24 results on '"Mathematics"'
Search Results
2. Algorithm 480 Procedures for Computing Smoothing and Interpolating Natural Splines [E1].
- Author
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Lyche, Tom and Schumaker, Larry L.
- Subjects
- *
SMOOTHING (Numerical analysis) , *NUMERICAL analysis , *ALGORITHMS , *MATHEMATICAL formulas , *EXTREMAL problems (Mathematics) , *MATHEMATICAL analysis , *MATHEMATICS , *SPLINES , *RINGS of integers - Abstract
The article presents the procedures for computing, smoothing and interpolating natural splines. Several mathematical formulae with mathematical and numerical analyses are cited. Problems indicating the application of smoothing and interpolating are discussed. The purpose of the procedure is to determine the coefficients in the representation of a natural spline of degrees in terms of a local basis. The procedure is based on algorithms. The parameters maxit should be a positive integer specifying the maximum number of iterations desired in computing the problem. Illustrations in applying such mathematical formulae are offered.
- Published
- 1974
- Full Text
- View/download PDF
3. Algorithms.
- Author
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Herriot, J. G.
- Subjects
- *
ALGORITHMS , *FOURIER transforms , *ORTHOGONALIZATION , *FOURIER series , *MATHEMATICAL analysis , *PERMUTATIONS , *HARMONIC functions , *MATHEMATICS , *ALGEBRA - Abstract
The article presents the procedures for computing the finite Fourier transform of a complex data vector which is based on the Cooley-Tukey algorithm. An overview of the complextransform and realtransform's procedures are offered. Either the complex Fourier transform or its inverse are the procedure computed by complextransform, while realtransform computes either the Fourier coefficients of a sequence of real data points or evaluates a fourier series. The topic also includes orthogonal transformation, permutations, and spectral analysis.
- Published
- 1968
4. Eliminating Monotonous Mathematics with FORMAC.
- Author
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Tobey, Robert G. and Graham, R. M.
- Subjects
- *
COMPILERS (Computer programs) , *COMPUTER software , *MATHEMATICAL analysis , *SYSTEMS software , *PROGRAMMING languages , *MATHEMATICS - Abstract
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]
- Published
- 1966
- Full Text
- View/download PDF
5. Applications of Differential Equations in General Problem Solving.
- Author
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Klopfenstein, R. W.
- Subjects
- *
DIFFERENTIAL equations , *NUMERICAL analysis , *CALCULUS , *BESSEL functions , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
A large class of problems leading to digital computer processing can be formulated in terms of the numerical solution of systems of ordinary differential equations. Powerful methods are in existence for the solution of such systems. A good general purpose routine for the solution of such systems furnishes a powerful tool for processing many problems. This is true from the point of view of ease of programming, ease of debugging, and minimization of computer time. A number of examples are discussed in detail. [ABSTRACT FROM AUTHOR]
- Published
- 1965
- Full Text
- View/download PDF
6. Finding Zeros of a Polynomial By the Q-D Algorithm.
- Author
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Henrici, P., Watkins, Bruce O., and Downing, Jr., A. C.
- Subjects
- *
NUMERICAL analysis , *POLYNOMIALS , *ALGEBRA , *MATHEMATICAL analysis , *MATHEMATICS , *STOCHASTIC convergence - Abstract
A method which finds simultaneously all the zeros of a polynomial, developed by H. Rutishauser, has been tested on a number of polynomials with real coefficients. This slowly converging method (the Quotient-Difference (Q-D) algorithm) provides starting values for a Newton or a Bairstow algorithm for more rapid convergence. Necessary and sufficient conditions for the existence of the Q-D scheme ore not completely known; however, failure may occur when zeros have equal, or nearly equal magnitudes. Success was achieved, in most of the cases tried, with the failures usually traceable to the equal magnitude difficulty. In some cases, computer roundoff may result in errors which spoil the scheme. Even if the Q-D algorithm does not give all the zeros, it will usually find a majority of them. [ABSTRACT FROM AUTHOR]
- Published
- 1965
- Full Text
- View/download PDF
7. Algorithms.
- Author
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Herriot, J. G.
- Subjects
- *
LEGENDRE'S functions , *ALGORITHMS , *MATHEMATICS , *MATHEMATICAL functions , *ALGEBRA , *DIFFERENTIAL equations , *COMPUTER systems , *ARITHMETIC , *MATHEMATICAL analysis - Abstract
The article presents mathematical procedures using Legendre functions for cases involving arguments larger than one. The integer Legendre 1 procedure is said to generate the associated Legendre functions of the first kind. Integer Legendre 2 generates associated Legendre functions of the second kind to d significant digits. The integer Legendre 3 generates the Legendre functions of the second kind to d significant digits and stores in the array Q. Legendre 1 procedure evaluates the Legendre functions to d significant digits. Other procedures include Legendre 2, conical and toroidal.
