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1.
Wolfgang Müller 《Monatshefte für Mathematik》2008,153(3):233-250
Let Q
1,…,Q
r
be quadratic forms with real coefficients. We prove that the set
is dense in
, provided that the system Q
1(x) = 0,…,Q
r
(x) = 0 has a nonsingular real solution and all forms in the real pencil generated by Q
1,…,Q
r
are irrational and have rank larger than 8r. Moreover, we give a quantitative version of the above assertion. As an application we study higher correlation functions
of the value distribution of a positive definite irrational quadratic form.
Author’s address: Institut für Statistik, Technische Universit?t Graz, A-8010 Graz, Austria 相似文献
2.
Let X, X
1, X
2,… be i.i.d.
\mathbbRd {\mathbb{R}^d} -valued real random vectors. Assume that E
X = 0 and that X has a nondegenerate distribution. Let G be a mean zero Gaussian random vector with the same covariance operator as that of X. We study the distributions of nondegenerate quadratic forms
\mathbbQ[ SN ] \mathbb{Q}\left[ {{S_N}} \right] of the normalized sums S
N
= N
−1/2 (X
1 + ⋯ + X
N
) and show that, without any additional conditions,
DN(a) = supx | \textP{ \mathbbQ[ SN - a ] \leqslant x } - \textP{ \mathbbQ[ G - a ] \leqslant x } - Ea(x) | = O( N - 1 ) \Delta_N^{(a)} = \mathop {{\sup }}\limits_x \left| {{\text{P}}\left\{ {\mathbb{Q}\left[ {{S_N} - a} \right] \leqslant x} \right\} - {\text{P}}\left\{ {\mathbb{Q}\left[ {G - a} \right] \leqslant x} \right\} - {E_a}(x)} \right| = \mathcal{O}\left( {{N^{ - 1}}} \right) 相似文献
3.
4.
Let K be either the rational number field
\Bbb Q{\Bbb Q} or an imaginary quadratic field. We give irrationality results for the number q = ?n=1¥rn/(qn-rl)\theta=\sum_{n=1}^{\infty}{r^n}/(q^n-r^l), where q (∣q∣ > 1) is an integer in K, r∈ K
× (∣r∣ < ∣q∣), and
1 £ l ? \Bbb Z1\le l\in{\Bbb Z} with q
n
≠ r
l (n ≥ 1). 相似文献
5.
For x = (x
1, x
2, …, x
n
) ∈ (0, 1 ]
n
and r ∈ { 1, 2, … , n}, a symmetric function F
n
(x, r) is defined by the relation
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