共查询到20条相似文献,搜索用时 46 毫秒
1.
Friedrich Roesler 《Archiv der Mathematik》1999,73(3):193-198
Abstract. For natural numbers n we inspect all factorizations n = ab of n with a £ ba \le b in \Bbb N\Bbb N and denote by n=an bnn=a_n b_n the most quadratic one, i.e. such that bn - anb_n - a_n is minimal. Then the quotient k(n) : = an/bn\kappa (n) := a_n/b_n is a measure for the quadraticity of n. The best general estimate for k(n)\kappa (n) is of course very poor: 1/n £ k(n) £ 11/n \le \kappa (n)\le 1. But a Theorem of Hall and Tenenbaum [1, p. 29], implies(logn)-d-e £ k(n) £ (logn)-d(\log n)^{-\delta -\varepsilon } \le \kappa (n) \le (\log n)^{-\delta } on average, with d = 1 - (1+log2 2)/log2=0,08607 ?\delta = 1 - (1+\log _2 \,2)/\log 2=0,08607 \ldots and for every e > 0\varepsilon >0. Hence the natural numbers are fairly quadratic.¶k(n)\kappa (n) characterizes a specific optimal factorization of n. A quadraticity measure, which is more global with respect to the prime factorization of n, is k*(n): = ?1 £ a £ b, ab=n a/b\kappa ^*(n):= \textstyle\sum\limits \limits _{1\le a \le b, ab=n} a/b. We show k*(n) ~ \frac 12\kappa ^*(n) \sim \frac {1}{2} on average, and k*(n)=W(2\frac 12(1-e) log n/log 2n)\kappa ^*(n)=\Omega (2^{\frac {1}{2}(1-\varepsilon ) {\log}\, n/{\log} _2n})for every e > 0\varepsilon>0. 相似文献
2.
We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables
ξ
1, … , ξ
n
and a vector of scalars x = (x
1, … , x
n
), and 1 ≤ k ≤ n, we provide estimates for
\mathbb E k-min1 £ i £ n |xixi|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and
\mathbb E k-max1 £ i £ n|xixi|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||yx||M{\|y_x\|_M} of the vector y
x
= (1/x
1, … , 1/x
n
). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ
1|,
G(t) = \mathbb P ({ |x1| £ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ
1 is the standard N(0, 1) Gaussian random variable, then
G(t) = ?{\tfrac2p}ò0t e-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds } and
M(s)=?{\tfrac2p}ò0se-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence. 相似文献
3.
For fixed k ≥ 3, let Ek(x) denote the error term of the sum
?n £ xrk(n)\sum_{n\le x}\rho_k(n)
, where
rk(n) = ?n=|m|k+|l|k, g.c.d.(m,l)=1\rho_k(n) = \sum_{n=|m|^k+|l|^k, g.c.d.(m,l)=1}
1. It is proved that if the Riemann hypothesis is true, then
E3(x) << x331/1254+eE_3(x)\ll x^{331/1254+\varepsilon}
,
E4(x) << x37/184+eE_4(x)\ll x^{37/184+\varepsilon}
. A short interval result is also obtained. 相似文献
4.
Given g { l\fracn2 g( lj x - kb ) }jezjezn ,where lj \left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L
2(R
n
) are given. For a class of functions g{ ezrib( j,x ) g( x - lk ) }jezn ,kez\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} to be a frame. 相似文献
5.
Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR+ ? \mathbbR+{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if
f (\fracx + y2)-\fracf(x) + f(y)2 £ a(||x - y||) for x, y ? V.f \left(\frac{x + y}{2}\right)-\frac{f(x) + f(y)}{2}\leq \alpha(\|x - y\|)\quad {\rm for} \, x, y \in V. 相似文献
6.
Let
W í \Bbb C\Omega \subseteq {\Bbb C}
be a simply connected domain in
\Bbb C{\Bbb C}
, such that
{¥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}]
is connected. If g is holomorphic in Ω and every derivative of g extends continuously on
[`(W)]\bar{\Omega}
, then we write g ∈ A∞ (Ω). For g ∈ A∞ (Ω) and
z ? [`(W)]\zeta \in \bar{\Omega}
we denote
SN (g,z)(z) = ?Nl=0\fracg(l) (z)l ! (z-z)lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l
. We prove the existence of a function f ∈ A∞(Ω), such that the following hold:
7.
Thomas Strömberg 《Archiv der Mathematik》2010,94(6):579-589
We present sharp Hessian estimates of the form D2 Se(t,x) £ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation
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