fundamental theorem of arithmetic
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11—20 of 166 matching pages
11: Bibliography M
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Foundations of Finite Precision Rational Arithmetic.
In Fundamentals of Numerical Computation (Computer-oriented
Numerical Analysis), G. Alefeld and R. D. Grigorieff (Eds.),
Comput. Suppl., Vol. 2, Vienna, pp. 85–111.
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The computation of elementary functions in radix
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Computing 53 (3-4), pp. 219–232.
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A -analog of the summation theorem for hypergeometric series well-poised in
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Adv. in Math. 57 (1), pp. 14–33.
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A -analog of the Gauss summation theorem for hypergeometric series in
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Adv. in Math. 72 (1), pp. 59–131.
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CODATA recommended values of the fundamental physical constants: 2002.
Rev. Mod.Phys. 77, pp. 1–107.
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12: Howard S. Cohl
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►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and -series.
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13: 24.10 Arithmetic Properties
14: Bibliography G
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A theorem on the numerators of the Bernoulli numbers.
Amer. Math. Monthly 97 (2), pp. 136–138.
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What every computer scientist should know about floating-point arithmetic.
ACM Computing Surveys 23 (1), pp. 5–48.
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Quantum mechanics: fundamentals.
Second edition, Springer-Verlag, New York.
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Multilateral summation theorems for ordinary and basic hypergeometric series in
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SIAM J. Math. Anal. 18 (6), pp. 1576–1596.
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15: 1.4 Calculus of One Variable
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Mean Value Theorem
… ►Fundamental Theorem of Calculus
… ►First Mean Value Theorem
… ►Second Mean Value Theorem
… ►§1.4(vi) Taylor’s Theorem for Real Variables
…16: 27.11 Asymptotic Formulas: Partial Sums
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►where , .
►Letting in (27.11.9) or in (27.11.11) we see that there are infinitely many primes if are coprime; this is Dirichlet’s theorem
on primes in arithmetic progressions.
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27.11.15
►Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3).
The prime number theorem for
arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if , then the number of primes with is asymptotic to as .
17: 3.1 Arithmetics and Error Measures
§3.1 Arithmetics and Error Measures
… ► … ►§3.1(ii) Interval Arithmetic
… ►§3.1(iii) Rational Arithmetics
… ►18: Leonard C. Maximon
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►Maximon published numerous papers on the fundamental processes of quantum electrodynamics and on the special functions of mathematical physics.
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19: 25.15 Dirichlet -functions
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►This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11).
Related results are:
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25.15.10
20: 18.39 Applications in the Physical Sciences
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►The fundamental quantum Schrödinger operator, also called the Hamiltonian, , is a second order differential operator of the form
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►An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with eigenfunctions vanishing at the end points, in this case see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem.
…Both satisfy Sturm’s theorem.
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►A major difficulty in such calculations, loss of precision, is addressed in Gautschi (2009) where use of variable precision arithmetic is discussed and employed.
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