共查询到20条相似文献,搜索用时 15 毫秒
1.
Artūras Dubickas 《Archiv der Mathematik》2010,95(2):151-160
Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α n + ν, n = 0, 1, 2, . . . , modulo ${\mathbb{Z}[i],}Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ
α
n
+ ν, n = 0, 1, 2, . . . , modulo
\mathbbZ[i],{\mathbb{Z}[i],} where i=?{-1}{i=\sqrt{-1}} and
\mathbbZ[i]=\mathbbZ+i\mathbbZ{\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}} is the ring of Gaussian integers. For any
z ? \mathbbC,{z\in \mathbb{C},} one may naturally call the quantity z modulo
\mathbbZ[i]{\mathbb{Z}[i]}
the fractional part of z and write {z} for this, in general, complex number lying in the unit square
S:={z ? \mathbbC:0 £ \mathfrakR(z),\mathfrakJ(z) < 1 }{S:=\{z\in\mathbb{C}:0\leq \mathfrak{R}(z),\mathfrak{J}(z) <1 \}}. We first show that if α is a complex non-real number which is algebraic over
\mathbbQ{\mathbb{Q}} and satisfies |α| > 1 then there are two limit points of the sequence {ξ
α
n
+ν}, n = 0, 1, 2, . . . , which are ‘far’ from each other (in terms of α only), except when α is an algebraic integer whose conjugates over
\mathbbQ(i){\mathbb{Q}(i)} all lie in the unit disc |z| ≤ 1 and
x ? \mathbbQ(a,i).{\xi\in\mathbb{Q}(\alpha,i).} Then we prove a result in the opposite direction which implies that, for any fixed
a ? \mathbbC{\alpha\in\mathbb{C}} of modulus greater than 1 and any sequence
zn ? \mathbbC,n=0,1,2,...,{z_n\in\mathbb{C},n=0,1,2,\dots,} there exists
x ? \mathbbC{\xi \in \mathbb{C}} such that the numbers ξ
α
n
−z
n
, n = 0, 1, 2, . . . , all lie ‘far’ from the lattice
\mathbbZ[i]{\mathbb{Z}[i]}. In particular, they all can be covered by a union of small discs with centers at
(1+i)/2+\mathbbZ[i]{(1+i)/2+\mathbb{Z}[i]} if |α| is large. 相似文献
2.
Nitis Mukhopadhyay 《Methodology and Computing in Applied Probability》2010,12(4):609-622
In this communication, we first compare z
α
and t
ν,α
, the upper 100α% points of a standard normal and a Student’s t
ν
distributions respectively. We begin with a proof of a well-known result, namely, for every fixed
0 < a < \frac120<\alpha <\frac{1}{2} and the degree of freedom ν, one has t
ν,α
> z
α
. Next, Theorem 3.1 provides a new and explicit expression b
ν
( > 1) such that for every fixed
0 < a < \frac120<\alpha < \frac{1}{2} and ν, we can conclude t
ν,α
> b
ν
z
α
. This is clearly a significant improvement over the result that is customarily quoted in nearly every textbook and elsewhere.
A proof of Theorem 3.1 is surprisingly simple and pretty. We also extend Theorem 3.1 in the case of a non-central Student’s
t distribution (Section 3.3). In the context of Stein’s (Ann Math Stat 16:243–258, 1945; Econometrica 17:77–78, 1949) 100(1 − α)% fixed-width confidence intervals for the mean of a normal distribution having an unknown variance, we have examined the
oversampling rate on an average for a variety of choices of m, the pilot sample size. We ran simulations to investigate this issue. We have found that the oversampling rates are approximated
well by tn,a/22za/2-2t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2} for small and moderate values of m( ≤ 50) all across Table 2 where ν = m − 1. However, when m is chosen large (≥ 100), we find from Table 3 that the oversampling rates are not approximated by tn,a/22za/2-2t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2} very well anymore in some cases, and in those cases the oversampling rates either exceed the new lower bound of tn,a/22za/2-2,t_{\nu ,\alpha /2}^{2}z_{\alpha /2}^{-2}, namely bn2,b_{\nu }^{2}, or comes incredibly close to bn2b_{\nu }^{2} where ν = m − 1. That is, the new lower bound for a percentile of a Student’s t distribution may play an important role in order to come up with diagnostics in our understanding of simulated output under
Stein’s fixed-width confidence interval method. 相似文献
3.
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
4.
In this paper we study the set of x ∈ [0, 1] for which the inequality |x − x
n
| < z
n
holds for infinitely many n = 1, 2, .... Here x
n
∈ [0, 1) and z
n
s> 0, z
n
→ 0, are sequences. In the first part of the paper we summarize known results. In the second part, using the theory of distribution
functions of sequences, we find the asymptotic density of n for which |x − x
n
| < z
n
, where x is a discontinuity point of some distribution function of x
n
. Generally, we also prove, for an arbitrary sequence x
n
, that there exists z
n
such that the density of n = 1, 2, ..., x
n
→ x, is the same as the density of n = 1, 2, ..., |x − x
n
| < z
n
, for x ∈ [0, 1]. Finally we prove, using the longest gap d
n
in the finite sequence x
1, x
2, ..., x
n
, that if d
n
≤ z
n
for all n, z
n
→ 0, and z
n
is non-increasing, then |x − x
n
| < z
n
holds for infinitely many n and for almost all x ∈ [0, 1]. 相似文献
5.
