Results 21 to 30 of about 157,767 (127)
Simultaneous approximation of a real number by all conjugates of an algebraic number
, 2007 Using a method of H. Davenport and W. M. Schmidt, we show that, for each positive integer n, the ratio 2/n is the optimal exponent of simultaneous approximation to real irrational numbers 1) by all conjugates of algebraic numbers of degree n, and 2) by ...
Alain, Guillaume
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Simultaneous rational approximations to algebraic numbers [PDF]
Bulletin of the American Mathematical Society, 1961 Let \(\beta_1, \beta_2, \dots,\beta_n\) be linearly independent numbers of a real algebraic number field of degree \(n+1\). By a study of the units of the field it is shown how to obtain all integral solutions of the set of inequalities: \[ q_0>0, \quad\gcd(q_0,q_1,\dots,q_n)=1,\quad | \beta_j/\beta_0-q_j/q_0| < Cq_0^{-1-1/n} \] for fixed \(C\).
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A Hyper-Dual Number Approach to Higher-Order Derivative Computation
AxiomsThis paper develops a theoretical framework for the computation of higher-order derivatives based on the algebra of hyper-dual numbers. Extending the classical dual number system, hyper-dual numbers provide a natural and rigorous mechanism for encoding ...
Ji Eun Kim
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The Polyhedron-Hitting Problem [PDF]
, 2014 We consider polyhedral versions of Kannan and Lipton's Orbit Problem (STOC '80 and JACM '86)---determining whether a target polyhedron V may be reached from a starting point x under repeated applications of a linear transformation A in an ambient vector ...
Chonev, Ventsislav +2 more
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Normal Shock Wave Approximations for Flight at Hypersonic Mach Numbers
AerospaceNormal shock pressure ratios in equilibrium air for Mach numbers up to 30 and altitudes to 300,000 feet are shown to be correlated by a simple power law which provides an accuracy of ±2%, thereby permitting direct calculation of corresponding enthalpy ...
Pasquale M. Sforza
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Chromatic homotopy theory is asymptotically algebraic
, 2019 Inspired by the Ax--Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects.
Barthel, Tobias +2 more
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Tropical Solution of Discrete Best Approximation Problems
MathematicsWe consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input–output pairs of an unknown function defined on a
Nikolai Krivulin
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Exact Real Arithmetic with Perturbation Analysis and Proof of Correctness [PDF]
, 2015 In this article, we consider a simple representation for real numbers and propose top-down procedures to approximate various algebraic and transcendental operations with arbitrary precision.
Groote, Jan Friso, Keshishzadeh, Sarmen
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Report on some recent advances in Diophantine approximation [PDF]
, 2009 A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex number, as well as
Waldschmidt, Michel
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Explicit Lower Bounds for Rational Approximation to Algebraic Numbers [PDF]
Proceedings of the London Mathematical Society, 1997 The author gives irrationality measures for some algebraic numbers of degree at least 4. A typical example is \(|{\root 4 \of{5}} -p/q |> 0.03 q^{-2.77}\) for \(p\), \(q\in \mathbb{N}\). This is obtained by using Padé approximation of the binomial function \((1+ x)^\alpha\), with a careful study of the divisibility properties of the coefficients of the
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