binary number system
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1: 24.1 Special Notation
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Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►Euler Numbers and Polynomials
… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. … ►Various systems of notation are summarized in Adrian (1959) and D’Ocagne (1904).2: 27.18 Methods of Computation: Primes
§27.18 Methods of Computation: Primes
… ►An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … ►An alternative procedure is the binary quadratic sieve of Atkin and Bernstein (Crandall and Pomerance (2005, p. 170)). … ►These algorithms are used for testing primality of Mersenne numbers, , and Fermat numbers, . …3: 3.1 Arithmetics and Error Measures
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§3.1(i) Floating-Point Arithmetic
►Computer arithmetic is described for the binary based system with base 2; another system that has been used is the hexadecimal system with base 16. ►A nonzero normalized binary floating-point machine number is represented as … ►In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) (, , , ), binary64 (previously double precision) (, , , ) and binary128 (previously quad precision) (, , , ) are as in Figure 3.1.1. … ►Computer algebra systems use exact rational arithmetic with rational numbers , where and are multi-length integers. …4: Bibliography B
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A Fortran-90 based multiprecision system.
ACM Trans. Math. Software 21 (4), pp. 379–387.
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Polynomials defined by a difference system.
J. Math. Anal. Appl. 2 (2), pp. 223–263.
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Error estimates for the solution of linear systems.
SIAM J. Sci. Comput. 21 (2), pp. 764–781.
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Density profiles in confined critical systems and conformal invariance.
Phys. Rev. Lett. 66 (7), pp. 895–898.
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The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms.
Dover Publications, New York.
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5: 27.15 Chinese Remainder Theorem
§27.15 Chinese Remainder Theorem
►The Chinese remainder theorem states that a system of congruences , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod ), where is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits. …6: 27.2 Functions
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►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing .
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►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).)
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►Such a set is a reduced
residue system modulo .
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§27.2(ii) Tables
…7: 20.12 Mathematical Applications
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§20.12(i) Number Theory
… ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). For an application of a generalization in affine root systems see Macdonald (1972). … ►The space of complex tori (that is, the set of complex numbers in which two of these numbers and are regarded as equivalent if there exist integers such that ) is mapped into the projective space via the identification . …8: Mourad E. H. Ismail
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► Suslov), Kluwer Academic Publishers, 2001; Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (with F.
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►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive
Approximation; Journal of Approximation Theory; and Integral Transforms
and Special Functions.
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9: Morris Newman
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►Department of Commerce Gold Medal in 1966 for his work on algorithms for solving integral linear systems exactly by using congruence techniques.
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►He served as Associate Editor for Combinatorics and Number Theory for the DLMF project.
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10: Barry I. Schneider
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►Schneider’s research interests spanned a broad number of areas of theoretical chemistry, atomic and molecular physics, numerical methods and high performance computing.
…Schneider served as Chair and Co-Chair of the APS Division of Computational Physics and the Topical Group on Few-Body Systems and Multipartical Dynamics and has been the organizer of a number of conferences and invited sessions in the US and abroad.
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