- Published
- 1965
8. Algorithm 467 Matrix Transposition in Place [Fl].
- Author
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Brenner, Norman
- Subjects
- *
MATRICES (Mathematics) , *ALGORITHMS , *PERMUTATIONS , *MATHEMATICS , *ALGEBRA , *MATHEMATICAL analysis , *COMBINATORIAL group theory , *VECTOR algebra , *VECTOR analysis - Abstract
The article discusses matrix transposition in place. Several algorithms have been used for solving the problem of transposing a rectangular matrix in place. A significantly speedier algorithm is described, based on a number of theoretical analysis, and experimentally compared with existing algorithms. The theory calls for the storing of a matrix in a vector in one of two ways, either rowwise or columnwise. The matrix is transposed by its conversion from one mode of storage to the other through permutation.
- Published
- 1973
- Full Text
- View/download PDF
9. Algorithm 481 Arrow to Precedence Network Transformation [H].
- Author
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Crandall, Keith C.
- Subjects
- *
BILINEAR transformation method , *ALGORITHMS , *PATH integrals , *RESOURCE allocation , *MATHEMATICAL analysis , *NUMERICAL analysis , *MATHEMATICS , *SCHEDULING , *MATHEMATICAL ability - Abstract
The article discusses the mathematical concept called arrow to precedence network transformation. Many of the application programs in the area of critical path scheduling and resource allocation are written for the precedence networking convention. The method of transforming arrow convention networks into precedence convention is required, because only few of these programs admit networks defined by arrow convention directly. The algorithm generates the required transformation by creating a list of followers for each non-dummy arrow. The logic used in the transformation can be used to create a list of predecessors if they are desirable.
- Published
- 1974
- Full Text
- View/download PDF
10. Algorithms.
- Author
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Herriot, J. G., Bray, T. A., and Witzgall, C.
- Subjects
- *
ALGORITHMS , *NUMERICAL analysis , *MATHEMATICAL analysis , *EQUATIONS , *MATHEMATICS , *ALGEBRA - Abstract
This article presents procedures for solving of equations related to algorithm. Procedure determines the least-cost flow over an upper and lower bound capacitated flow network. Each directed network arc a is defined by nodes I[a] and J[a], has upper and lower flow bounds hi{a] and lo[a], and cost per unit of flow cost[a]. Costs and flow bounds may be any positive or negative integers. An upper flow bound must be greater than or equal to its corresponding lower flow bound for a feasible solution to exist. There may be any number of parallel arcs connecting any two nodes.
- Published
- 1968
11. Values and Shooting Times in Noisy Duels.
- Author
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Fox, Martin and Kimeldorf, George S.
- Subjects
- *
CIRCLE-squaring , *DUELING , *DISTRIBUTION (Probability theory) , *PROBABILITY theory , *ESTIMATION theory , *MATHEMATICS , *MATHEMATICAL combinations , *MATHEMATICAL statistics , *STATISTICS , *MATHEMATICAL analysis - Abstract
A noisy duel is a zero-sum, two-person game with the following structure. Each player has bullets which he can fire at any time in [0, 1]. If Player i fires at time t, he hits with probability Pi(t). The functions Pi are continuous and nondecreasing with P[sub i](0) = 0 and P[sub i](l) = 1. The number of bullets each player possesses at any time and the functions Pi are known to both. The payoff is I to the sole survivor, otherwise 0. This article states the authors' earlier result on the existence of a value of a noisy duel and presents a detailed discussion of the structure of epsilon-good strategies and criteria for the existence of good first-shot times. We present tables of values and shooting times for noisy duels, which, in some cases, can be used to trace the play of the game. An additional table illustrates how large an arsenal is necessary to overcome the effects of an opponent's superior accuracy. [ABSTRACT FROM AUTHOR]
- Published
- 1970
- Full Text
- View/download PDF
12. An Alternative to Satterthwaite's's Test Involving Positive Linear Combinations of Variance Components.
- Author
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Howe, Richard B. and Myers, Raymond H.
- Subjects
- *
APPROXIMATION theory , *REGRESSION analysis , *ANALYSIS of variance , *LINEAR statistical models , *DEGREES of freedom , *STATISTICS , *MATHEMATICS , *STATISTICAL hypothesis testing , *MATHEMATICAL analysis , *ALGEBRA - Abstract
This article proposes a refinement of the usual approximation due to Satterthwaite for making a significance test when the test statistic involves a single mean square in the numerator and a linear combination of mean squares in the denominator, that linear combination involving known positive constants. The test procedure is outlined in detail, and its derivation given in a separate section. The development of the technique essentially involves determining, by successive approximations, a function which serves as the test criterion. This function is given through terms in the cubes of the reciprocals of the degrees of freedom involved in the problem, thereby giving a deviation from the advertised significance level which is smaller (in magnitude) than the deviation provided by the Satterthwaite procedure. An expression approximating the power of the test is also developed. [ABSTRACT FROM AUTHOR]
- Published
- 1970
- Full Text
- View/download PDF
13. Ideal Multiple-choice Items.
- Author
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Weitzman, R. A.
