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The article presents several mathematical games related to ellipse. This game is tracing a perfect ellipse. One example is the two thumbtacks that stick in a sheet of paper, putting a loop of string around them and keep the string taut with the point of a pencil. Moving a pencil around the tacks will trace a perfect ellipse. Another method can be demonstrated with a circular cake pan and a cardboard disk having half the diameter of the pan.

The article provides information about Origami, the Japanese art of folding objects made of paper. It is traditionally an art of folding realistic animals without cutting or decorating from a single of sheet of paper. Many mathematicians became fascinated with paper-folding, in view of its geometrical aspect. An interesting mathematical question is raised by the very act of folding a paper.

The article focuses on the procedures in making hexaflexagons and tetraflexagon out of paper. Diverting six-sided paper structures that can be flexed to bring different surfaces into view is known as hexaflexagons. Tetraflexagon is a three-faced structure. It suggested that papers must be folded according to the number of sides it requires. Tetraflexagon can also be made with four or more faces. It has been stated that all folds of paper are made to conform with the way the lines were originally creased.

Comments on author M.L. Patrick's paper related to numerical mathematics. Suggestion of a modification on parallel algorithm.

The article presents information on the publication of a book "Applied Statistics, Algorithms," relevant to statistics, by the Royal Statistical Society in cooperation with the Science Research Council's Working Party on Statistical Computing. A policy statement describing the editorial policy appears in "Applied Statistics," Vol. 1. No. 1 (1968). A support paper describing the expected contents of the external specification and making recommendation for the layout of algorithms and for programming strategy will appear in the following issue.

A calendar of events for the Fall Joint Computer Conference 1968 technical program is presented. Information about several papers that will be discussed at a symposium including doctor Sanford Elkin's "Reliability, Maintenance, Error Recovery in Third Generation Systems" and doctor Glen Lewis "Applied Mathematics" is presented.

Two recent papers, one by Hansen and one by Hansen and R. R. Smith, have shown how interval Arithmetic (I.A.) can be used effectively to bound errors in matrix computations. In the present paper a method proposed by Hansen and R. R. Smith is compared with straightforward use of I.A. in determinant evaluation. Computational results show the accuracy and running times that can be expected when using I.A. for determinant evaluation. An application using I.A. determinants in a program to test a set of functions to see if they form a Chebyshev system is then presented. [ABSTRACT FROM AUTHOR]

The article presents several mathematical questions.

Neely has discussed computational error generated by some algorithms used to compute various statistics. In the present paper methods are described which are error-free, simple in concept, and usually less costly in machine time than those mentioned by Neely. [ABSTRACT FROM AUTHOR]

The article reports on finding all subgraphs of an undirected graph [H]. A maximal complete subgraph is also called clique which refers to a complete subgraph that is not included in any other complete subgraph. According to the authors, a research paper was released wherein descriptions of techniques to find maximal complete subgraphs are provided. The authors discussed two algorithms using a branch-and-bound technique so that there will be no clique. The first algorithm produces cliques in lexicographic order while the second is derived from the first algorithm.

This paper presents the statistical results of tests of the accuracy of certain arithmetic systems in evaluating sums, products and inner products, and analytic error estimates for some of the computations. The arithmetic systems studied are 6-digit hexadecimal and 22-digit binary floating point number representations combined with the usual chop and round modes of arithmetic with various numbers of guard digits, and with a modified round mode with guard digits. In a certain sense, arithmetic systems differing only in their use of binary or hexadecimal number representations are shown to be approximately statistically equivalent in accuracy. Further, the usual round mode with guard digits is shown to be statistically superior in accuracy to the usual chop mode in all cases save one. The modified round mode is found to be superior to the chop mode in all cases. [ABSTRACT FROM AUTHOR]

