共查询到20条相似文献,搜索用时 31 毫秒
1.
A. V. Zheleznyak 《Vestnik St. Petersburg University: Mathematics》2009,42(4):269-274
In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f
−1(x) = $
\sum\limits_{n = 0}^\infty {b_n x^n }
$
\sum\limits_{n = 0}^\infty {b_n x^n }
b
n
x
n
be such that b
0 > 0 and b
n
≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x
1, x
2, …, x
m
) with positive coefficients in order that the series f
−1(x
1, x
2, …, x
m
) = $
\sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } }
$
\sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } }
satisfies the property b
0, …, 0 > 0, $
bi_1 ,i_2 , \ldots ,i_m
$
bi_1 ,i_2 , \ldots ,i_m
≤ 0, i
12 + i
22 + … + i
m
2 > 0, which is similar to the one-dimensional case. 相似文献
2.
Wlodzimier Greblicki Miroslaw Pawlak 《Annals of the Institute of Statistical Mathematics》1985,37(1):443-454
Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X
1,Y
1),…, (X
n, Yn) of a pair (X, Y) of random variables. We examine an estimate of a type
, whereN depends onn andϕ
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 tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then
converges tom(x) as rapidly asO(nC−(2s−1)/4s) in probability andO(n
−(2s−1)/4s logn) almost completely. 相似文献
3.
Let ( Y,d,dl )\left( {\mathcal{Y},d,d\lambda } \right) be (ℝ
n
, |·|, μ), where |·| is the Euclidean distance, μ is a nonnegative Radon measure on ℝ
n
satisfying the polynomial growth condition, or the Gauss measure metric space (ℝ
n
, |·|, d
λ
), or the space (S, d, ρ), where S ≡ ℝ
n
⋉ ℝ+ is the (ax + b)-group, d is the left-invariant Riemannian metric and ρ is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces
{ Xs ( Y ) }0 < s \leqslant ¥\left\{ {X_s \left( \mathcal{Y} \right)} \right\}_{0 < s \leqslant \infty } and the BMO-type spaces
{ BMO( Y, s ) }0 < s \leqslant ¥\left\{ {BMO\left( {\mathcal{Y}, s} \right)} \right\}_{0 < s \leqslant \infty }. Let H
1
( Y )\left( \mathcal{Y} \right) be the known atomic Hardy space and L
01
( Y )\left( \mathcal{Y} \right) the subspace of f ∈ L
1
( Y )\left( \mathcal{Y} \right) with integral 0. The authors prove that the dual space of X
s
( Y )\left( \mathcal{Y} \right) is BMO( Y,s )BMO\left( {\mathcal{Y},s} \right) when s ∈ (0,∞), X
s
( Y )\left( \mathcal{Y} \right) = H
1
( Y )\left( \mathcal{Y} \right) when s ∈ (0, 1], and X
∞
( Y )\left( \mathcal{Y} \right) = L
01
( Y )\left( \mathcal{Y} \right) (or L
1
( Y )\left( \mathcal{Y} \right)). As applications, the authors show that if T is a linear operator bounded from H
1
( Y )\left( \mathcal{Y} \right) to L
1
( Y )\left( \mathcal{Y} \right) and from L
1
( Y )\left( \mathcal{Y} \right) to L
1,∞
( Y )\left( \mathcal{Y} \right), then for all r ∈ (1,∞) and s ∈ (r,∞], T is bounded from X
r
( Y )\left( \mathcal{Y} \right) to the Lorentz space L
1,s
( Y )\left( \mathcal{Y} \right), which applies to the Calderón-Zygmund operator on (ℝ
n
, |·|, μ), the imaginary powers of the Ornstein-Uhlenbeck operator on (ℝ
n
, |·|, d
γ
) and the spectral operator associated with the spectral multiplier on (S, d, ρ). All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces. 相似文献
4.
Daniel Wulbert 《Israel Journal of Mathematics》2001,126(1):363-380
LetX be a Borel subset of a separable Banach spaceE. Letμ be a non-atomic,σ-finite, Borel measure onX. LetG ⊆L
1 (X, Σ,μ) bem-dimensional.
Theorem:There is an l ∈ E* and real numbers −∞=x
0<x
1<x
2<…<x
n<x
n+1=∞with n≤m, such that for all g ∈ G,
相似文献
5.
Y. C. Wang 《Acta Mathematica Hungarica》2012,135(3):248-269
Let Hk\mathcal{H}_{k} denote the set {n∣2|n,
n\not o 1 (mod p)n\not\equiv 1\ (\mathrm{mod}\ p) ∀ p>2 with p−1|k}. We prove that when
X\frac1120(1-\frac12k) +e\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X, almost all integers
n ? \allowbreak Hk ?(X, X+H]n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]} can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when
X\frac1120(1-\frac1k) +e\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n∈(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3. 相似文献
6.
LetX andY denote two complex Banach spaces and letB(Y, X) denote the algebra of all bounded linear operators fromY toX. ForA∈B(X)
n
,B∈B(Y)
n
, the elementary operator acting onB(Y, X) is defined by
. In this paper we obtain the formulae of the spectrum and the essential spectrum of Δ(A, B) by using spectral mapping theorems. Forn=1, we prove thatS
p
(L
A
,R
B
)=σ(A)×σ(B) and
. 相似文献
7.
