共查询到20条相似文献,搜索用时 31 毫秒
1.
If p(z) is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then it is proved[5] that max |z|=1 |p′(z)| ≤ kn1n + kn m|z|=ax1 |p(z)|. In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type p(z) = cnzn + ∑n j=μ cn jzn j, 1 ≤μ≤ n. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros. 相似文献
2.
We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than
( n - 1 ) \mathord | / |
\vphantom ( n - 1 ) ?{1 - x2} ?{1 - x2} {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} for all x ∈ (−1, 1) such that
| x | ? èk = 0[ n \mathord | / |
\vphantom n 2 2 ] [ cos\frac2k + 12( n - 1 )p, cos\frac2k + 12np ] \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} . 相似文献
3.
Let V( z) be a complex-valued function on the complex plane ℂ satisfying the condition | V( z) − V(ζ)| ≤ w| z − ζ|, z, ζ ε ℂ; ω ≥ 0 be a Muckenhoupt A
p
weight on ℂ; i.e., the inequality
$
\left( {\frac{1}
{{\left| B \right|}}\int\limits_B {\omega d\sigma } } \right)\left( {\frac{1}
{{\left| B \right|}}\int\limits_B {\omega ^{ - \frac{1}
{{p - 1}}} d\sigma } } \right)^{p - 1} \leqslant c_0
$
\left( {\frac{1}
{{\left| B \right|}}\int\limits_B {\omega d\sigma } } \right)\left( {\frac{1}
{{\left| B \right|}}\int\limits_B {\omega ^{ - \frac{1}
{{p - 1}}} d\sigma } } \right)^{p - 1} \leqslant c_0
相似文献
4.
Summary Let S
i have the Wishart distribution W
p(∑ i,n i) for i=1,2. An asymptotic expansion of the distribution of
for large n=n 1+n 2 is derived, when ∑
1∑
2
−1
=I+n −1/2θ, based on an asymptotic solution of the system of partial differential equations for the hypergeometric function 2
F
1, obtained recently by Muirhead [2]. Another asymptotic formula is also applied to the distributions of −2 log λ and −log| S
2( S
1+ S
2) −1| under fixed ∑
1∑
2
−1
, which gives the earlier results by Nagao [4]. Some useful asymptotic formulas for 1
F
1 were investigated by Sugiura [7]. 相似文献
5.
Considering the positive d-dimensional lattice point Z
+
d
( d ≥ 2) with partial ordering ≤, let { X
k: k ∈ Z
+
d
} be i.i.d. random variables taking values in a real separable Hilbert space ( H, ‖ · ‖) with mean zero and covariance operator Σ, and set $
S_n = \sum\limits_{k \leqslant n} {X_k }
$
S_n = \sum\limits_{k \leqslant n} {X_k }
, n ∈ Z
+
d
. Let σ
i
2, i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ
2. Let log x = ln( x ∨ e), x ≥ 0. This paper studies the convergence rates for $
\sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right)
$
\sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right)
. We show that when l ≥ 2 and b > − l/2, E[‖ X‖ 2(log ‖ X‖)
d−2(log log ‖ X‖)
b+4] < ∞ implies $
\begin{gathered}
\mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\
= \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}}
{2}} \Gamma (b + l/2)}}
{{\Gamma (l/2)(d - 1)!}} \hfill \\
\end{gathered}
$
\begin{gathered}
\mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\
= \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}}
{2}} \Gamma (b + l/2)}}
{{\Gamma (l/2)(d - 1)!}} \hfill \\
\end{gathered}
, where Γ(·) is the Gamma function and $
\prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }
$
\prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }
. 相似文献
6.
Summary In the paper we estimate a regression m(x)=E { Y| X=x} from a sequence of independent observations ( X
1, Y
1),…, ( X
n, Y n) of a pair ( X, Y) of random variables. We examine an estimate of a type
, where N depends on n and ϕ
N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E| Y|<∞ and | Y|≦γ≦∞, we give condition for
to converge to m(x) at almost all x, provided that X has a density. if the regression has s derivatives, then
converges to m(x) as rapidly as O(nC −(2s−1)/4s) in probability and O(n
−(2s−1)/4s log n) almost completely. 相似文献
7.
Let 0 < c < s be fixed real numbers such that
, and let f : E 2 → E
d
for d ≥ 2 be a function such that for every p, q ∈ E
2 if | p − q| = c, then | f( p) − f( q)| ≤ c, and if | p − q| = s, then | f( p) − f( q)| ≥ s. Then f is a congruence. This result depends on and expands a result of Rádo et. al. [9], where a similar result holds, but for
replacing
. We also present a further extensions where E 2 is replaced by E
n
for n > 2 and where the range of c/s is enlarged.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
8.
Let X1, X2, ... be i.i.d. random variables with EX1 = 0 and positive, finite variance σ2, and set Sn = X1 + ... + Xn. For any α > −1, β > −1/2 and for κn(ε) a function of ε and n such that κn(ε) log log n → λ as n ↑ ∞ and
, we prove that
*Supported by the Natural Science Foundation of Department of Education of Zhejiang Province (Grant No. 20060237 and 20050494). 相似文献
9.