Because of the problems created by the explosion of papers in the mathematical sciences and the drawbacks that this places on research, it is suggested that a tree of all mathematical results and terminology be maintained in a multiterminal computer system. Users of the system can store in the computer an updated file of their current knowledge, and on selecting a paper to read, they can obtain from the computer the minimum subtree of theorems required to bring them from what they already know to the background knowledge which the paper assumes. Under certain conditions, means are also provided for the contribution of useful comments by the readers of a work and for interaction between commentators and with the author. This paper describes how the system can be organized and the role required of readers, writers, and commentators. [ABSTRACT FROM AUTHOR]

Examines the phenomenon of the peculiar distribution of first digits. Discussion on the paper by physicist Frank Benford on the distribution of first digits; Papers which appeared following the publication of Benford's law which deny that the first-digit phenomenon is a law of nature; Information Roger Pinkham's theorem on the distribution of first digits.

Presents an analysis of the normalization requirement of the divisor in a divide-and-correct method. Choice of proper higher precision segments of operands for the basic division; Economy of the scheme for small radices; Normalization of the divisor and the final multiplication of the quotient; Computation of the remainder from the final quotient, the dividend, and the original divisor.

The FORMAC (FORmula MAnipulation Compiler) programming system provides a powerful tool for performing mathematical analysis. It is an extension of FORTRAN IV which permits the use of the computer to perform the tedious algebraic computations that arise in many different fields. Among the areas in which it has been successfully used are: differentiation of complicated expressions, expansion of truncated power series, solution of simultaneous equations with literal coefficients, nonlinear maximum likelihood estimation, tensor analysis, and generation of the coefficients of equations in Keplerian motion. These types of analysis—which arose in the solution of specific practical problems in physics, engineering, astronomy, statistics and astronautics—are discussed in the paper. In addition to its usage for specific problem solutions, FORMAC can also be used to automate the analysis phase in certain production programming. Several such applications are presented. [ABSTRACT FROM AUTHOR]

This article presents several developments related to Association of Computing Machinery (ACM) in different parts of the U.S. The Los Angeles Chapter of ACM recently devoted a meeting to short talks by representatives from some of the local Special Interest Groups. The purpose of this "Sigsampler" was to acquaint members with the purposes and activities of the local SIGs. It is hoped that this will spark increased membership in the SIGs and enable members to select the Group most closely aligned with their interests. The article includes details of meetings and symposiums. One of them was a Symposium on Statistics, Governments, and the Analysis of Social Problems that will be held on December 27, 1969 at the Sheraton Boston Hotel. The purpose of the symposium is to explore ways to improve the design of statistical systems for economic and social data so that information about the relationships between various socio-economic conditions may he more readily available. Another event was the 1970 National Telemetering Conference that will be held in Los Angeles, April 27-30, 1970 under the sponsorship of two IEEE professional groups, Communications Technology, Aerospace and Electronic Systems.

The article focuses on the general conference of the Association for Computing Machinery (ACM) for the year 1965. It reports conference chairman professor Jack Belzer's statement on aims and purpose of the conference, and the technical sessions held during the conference. It also informs that ACM has expanded its activities to new areas, establishing several special interest committees (SICs) which include SIC on Numerical Mathematics, Symbol and Algebraic Manipulation and Design Automation.

This article discusses several books on scientific literature and presents developments in the field of science, as of August 29, 1907. The seventh annual meeting of the International Zoological Congress, held at the Harvard Medical School in August 1907, emphasized the new features, which have come to characterize zoology-physiology and mathematics. Professor H.F. Osborn of Columbia University delivered an address on "Evolution as it Appears to the Paleontologist," in which he enunciated some general laws of great significance. The book "Practical Health," L.E. Whipple, principle of the American School of Metaphysics, seeks to simplify the application of doctrines presented for some years past in several editions of his widely read book "Mental Health."