We develop a theory of “special functions” associated with a certain fourth-order differential operator Dm,n\mathcal{D}_{\mu,\nu} on ℝ depending on two parameters μ,ν. For integers μ,ν≥−1 with μ+ν∈2ℕ0, this operator extends to a self-adjoint operator on L
2(ℝ+,x
μ+ν+1 dx) with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive
basic properties of the eigenfunctions such as orthogonality, completeness, L
2-norms, integral representations, and various recurrence relations. 相似文献
6.
L. P. Bedratyuk 《Ukrainian Mathematical Journal》2011,63(6):880-890
We deduce formulas for finding the Poincaré multiseries P( Cd,z1,z2, ?, zn,t ) \mathcal{P}\left( {{\mathcal{C}_d},{z_1},{z_2}, \ldots, {z_n},t} \right) and P( Id,z1,z2, ?, zn ) \mathcal{P}\left( {{\mathcal{I}_d},{z_1},{z_2}, \ldots, {z_n}} \right) , where Cd {\mathcal{C}_d} and Id {\mathcal{I}_d} , d = (d
1, d
2, . . . , d
n
), are multigraded algebras of joint covariants and joint invariants for n binary forms of degrees d
1, d
2, . . . , d
n
. 相似文献
7.
P. H. Sach L. J. Lin L. A. Tuan 《Journal of Optimization Theory and Applications》2010,147(3):607-620
This paper deals with the generalized vector quasivariational inclusion Problem (P1) (resp. Problem (P2)) of finding a point (z
0,x
0) of a set E×K such that (z
0,x
0)∈B(z
0,x
0)×A(z
0,x
0) and, for all η∈A(z
0,x
0),
lF(z0,x0,h) ì G(z0,x0,x0)+C(z0,x0) [resp.F(z0,x0,x0) ì G(z0,x0,h)+C(z0,x0)],\begin{array}{l}F(z_0,x_0,\eta)\subset G(z_0,x_0,x_0)+C(z_0,x_0)\cr \mathrm{[resp.}F(z_0,x_0,x_0)\subset G(z_0,x_0,\eta)+C(z_0,x_0)],\end{array} 相似文献
8.
C. Boldrighini R. A. Minlos A. Pellegrinotti 《Probability Theory and Related Fields》1997,109(2):245-273
Summary We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ
t
(x): (t,x)∈ℤν+1}. The random walk transition probabilities are taken as P(X
t
+1= y|X
t
= x,ξ
t
=η) =P
0( y−x)+ c(y−x;η(x)). We assume that the variables {ξ
t
(x):(t,x) ∈ℤν+1} are i.i.d., that both P
0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P
0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X
t
, and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X
t
and a corresponding correction of order to the C.L.T.. Proofs are based on some new L
p
estimates for a class of functionals of the field.
Received: 4 January 1996/In revised form: 26 May 1997 相似文献
9.
We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself,
if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = [`(f° [`(j)] )]\overline {f^\circ \bar \varphi } for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z
1, ..., z
n
) = (l1 zi1 ,...,ln zin )(\lambda _1 z_{i_1 } ,...,\lambda _n z_{i_n } ) for |λ
j
| = 1, 1 ≤ j ≤ n, and (i
1; ..., i
n
)is some permutation of the integers from 1through n in the case of the n-dimensional polydisk. 相似文献
10.
Jie Ming Wang 《数学学报(英文版)》2009,25(5):741-758
The stochastic comparison and preservation of positive correlations for Levy-type processes on R^d are studied under the condition that Levy measure v satisfies f{0〈|z|≤1)|z||v(x, dz) - v(x, d(-z))| 〈 ∞, x∈ R^d, while the sufficient conditions and necessary ones for them are obtained. In some cases the conditions for stochastic comparison are not only sufficient but also necessary. 相似文献
11.
Artūras Dubickas 《Monatshefte für Mathematik》2009,158(3):271
We prove that, for any real numbers ξ ≠ 0 and ν, the sequence of integer parts [ξ2 n + ν], n = 0, 1, 2, . . . , contains infinitely many composite numbers. Moreover, if the number ξ is irrational, then the above sequence contains infinitely many elements divisible by 2 or 3. The same holds for the sequence [ξ( ? 2) n + ν n ], n = 0, 1, 2, . . . , where ν 0, ν 1, ν 2, . . . all lie in a half open real interval of length 1/3. For this, we show that if a sequence of integers x 1, x 2, x 3, . . . satisfies the recurrence relation x n+d = cx n + F(x n+1, . . . , x n+d-1) for each n ≥ 1, where c ≠ 0 is an integer, \({F(z_1,\dots,z_{d-1}) \in \mathbb {Z}[z_1,\dots,z_{d-1}],}\) and lim n→ ∞|x n | = ∞, then the number |x n | is composite for infinitely many positive integers n. The proofs involve techniques from number theory, linear algebra, combinatorics on words and some kind of symbolic computation modulo 3. 相似文献
12.
Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475
Let
\mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 1, j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let
[`(\mathbbD)]n{\overline{\mathbb{D}}^n} denote its closure in
\mathbb Cn{\mathbb {C}^n}. Consider the ring
|