is called a bond from a mathematical standpoint. The principal mentioned in the contract is its face value, or par value. The amount redeemed, or the redemption value, is denoted by C, the dividends by D, and the principal by F. A bond is redeemed at a par if C and F are the same, and at a premium if C is greater than F. The divided rate, or bond rate, is the interest rate named in the bond. The divided is the interest on the par value at the divided rate. A bond matures on its redemption date.

The purpose of this study is to investigate the nature of the coefficients of certain power series. In particular, it is desired to know what characteristics the coefficients must possess in order that the series shall represent an entire function.

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.

The simplest non-parametric problem of the calculus of variation, the so-called direct problem of the plane, is the problem of finding that arc Co of a family of admissible arcs y=y (x) joining two fixed pointed (x1 , y1 ), (x1, y2) in the x,y-plane such that along the Co the integral takes on a minimum value.

The purpose of this investigation is to prove that the sum of a series of variable terms of a certain type is equal to a constant plus the sum of another series of variable terms. As a result of this proof it may be hoped that certain series, which so far have not been summed, may be shown equal to a constant plus another series for which the sum is known.

It is the purpose of this study to consider certain aspects of the theorem due to Walter B. Ford, may be considered as a function g(w) of the complex variable w=x+iy and as such satisfies two conditions chosen by the author.

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 problem of this study is to determine the relative efficiency of the more important formulae for approximating the rate of interest involved in an amortization plan when the amount of the debt, the amount of the periodic payment, and the number of periods are known. By efficiency this meant the degree of accuracy obtainable, and the time required for solution.

Problems of the Calculus of Variations are generalizations of the familiar minimum problems treated in the differential calculus. The relationships between the ordinary minimum problems of the calculus and the generalizations dealt with in the Calculus of Variations is possibly best seen by examining the general Hilbert-Moore minimum problem, and the special examples of such problems formulated in the table below.

In mathematics, generalization is progress; so much so that oftentimes one loses sight of the fact that generalization is the result of arduous work in the consideration of the particular. In no other branch of mathematics is this better exemplified than in the Calculus of Variations. The beginning of a systematic development of the theory of the Calculus of Variations really started with the two Bernoulli brothers (1654-1748) in their discussion of the brachistochrone problem in 1696. The method devised by them were sufficiently powerful in the attack of a large number of problems. Euler (1707-85) further elaborated the geometrical analytical method of James Bernoulli and it was Langrange (1736-1813) who made it possible to deduce readily the differential equations of the minimizing curves for very general problems.

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 importance of the classical orthogonal polynomials has long been acknowledged. It has not been possible, however, to represent them in such a way that all of their important properties are immediately evident. In particular, the location of the zeros of these polynomials is of considerable interest. This thesis is primarily concerned with a different technique in which Kamke's transformation is applied to the differential equations frequently used to define these polynomials. The resulting trigonometric differential equations cannot be explicitly solved either, but certain characteristics of these solutions facilitate the derivation of approximations to the zeroes of the solutions.

The purpose of this study is the interpretation of α3 by means of a relatively simple formula which will predict the amount of shift in the effective limits of the Type III curve for a given change in the skewness. The methods used are chiefly empirical.

It is the purpose of this study to discuss completely the median of a frequency distribution: first, by showing clearly what a median is; second, by showing the present and possible future applications for the median in the field of mathematical statistics; third, by taking all methods for interpolating for the median which have been advocated in the literature, or which may not have been advocated to date but appear plausible, and discuss these methods as to the way they are used, the plausibility of their use, and the accuracy obtained by their use.

[From the Introduction] One of the most amazing stories to come from the foods business during the past decade is the growth of the frozen foods industry. Its rapid development, after having partially solved many perplexing problems of production and distribution, was somewhat accelerated by higher consumer incomes and wartime scarcities. Equally important is the fact that housewives have been relatively quick to adopt frozen foods. Therefore, today frozen foods are occupying an increasingly important position and the industry no longer considers itself as being in the "infant" stage of development.

Statement of the Problem: It is the problem of this investigation to analyze by use of statistical methods the number of students enrolled at The University of Tennessee from 1899 to 1950, and with the aid of such analysis to forecast possible future enrollments. The data used in this investigation will be limited in that only the resident students of that portion of The University of Tennessee which is located at Knoxville, Tennessee, will be used. Thus, the Colleges, Schools and Experiment Stations located in Memphis, Martin, Jackson, Columbia, Greeneville, Nashville, Crossville and Springfield, and the University extension work carried on throughout the counties of the state are not to furnish data for this study. The analysis of the enrollment data will cover the period beginning 1899 and ending 1950 with particular emphasis being given to the data in the past two decades. With the aid of the results of the analysis, the possible future enrollment will be forecast through 1970. It is essential that, continuously, recognition be given to the vast number of unknown variables entering into a forecast of future college enrollment.