James East 《Semigroup Forum》2010,81(2):357-379
The (full) transformation semigroup Tn\mathcal{T}_{n} is the semigroup of all functions from the finite set {1,…,n} to itself, under the operation of composition. The symmetric group Sn í Tn{\mathcal{S}_{n}\subseteq \mathcal{T}_{n}} is the group of all permutations on {1,…,n} and is the group of units of Tn\mathcal{T}_{n}. The complement Tn\Sn\mathcal{T}_{n}\setminus \mathcal{S}_{n} is a subsemigroup (indeed an ideal) of Tn\mathcal{T}_{n}. In this article we give a presentation, in terms of generators and relations, for Tn\Sn\mathcal{T}_{n}\setminus \mathcal{S}_{n}, the so-called singular part of Tn\mathcal{T}_{n}. 相似文献
8.
Let {S
n
, n=0, 1, 2, …} be a random walk (S
n
being thenth partial sum of a sequence of independent, identically distributed, random variables) with values inE
d
, thed-dimensional integer lattice. Letf
n
=Prob {S
1 ≠ 0, …,S
n
−1 ≠ 0,S
n
=0 |S
0=0}. The random walk is said to be transient if
and strongly transient if
. LetR
n
=cardinality of the set {S
0,S
1, …,S
n
}. It is shown that for a strongly transient random walk with p<1, the distribution of [R
n
−np]/σ √n converges to the normal distribution with mean 0 and variance 1 asn tends to infinity, where σ is an appropriate positive constant. The other main result concerns the “capacity” of {S
0, …,S
n
}. For a finite setA inE
d
, let C(A=Σ
x∈A
) Prob {S
n
∉A, n≧1 |S
0=x} be the capacity ofA. A strong law forC{S
0, …,S
n
} is proved for a transient random walk, and some related questions are also considered.
This research was partially supported by the National Science Foundation. 相似文献
9.
R. V. Hrushevoi 《Ukrainian Mathematical Journal》2008,60(4):540-550
We describe the set
of parameters γ for which there exists a decomposition of the operator γI
H in a sum of n self-adjoint operators with spectra from the sets M
1, …, M
n, M
i = 0, 1, …, k
i, for n ≥ 4 and, in some cases, for n = 3.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 470–477, April, 2008. 相似文献
10.
Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula 总被引:2,自引:0,他引:2
For a family of compact Riemann surfaces Xt of genus g > 1, parameterized by the Schottky space
we define a natural basis of
which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n = 1. We introduce a holomorphic function F(n) on
which generalizes the classical product
for n = 1 and g = 1. We prove the holomorphic factorization formula
where det'Δ n is the zeta-function regularized determinant of the Laplace operator Δn in the hyperbolic metric acting on n-differentials, Nn is the Gram matrix of the natural basis with respect to the inner product given by the hyperbolic metric, S is the classical Liouville action –a K?hler potential of the Weil–Petersson metric on
– and cg,n is a constant depending only on g and n. The factorization formula reduces to Kronecker’s first limit formula when n = 1 and g = 1, and to Zograf’s factorization formula for n = 1 and g > 1.
Received: April 2005. Accepted: October 2005 相似文献
11.
Fix any n≥1. Let X
1,…,X
n
be independent random variables such that S
n
=X
1+⋅⋅⋅+X
n
, and let
S*n=sup1 £ k £ nSkS^{*}_{n}=\sup_{1\le k\le n}S_{k}
. We construct upper and lower bounds for s
y
and
sy*s_{y}^{*}
, the upper
\frac1y\frac{1}{y}
th quantiles of S
n
and
S*nS^{*}_{n}
, respectively. Our approximations rely on a computable quantity Q
y
and an explicit universal constant γ
y
, the latter depending only on y, for which we prove that
${l}\displaystyle s_y\le s_y^*\le Q_y\quad\mbox{for }y>1,\\[4pt]\displaystyle \gamma_{3y/16}Q_{3y/16}-Q_1\le s_y^*\quad\mbox{for }y>\frac{32}{3},$\begin{array}{l}\displaystyle s_y\le s_y^*\le Q_y\quad\mbox{for }y>1,\\[4pt]\displaystyle \gamma_{3y/16}Q_{3y/16}-Q_1\le s_y^*\quad\mbox{for }y>\frac{32}{3},\end{array} 相似文献
12.
LetK be a field, charK=0 andM
n
(K) the algebra ofn×n matrices overK. If λ=(λ1,…,λ
m
) andμ=(μ
1,…,μ
m
) are partitions ofn
2 let
wherex
1,…,x
n
2,y
1,…,y
n
2 are noncommuting indeterminates andS
n
2 is the symmetric group of degreen
2.
The polynomialsF
λ, μ
, when evaluated inM
n
(K), take central values and we study the problem of classifying those partitions λ,μ for whichF
λ, μ
is a central polynomial (not a polynomial identity) forM
n
(K).
We give a formula that allows us to evaluateF
λ, μ
inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF
λ, μ
is a polynomial identity forM
n
(K). As an application, we exhibit a new class of central polynomials forM
n
(K).
In memory of Shimshon Amitsur
Research supported by a grant from MURST of Italy. 相似文献
13.
Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f, we prove the following weighted strong law of large numbers: if X,X
1,X
2, … is any sequence of integrable i.i.d. random variables, then
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