Let M
n
be an n-dimensional compact C
∞-differentiable manifold, n ≥ 2, and let S be a C
1-differential system on M
n
. The system induces a one-parameter C
1 transformation group φ
t
(−∞ < t < ∞) over M
n
and, thus, naturally induces a one-parameter transformation group of the tangent bundle of M
n
. The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group.
Among various results established in the paper, we mention here only the following, which might describe quite well the nature
of our study.
(A) Let M be the set of regular points in M
n
of the differential system S. With respect to a given C
∞ Riemannian metric of M
n
, we consider the bundle
of all ( n−2) spheres Q
x
n−2, x∈ M, where Q
x
n−2 for each x consists of all unit tangent vectors of M
n
orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ
t
#
(−∞< t<∞) of
. For an l-frame α = ( u
1, u
2,⋯, u
l
) of M
n
at a point x in M, 1 ≥ l ≥ n−1, each u
i
being in
, we shall denote the volume of the parallelotope in the tangent space of M
n
at x with edges u
1, u
2,⋯, u
l
by υ( α), and let
. This is a continuous real function of t. Let
α is said to be positively linearly independent of the mean if I
+
*( α) > 0. Similarly, α is said to be negatively linearly independent of the mean if I
−
*( α) > 0.
A point x of M is said to possess positive generic index κ = κ
+
*( x) if, at x, there is a κ-frame
,
, of M
n
having the property of being positively linearly independent in the mean, but at x, every l-frame
, of M
n
with l >
κ does not have the same property. Similarly, we define the negative generic index κ
−
*( x) of x. For a nonempty closed subset F of M
n
consisting of regular points of S, invariant under φ
t
(−∞ < t < ∞), let the (positive and negative) generic indices of F be defined by
Theorem
κ
+
*(F)= κ
−
*(F).
(B) We consider a nonempty compact metric space x and a one-parameter transformation group ϕ
t
(−∞ < t < ∞) over X. For a given positive integer l ≥ 2, we assume that, to each x∈ X, there are associated l-positive real continuous functions of −∞ < t < ∞. Assume further that these functions possess the following properties, namely, for each of k = 1, 2,⋯, l,
(i*) |
h
k
(x, t) = h
xk
(t) is a continuous function of the Cartesian product X×(−∞, ∞).
|
(ii*) |
|
for each x∈ X, each −∞ < s < ∞, and each −∞ < t < ∞.
Theorem
With X, etc., given above, let μ
be a normal measure of X that is ergodic and invariant under ϕ
t
(− ∞
< t < ∞). Then, for a certain permutation k→p( k) of k= 1, 2,⋯, l, the set W of points x of X such that all the inequalities
(I
k
)
(II
k
)
( k= 2, 3,⋯ , l) hold is invariant under ϕ
t
(− ∞
< t < ∞) and is μ-measurable with μ-measure1.
In practice, the functions h
xk
( t) will be taken as length functions of certain tangent vectors of M
n
. This theory, established such as in this paper, is expected to be used in the study of structurally stable differential
systems on M
n
.
Translated from Qualitative Theory of Differentiable Dynamical Systems, Beijing, China: Science Press, 1996, by Dr. SUN Wen-xiang, School of Mathematical Sciences, Peking University, Beijing 100871,
China. The Chinese version of this paper was published in Acta Scientiarum Naturalium Universitatis Pekinensis, 1963, 9: 241–265, 309–326 相似文献
10.
We consider the space
A(\mathbb T)A(\mathbb{T}) of all continuous functions f on the circle
\mathbb T\mathbb{T} such that the sequence of Fourier coefficients
[^( f)] = { [^( f)]( k ), k ? \mathbb Z }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l
1(ℤ). The norm on
A(\mathbb T)A(\mathbb{T}) is defined by
|| f || A(\mathbbT) = || [^( f)] || l1 (\mathbbZ)\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if
f:\mathbb T ? \mathbb T\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that
|| einf || A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that
|| einf || A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/
{\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right.
\kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/
{\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right.
\kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear. 相似文献
11.
Let σ(k, n) be the smallest even integer such that each n-term positive graphic sequence with term sum at least σ(k, n) can be realized by a graph containing a clique of k + 1 vertices. Erdos et al. (Graph Theory, 1991, 439-449) conjectured that σ(k, n) = (k - 1)(2n- k) + 2. Li et al. (Science in China, 1998, 510-520) proved that the conjecture is true for k 〉 5 and n ≥ (k2) + 3, and raised the problem of determining the smallest integer N(k) such that the conjecture holds for n ≥ N(k). They also determined the values of N(k) for 2 ≤ k ≤ 7, and proved that [5k-1/2] ≤ N(k) ≤ (k2) + 3 for k ≥ 8. In this paper, we determine the exact values of σ(k, n) for n ≥ 2k+3 and k ≥ 6. Therefore, the problem of determining σ(k, n) is completely solved. In addition, we prove as a corollary that N(k) -= [5k-1/2] for k ≥6. 相似文献
12.