- Subjects
- *
MULTIPLE choice examinations , *MULTIPLE comparisons (Statistics) , *EXAMINATIONS , *MATHEMATICAL analysis , *PROBABILITY measures , *MATHEMATICS , *MATHEMATICAL economics , *LOGIC ,QUESTIONS & answers - Abstract
A multiple-choice item is called ideal if all its alternatives are equally attractive to every person who cannot answer the item correctly without guessing. This paper discusses reasons for attempting to construct ideal items, develops methods for testing whether items are ideal or at least more nearly ideal than other items, and presents examples of both ideal and nonideal items. [ABSTRACT FROM AUTHOR]
- Published
- 1970
- Full Text
- View/download PDF
14. Note on the Cochran Q Test.
- Author
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Tate, Merle W. and Brown, Sara M.
- Subjects
- *
STATISTICAL correlation , *STATISTICS , *STATISTICAL sampling , *MATHEMATICS , *PROBABILITY theory , *MATHEMATICAL combinations , *STATISTICAL hypothesis testing , *MATHEMATICAL analysis , *LOGIC - Abstract
Cochran's Q test for differences between related-sample percentages or proportions has generally been incorrectly presented in secondary sources. The most common mistake results from failure to recognize that rows containing only 1's or only O's, i.e., only successes or only failures, do not affect the value of Q. The F test, however, is affected by such rows. The probabilities from the x[sup 2] and F approximations are compared with the exact probabilities in three sets of data. A rule of thumb, based on extensive study of the distribution of Q in small samples, is given as an aid in judging when the x[sup 2] approximation is satisfactory for practical purposes. [ABSTRACT FROM AUTHOR]
- Published
- 1970
- Full Text
- View/download PDF
15. ON AN EXTREME RANK SUM TEST WITH EARLY DECISION.
- Author
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Chun, D.
- Subjects
- *
MATHEMATICAL analysis , *HYPOTHESIS , *STATISTICS , *NUMERICAL analysis , *MATHEMATICAL sequences , *MATHEMATICS - Abstract
Where Youden's extreme rank sum test is used, an early decision is often possible by performing the trials in sequence with the following method. After each trial is completed, the greatest lower bound and least upper bound of both extreme rank sums are calculated for all remaining trials. If the critical regions cover either or both intervals of the statistics or the intervals lie completely outside the critical regions, the round-robin test is terminated with the acceptance of the proper hypothesis. [ABSTRACT FROM AUTHOR]
- Published
- 1965
- Full Text
- View/download PDF
16. A Localization Theorem for Optimal Facility Placement.
- Author
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Goldman, A. J. and Witzgall, C. J.
- Subjects
- *
LOCALIZATION theory , *OPTIMAL designs (Statistics) , *DISTANCES , *CONSUMERS , *STATISTICAL hypothesis testing , *FACILITY management , *MATHEMATICS , *MATHEMATICAL analysis - Abstract
The problem is that of locating a facility in a region, so as to minimize the weighted sum of distances between customers and facility. This note shows that a subregion containing half or more of the total 'customer-weight,' if it obeys an additional restrictive hypothesis, must include at least one optimal location for the facility. Thus the search for an optimal location can be limited to such a subregion, once identified. [ABSTRACT FROM AUTHOR]
- Published
- 1970
- Full Text
- View/download PDF
17. Algorithm 466 Four Combinatorial Algorithms [G6].
- Author
-
Ehrlich, Gideon
- Subjects
- *
ALGORITHMS , *PERMUTATIONS , *COMBINATORICS , *MATHEMATICS , *MATHEMATICAL combinations , *PERMANENTS (Matrices) , *ALGEBRA , *MATHEMATICAL analysis , *COMBINATORIAL group theory - Abstract
The article discusses four combinatorial algorithms. The different algorithms, by successive calls, produce a sequence of all combinatorial configurations belonging to the appropriate type, such as PERMU for permutations, COMBI for combinations of natural numbers, COMPOMIN for the composition of an integer, and COMPOMAX, which is the same as COMPOMIN but with maximum for each term. The four algorithms uses a single operation for producing a new configuration from the old one, in which PERMU is transformed by a single transposition of two adjacent elements, COMBI by replacing a single element x by a y, and COMPOMIN by changing the values of two adjacent terms.
- Published
- 1973
- Full Text
- View/download PDF
18. Algorithm 482 Transitivity Sets [G7].
- Author
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McKay, John and Regener, E.
