Displaying similar documents to “On inhomogeneous diophantine approximation with some quasi-periodic expressions, II”

More on inhomogeneous diophantine approximation

Christopher G. Pinner (2001)

Journal de théorie des nombres de Bordeaux

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For an irrational real number α and real number γ we consider the inhomogeneous approximation constant M ( α , γ ) : = lim inf | n | → ∞ | n | | | n α - γ | | via the semi-regular negative continued fraction expansion of α α = 1 a 1 - 1 a 2 - 1 a 3 - ⋯ and an appropriate alpha-expansion of γ . We give an upper bound on the case of worst inhomogeneous approximation, ρ ( α ) : = sup γ ∉ 𝐙 + α 𝐙 M ( α , γ ) , which is sharp when the partial quotients ai are almost all even and at least four. When the...

The diophantine equation a x 2 + b x y + c y 2 = N , D = b 2 - 4 a c > 0

Keith Matthews (2002)

Journal de théorie des nombres de Bordeaux

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We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of a x 2 + b x y + c y 2 = N in relatively prime integers x , y , where N ≠ 0 , gcd ( a , b , c ) = gcd ( a , N ) = 1 et D = b 2 - 4 a c > 0 is not a perfect square. In the case of solubility, solutions with least positive y, from each equivalence class, are also constructed. Our paper is a generalisation of an earlier paper by the author on the equation x 2 - D y 2 = N . As in that paper, we use a lemma on unimodular matrices that gives a much simpler proof than Lagrange’s...

«Approximate approximations» and the cubature of potentials

Gunther Schmidt (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The paper discusses new cubature formulas for classical integral operators of mathematical physics based on the «approximate approximation» of the density with Gaussian and related functions. We derive formulas for the cubature of harmonic, elastic and diffraction potentials approximating with high order in some range relevant for numerical computations. We prove error estimates and provide numerical results for the Newton potential.