This paper describes an alternate method for summing a set of floating-point numbers. Comparison of the error bound for this method with that of the standard summation method shows that it is considerably less sensitive to propagation of round-off error. [ABSTRACT FROM AUTHOR]

The author comments regarding the method for performing Boolean OR, AND and NOT operations using arithmetic and conditional transfer operations presented by M. Morris Mano. He states that the operations are useful in many cases but they are seriously limited by the OR operation in many ways. He notes that in a Fortran program, not all Boolean functions that contain an OR operation can be placed in the parenthesis of an IF statement. He also reveals that because of the number of necessary conditional transfers, a function which includes all three operations is more difficult to program.

In computer assisted typesetting by means of photocomposition, special problems arise in highly technical material such as mathematical formulas. New solutions to several of these problems have been devised in the information system of the American Institute of Physics. They include: the representation of special characters (foreign alphabets, mathematical symbols. etc.) not available on input keyboards or on the photocomposer; the generation of such symbols. e.g. by overprinting; the precise positioning of accent marks (floating diacritics); line breaks, i.e. words or formulas placed partly at the end of one line and partly at the beginning of the next; and certain aspects of error correction. [ABSTRACT FROM AUTHOR]

The article proposes a mathematical approach that can make numismatics more of an exact science. With this approach the archaeologist can add a degree of precision to his study of coins, and modern governments can improve the procedures with which they control their coinages and paper currency. The designs on coins reveal something of the society's religion or mythology and reflect the evolution of its art.

The article presents various mathematical board games. The Military Game is an example of a two-player game that combines extreme simplicity with extraordinary strategic subtlety. It was popular in French military circles during and after the Franco-Prussian War of 1870-1871. The topological board game Black is easily played as a pencil-and-paper game on a checkered field.

The article explores how public support is achieved for modern programs in mathematics in Hawaii. Hawaii has the ninth largest school system in the nation. The fact that Hawaii is geographically isolated from continental United States has its disadvantages certainly, but there are also some psychological advantages. High transportation costs make attendance at mainland conferences, seminars, institutes and other professional meetings difficult. Teacher interest is the focal point in winning public support for modern programs in mathematics. When schools are planning new programs, the matter of articulation from the elementary school to the intermediate school, from the intermediate school to the high school must be worked out. The Honolulu Advertiser and the Honolulu Star Bulletin, both daily papers, have manifested an interest in the schools. The Head of the Mathematics Department at one of the intermediate school sponsored a mathematics fair which she introduced to the parents through a letter. Modern programs in mathematics have won public support in Hawaii through interested teachers, cooperative administrators, informed parents, enthusiastic students, and an enlightened public.

The article presents a simulation of the computer logic equations using Fortran arithmetic expressions. The simulation involved methods such as writing the equations for the logic for each side of the flip-flop which is an A-C coupled, set-reset device. The logic was implemented using the representative terms A, AN, ONA, and OFA, where A and AN represent the outputs of the flip-flop while ONA and OFA represent the set and reset impulses to A. A and AN are unaltered during an entire pass and at the end of the pass, a subroutine SET is called. A logical presentation of the simulation is presented.

A calendar of events for the Association for Computing Machinery (ACM) from August 1965 to November 1967 is presented. The ACM National Conference will be held in Cleveland, Ohio on August 24-26, 1965. The American Mathematical Society Annual Meeting will be held in Chicago, Illinois on January 24-28, 1966. The Spring Joint Computer Conference will be held in Philadelphia on May 9-11, 1967.

The article announces that Charles Fefferman has been promoted to the professor of mathematics at the University of Chicago.

The article presents information on the experimental results presented by researchers J.J. Kohfeld and G.T. Thompson in their paper on a modification of Reinhard Nordsieck's method for the numerical solution of ordinary differential equations using a multiple precision arithmetic package available on the IBM 7094 at the Ohio State University Computer Center in Columbus, Ohio. The experimental results agree with those presented by Kohfeld and Thompson at h = 0.10 and 0.15 for the Nordsieck and the GSN methods, which is to be expected since round-off error is not critical at those interval lengths.