- Subjects
- *
SET theory , *ALGORITHMS , *CONCEPTS , *NUMERICAL analysis , *MATHEMATICAL analysis , *RINGS of integers , *MATHEMATICAL formulas , *MATHEMATICS , *MATHEMATICAL ability - Abstract
The article discusses the mathematical concept called Transitivity Sets. Several mathematical formulae with mathematical and numerical analyses are cited. Problems indicating the application of Transitivity Sets are discussed. The algorithm can cause an infinite integer. It can manifest whenever the corrected point called the centroid of the remaining complex points is given, and every point on the line segment has functional values lower than the functional values at each of the remaining complex points.
- Published
- 1974
- Full Text
- View/download PDF
19. Algorithms.
- Author
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Fosdick, L. D.
- Subjects
- *
ALGORITHMS , *ALGEBRA , *FOUNDATIONS of arithmetic , *MATHEMATICS , *MATHEMATICAL analysis , *COMBINATORICS - Abstract
The article comments on algorithms that offer symbolic expansion of algebraic expressions and prime number generator. The first algorithm expands arbitrarily parenthesized expressions into monomials and has direct distribution without intermediate expansion of lower level expressions. It has also been used as part of algebra programs in theoretical physics. The second algorithm has the intention of giving only the number of primes less than or equal to m. The algorithm has been tested for a large range of values including m =5, 10, 501 and 2000.
- Published
- 1970
20. Truncated Version of a Play-the-Winner Rule for Choosing the Better of Two Binomial Populations.
- Author
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Kiefer, James E. and Weiss, George H.
- Subjects
- *
NUMERICAL analysis , *PROBABILITY theory , *BINOMIAL distribution , *BINOMIAL theorem , *MATHEMATICS , *MATHEMATICAL analysis - Abstract
Results are developed for play-the-winner sampling for choosing the better of two binomial populations when the maximum number of tests is specified. It is shown that for a fixed number of tests the probability of correct selection with alternating assignment exceeds that for play- the-winner sampling. However, when the probability of correct selection is fixed, neither sampling method is uniformly better than the other as measured by the expected number of tests on the poorer population, or on the total expected number of tests. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
21. Negative Moments of Positive Random Variables.
- Author
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Chao, M. T. and Strawderman, W. E.
- Subjects
- *
RANDOM variables , *COMBINATORICS , *NONNEGATIVE matrices , *MATHEMATICAL combinations , *PROBABILITY theory , *MATHEMATICS , *BINOMIAL distribution , *MATHEMATICAL analysis , *POISSON distribution , *MATHEMATICAL variables - Abstract
We investigate the problem of finding the expected value of functions of a random variable X of the form f(X) = (X + A)[sup -n] where X + A > 0 a.s. and n is a non-negative integer. The technique is to successively integrate the probability generating function and is suggested by the well-known result that successive differentiation leads to the positive moments. The technique is applied to the problem of finding E[1/(X + A)] for the binomial and Poisson distributions. [ABSTRACT FROM AUTHOR]
- Published
- 1972
- Full Text
- View/download PDF
22. Algorithm 469 Arithmetic Over a Finite Field [A 1].
- Author
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Lam, C. and McKay, J.
- Subjects
- *
MATHEMATICS , *CALCULATORS , *ALGORITHMS , *NUMERICAL analysis , *NUMERICAL integration , *ALGEBRA , *MATHEMATICAL analysis , *FINITE fields , *IRREDUCIBLE polynomials - Abstract
The article focuses on arithmetic over a finite field. An algorithm for the rational operations of arithmetic over a finite field of elements is presented. The detailed procedure of the algorithmic operation is described. For small values, it is suggested that multiplication tables and addition be generated by the algorithm. The method and its generalization to a multi-step process is also described. A list of primitive irreducible polynomials is provided, as well as other useful information on different functions.
- Published
- 1973
- Full Text
- View/download PDF
23. REMARKS ON SIMULATION OF BOOLEAN FUNCTIONS.
- Author
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Dodd, George G.
- Subjects
- *
BOOLEAN algebra , *MATHEMATICS , *COMPUTER logic , *COMPUTER arithmetic , *EQUATIONS , *ARITHMETIC , *MATHEMATICAL functions , *DIFFERENTIAL equations , *MATHEMATICAL analysis - Abstract
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.
- Published
- 1965
- Full Text
- View/download PDF
24. Mathematical Analysis of Observations (Book).
- Author
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I.R.S.
- Subjects
- *
MATHEMATICAL analysis , *MATHEMATICS , *NONFICTION - Abstract
Reviews the book "Mathematical Analysis of Observations," by B.M. Shchigolev and edited by H. Eagle.
- Published
- 1965
- Full Text
- View/download PDF
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