The following regularity of weak solutions of a class of elliptic equations of the form are investigated. 相似文献
13.
Approximation to the function |x| plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals |x| if α = 1. We construct a Newman Type Operator rn(x) and prove max |x|≤1|xαsgn x-rn(x)|~Cn1/4e-π1/2(1/2)αn. 相似文献
14.
We consider the weighted Hardy integral operator T: L
2( a, b) → L
2( a, b), −∞≤ a< b≤∞, defined by
. In [EEH1] and [EEH2], under certain conditions on u and v, upper and lower estimates and asymptotic results were obtained for the approximation numbers a
n(T) of T. In this paper, we show that under suitable conditions on u and v,
where ∥ w∥ p=(∫
a
b
| w( t)| p
dt) 1/p.
Research supported by NSERC, grant A4021.
Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic. 相似文献
15.
Let X
1, X
2, ..., X
n
be a sequence of nonnegative independent random variables with a common continuous distribution function F. Let Y
1, Y
2, ..., Y
n
be another sequence of nonnegative independent random variables with a common continuous distribution function G, also independent of { X
i
}. We can only observe Z
i
=min( X
i
, Y
i
), and
. Let H=1−(1−F)(1−G) be the distribution function of Z. In this paper, the limit theorems for the ratio of the Kaplan-Meier estimator
or the Altshuler estimator
to the true survival function S(t) are given. It is shown that (1) P(δ (n)=1 i.o.)=0 if F(τ
H
) < 1 and P(δ
n
=0 i.o. )=0 if G(τ H) > 1 where δ (n) is the corresponding indicator function of
and
have the same order
a.s., where { T
n
} is a sequence of constants such that 1− H(T
n
)= n
−α(log n) β(log log n) γ. 相似文献
16.
The closed neighborhood NG[ e] of an edge e in a graph G is the set consisting of e and of all edges having an end-vertex in common with e. Let f be a function on E( G), the edge set of G, into the set {−1, 1}. If
for each e ∈ E( G), then f is called a signed edge dominating function of G. The signed edge domination number γ s′( G) of G is defined as
. Recently, Xu proved that γ s′( G) ≥ | V( G)| − | E( G)| for all graphs G without isolated vertices. In this paper we first characterize all simple connected graphs G for which γ s′( G) = | V( G)| − | E( G)|. This answers Problem 4.2 of [4]. Then we classify all simple connected graphs G with precisely k cycles and γ s′( G) = 1 − k, 2 − k.
A. Khodkar: Research supported by a Faculty Research Grant, University of West Georgia.
Send offprint requests to: Abdollah Khodkar. 相似文献
17.
Let x
1,..., x
m be points in the solid unit sphere of E
n and let x belong to the convex hull of x
1,..., x
m. Then
. This implies that all such products are bounded by (2/ m)
m
( m −1)
m−1. Bounds are also given for other normed linear spaces. As an application a bound is obtained for | p( z
0)| where
and p′( z
0)=0. 相似文献
18.
In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{( φp(u'))'+q(t)f(t,u,u')=0,0〈t〈1,u(0)=∑i=1^nαiu(ξi),u'(1)=∑i=1^nβiu'(ξi),whereφp(s)=|s|^p-2s,p≥2;ξi∈(0,1)(i=1,2,…,n),0≤αi,βi〈1(i=1,2,…n),0≤∑i=1^nαi,∑i=1^nβi〈1,and q(t) may be singular at t=0,1,f(t,u,u')may be singular at u'=0 相似文献
19.
Let G be a permutation group on a set Ω with no fixed points in,and m be a positive integer.Then the movement of G is defined as move(G):=sup Γ {|Γg\Γ| | g ∈ G}.It was shown by Praeger that if move(G) = m,then |Ω| 3m + t-1,where t is the number of G-orbits on.In this paper,all intransitive permutation groups with degree 3m+t-1 which have maximum bound are classified.Indeed,a positive answer to her question that whether the upper bound |Ω| = 3m + t-1 for |Ω| is sharp for every t > 1 is given. 相似文献
20.
Let U
n
be the unit polydisk in C
n
and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = ( ω
1, ..., ω
n
), ω
j
∈ S(1 ≤ j ≤ n) and f ∈ H( U
n
). The function f is said to be in holomorphic Besov space B
p
( ω) if
$
\left\| f \right\|_{B_p (\omega )}^p = \int_{U^n } {\left| {Df(z)} \right|^p \prod\limits_{j = 1}^n {\frac{{\omega _j (1 - |z_j |)}}
{{(1 - |z_j |^{2 - p} )}}} dm_{2n} (z) < + \infty }
$
\left\| f \right\|_{B_p (\omega )}^p = \int_{U^n } {\left| {Df(z)} \right|^p \prod\limits_{j = 1}^n {\frac{{\omega _j (1 - |z_j |)}}
{{(1 - |z_j |^{2 - p} )}}} dm_{2n} (z) < + \infty }
相似文献
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