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Rational points on the unit sphere

100%

Schmutz E.

Open Mathematics

2008

tom 6

nr 3

482-487

It is known that the unit sphere, centered at the origin in ℝn, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝn, and every ν > 0; there is a point r = (r 1; r 2;…;r n) such that: ⊎ ‖r-v‖∞ < ε.⊎ r is also a point on the unit sphere; Σ r i 2 = 1.⊎ r has rational coordinates; $$ r_i = \frac{{a_i }} {{b_i }} $$ for some integers a i, b i.⊎ for all $$ i,0 \leqslant \left| {a_i } \right| \leqslant b_i \leqslant (\frac{{32^{1/2} \left\lceil {log_2 n} \right\rceil }} {\varepsilon })^{2\left\lceil {log_2 n} \right\rceil } $$ . One consequence of this result is a relatively simple and quantitative proof of the fact that the rational orthogonal group O(n;ℚ) is dense in O(n;ℝ) with the topology induced by Frobenius’ matrix norm. Unitary matrices in U(n;ℂ) can likewise be approximated by matrices in U(n;ℚ(i))

Mathematical model and optimal control of flow induced vibration of pipelines

86%

Ahmed N.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1999

tom 19

nr 1-2

67-84

In this paper we consider a dynamic model for flow induced vibration of pipelines. We study the questions of existence and uniqueness of solutions of the system. Considering the flow rate as the control variable, we present three different necessary conditions of optimality. The last one with state constraint involves Differential Inclusions. The paper is concluded with an algorithm for computing the optimal controls.

Ograniczanie wyników

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Open Mathematics

1 Ahmed N.

1 Schmutz E.

1 2008

1 1999

Wyniki wyszukiwania

Rational points on the unit sphere

Mathematical model and optimal control of flow induced vibration of pipelines