The author provides an economic outlook for the second half of 1962. He notes that 85 billion U.S. dollars were lost by 17 million investors due to market decline. The author believes that the market decline will have a significant effect on consumer and corporate spending, adding that one of the mathematically unmeasurable factors determining the economic health is the psychological one. He points out that the losses of many investors are still on paper but the psychological impact on their thinking and planning in terms of spending is real.

The article presents several studies conducted in relationship to the mathematics laboratory as a teaching technique. The studies cited, include "A Comparison of Effects of Laboratory and Conventional Mathematics Teaching upon Underachieving Middle School Boys," by M.S. Cohen, and "Arithmetic Achievement as a Function of Concrete, Semi-Concrete, and Abstract Teaching Methods," by R.D. Curry. Also included are "A Study of the Relative Effectiveness of a Meaningful Concrete and a Meaningful Symbolic Model in Learning a Selected Mathematical Principle," by Elizabeth Hammer Fennema, and others.

The fourth International Congress of Mathematicians opened on April 6 in the University of Rome in Rome, Italy, as Senator P. Blaserna, president of the Accademia dei Lincei, the scientific society at whose invitation this congress met at Rome, presided. After the opening exercises, the congress met at the headquarters of the Accademia dei Lincei in the Corsini Palace and chose Blaserna as president. The programme was divided into four sections, namely, arithmetic algebra and analysis; geometry; mechanics; physical mathematics; and Geodesy, with a subsection of applied mathematics; philosophical, historical and didactic subjects.

The article offers information on the findings of Opinion Research Corp., a private testing organization, regarding the ability of clerical workers of business concerns in solving a sum correctly.

Presents a note on author C.J. Cheney's paper 'Nonrecursive List-Compacting Algorithm,' which appeared in the November 1970 issue of the periodical 'Communications of the ACM.' Effect of circular list consisting exclusively of non-items on the flow of the list-compacting algorithm; Structural meaning of the list-compacting algorithm.

The annual general meeting of the American Philosophical Society, the oldest scientific society in America, was held this year in Philadelphia, Pennsylvania, from April 22 to April 24. The meeting was opened on Thursday afternoon by the President, Dr. W.W. Keen, who, with Vice-Presidents A.A. Michelson, W.B. Scott and C.L. Doolittle, presided over the various sessions. The session on Thursday afternoon was devoted to mathematics and physics. Prof. A.A. Michelson, of Chicago, described the recent progress made by him in the ruling of diffraction gratings.

The article offers information on the Marienbad match game in the U.S. It mentions that the game is a variation of one of the most ancient of all two-person mathematical divertissements, and it uses move combinations based on binary numbers. It notes that Harvard mathematician Charles Leonard Bouton called the game Nim, in which the successful player has to memorize the numbers.

Some observations are made on steps to betaken toward the creation of better mathematical software. These steps suggest the need for a coordinated effort and the creation of a center to focus activities in this area. [ABSTRACT FROM AUTHOR]

Presents a computer system to teach portions of an undergraduate course in numerical methods. Classroom-independent course of study; Measures of effectiveness, student acceptance and operational cost; Use of a CDC 6500 with teletype terminals.

Considers the applications of automated algebraic manipulation which have been made in celestial mechanics. Use of Poisson series; Description of a typical problem in perturbation theory; Requirements of processors for use in celestial mechanics.

The article presents various mathematical analysis regarding algorithms. Algorithm 401 procedure generates complex prime numbers while Algorithm 402 procedure sorts mathematical elements into nondescending order. Algorithm 343 contains certain failures during its application to pathological matrices which include incorrect computed eigenvalues resulting to 2 or more percent maximum error. Remarks about Algorithm 372 and 401 are also discussed. Diagrams of the algorithm procedures are also presented.

This article discusses several algorithms applicable to computer programming. In programs for special series summation, the algorithm requires only integer arithmetic and can be implemented in algebraic languages. Changes in Coulomb wave functions that will not completely eliminate the overflow problems encountered in the generation of intermediate arrays are presented.

A new version of the Euclidean algorithm for finding the greatest common divisor of n integers 01 and multipliers xi such that gcd = x1 d1 + … + xn an is presented. The number of arithmetic operations and the number of storage locations are linear in n. A theorem of Lamé that gives a bound for the number of iterations of the Euclidean algorithm for two integers is extended to the case of it integers. An algorithm to construct a minimal set of multipliers is presented. A Fortran program for the algorithm appears as Comm. ACM Algorithm 386. [ABSTRACT FROM AUTHOR]

The article presents algorithms related to matrices and random vectors. One of the algorithms generates the components of random unit vectors distributed uniformly in solid angle. A modification of this algorithm could be used to generate vectors which were azimuthally uniform but have a specific non-uniform distribution in the cosine of the polar angle. Another algorithm presents a set of instructions to perform in-situ transposition of a rectangular matrix.

A method of optimizing the computation of arithmetic and indexing expressions of a Fortran program is presented. The method is based on a linear analysis of the definition points of the variables and the branching and DO loop structure of the program. The objectives of the processing are (1) to eliminate redundant calculations when references are made to common sub-expression values, (2) to remove invariant calculations from DO loops, (3) to efficiently compute subscripts containing DO iteration variables, and (4) to provide efficient index register usage. The method presented requires at least a three-pass compiler, the second of which is scanned backward. It has been used in the development of several FORTRAN compilers that have proved to produce excellent object code without significantly reducing the compilation speed. [ABSTRACT FROM AUTHOR]

Optical and digital holography are reviewed. The mathematical model and computational techniques of the authors' digital holographic process are discussed, and applications of computer holography are suggested. Computer holograms have been made of three-dimensional objects which give faithful reconstructions, even in white light. A new approach based on point apertures for the image is discussed. Photographs of the images reconstructed from digital holograms are presented. [ABSTRACT FROM AUTHOR]

Algorithms are presented for multiplication and division of unsigned integer operands in which the digits normally reserved for signs participate as significant arithmetic digits with positive weight. [ABSTRACT FROM AUTHOR]

In sorting by replacement selecting, the expected length of a sequence beginning with the ith element (i > 1) is proved to be 2F, in accordance with a conjecture of E. H. Friend, where F is the number of memory cells used. The expected length of the jth sequence is determined to be F times a jth degree polynomial in e, such that the value of this polynomial approaches 2 as j approaches infinity. Recursive formulas are obtained for both the mean and the standard deviation of the length of the jth sequence. The mathematical proofs of these results are based upon the assumption that n, the number of items to be sorted, is infinite, but it is shown that the error due to the finiteness of n approaches zero rapidly as n increases. [ABSTRACT FROM AUTHOR]

The article reports on several algorithms with their specific computational dynamics. A specific algorithm with its corresponding formula which provides for the procedure for deriving a certain value is provided. The algorithms include, among others, algorithm 290 for linear equations, algorithm 291 for logarithm of gamma functions, and algorithm 178 for direct search. Moreover, the algorithms are written in the ALGOL 60 Reference Language, Standard FORTRAN, and Basic FORTRAN. Specific instructions on how to submit algorithms are also included.

Various facets of the design and implementation of mathematical expression manipulation algorithms are discussed. Concrete examples are provided by the FORMAC EXPAND and differentiation algorithms, a basic FORMAC utility routine, and an experiment in the extraction of the skeletal structure of on expression. One recurrent theme is the need to avoid excessive intermediate expression swell in order to minimize core storage requirements. Although many details from the FORMAC implementation are presented, an attempt is made to stress principles and ideas of general relevance in the design of algorithms for manipulating mathematical expressions. [ABSTRACT FROM AUTHOR]