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1.
For positive integersn, m and realp≥1, let
Upper and lower bounds for this quantity are derived, extending results of Brown and Spencer forB
1(n,n), corresponding to the Gale-Berlekamp switching problem. For a Minkowski spaceM of dimensionm, define
a quantity investigated by Dvoretzky and Rogers. 相似文献
2.
W. Zudilin 《Journal of Mathematical Sciences》2006,137(2):4673-4683
We construct simultaneous rational approximations to q-series L1(x1; q) and L1(x2; q) and, if x = x1 = x2, to series L1(x; q) and L2(x; q), where
. Applying the construction, we obtain quantitative linear independence over ℚ of the numbers in the following collections:
1, ζq(1) = L1(1; q),
and 1, ζq(1), ζq(2) = L2(1; q) for q = 1/p, p ε ℤ \ {0,±1}. Bibliography: 14 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 107–124. 相似文献
3.
Deh-phone Kung Hsing 《Annali di Matematica Pura ed Applicata》1976,109(1):235-245
Summary We consider the system(L):
, t ⩾ p, y(t)=f(t), t⩽0, where y is an n-vector and each Ai, B(t) are n × n matrices. System(L) generates a semigroup by means of Ttf(s)=y (t+s, f), f(s) ∈ BCl(− ∞, 0]. Under some hypotheses concerning the roots ofdet
where
is the Laplace transform of B(t), the asymptotic behavior of y(t) is discussed. Two typical results are: Theorem 3.1: suppose
∥B(t)∥ ɛ L1[0, ∞),
thendet
forRe λ>0 iff for every ɛ>0 there is an Mɛ>0 such that ∥Ttf∥l ⩽ ⩽ Mɛ
exp [ɛt]∥f∥l for t ⩾ 0. Corollary 3.1.1: suppose
exp [at]B(t) ∈ ∈ L1[0, ∞) for some a>0 anddet
forRe λ>−a. Then the solution of(L) is exponentially asymptotically stable.
Entrata in Redazione il 21 marzo 1975.
The author is grateful to ProfessorC. Corduneanu for suggesting this problem and for many helpful discussions during the preparation of the paper. 相似文献
4.
V. A. Rodin 《Analysis Mathematica》1992,18(4):295-305
Пустьf— интегрируем ая на группе \(G = \mathop \prod \limits_{k = 0}^\infty Z(p_k )\) функция и lim supp k <∞. Дана характериза ция точекμ-сильной сумми руемости ряда Фурье ф ункцииf по системе Прайса. Это позволило установит ь слабый тип (1,1) мажоран тных операторов, связанны х с ВМО-сильнымн средними рядов по сис темам Крестенсонаp k ≡p и Уолшаp k ≡2. Как следствие, для частных сумм рядов по этим системам получе но: еслиf∈L [0, 1], то для любой конс тантыA>0 справедливо $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\mathop \sum \limits_{k = 0}^{n - 1} \exp A|s_k (f,x) - f(x)| = 1$$ для почти всехx∈[0,1]. Тем самым установлена справедливость гипо тезы В. Тотика для этих систем. 相似文献
5.
V. V. Vysotsky 《Journal of Mathematical Sciences》2007,147(4):6873-6883
Let Si be a random walk with standard exponential increments. The sum ∑
i=1
k
Si is called the k-step area of the walk. The random variable
∑
i=1
k
Si plays an important role in the study of the so-called one-dimensional sticky particles model. We find the distribution of
this variable and prove that
for 0 ≤ t ≤ 1. We also show that
, where the Ui,n are order statistics of n i.i.d. random variables uniformly distributed on [0, 1]. Bibliography: 6 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 48–67. 相似文献
6.
M. N. Yakovlev 《Journal of Mathematical Sciences》2009,157(5):784-788
The paper presents an error bound of the Ritz method for the problem of minimizing the functional
in the space
in the case where the standard assumption on the continuity of q(t) is replaced by the condition q2(t)t(1-t) ∈ L(0,1). In the case where q(t) is continuous, the new bound is sharper than the known one. Bibliography: 5 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 208–215. 相似文献
7.
Jürgen Weyer 《manuscripta mathematica》1979,28(4):305-316
In a Hilbert space H, we consider operators of type A=L*ϕ·L, where L is a closed, linear operator and ϕ is a maximal cyclically monotone, coercive operator. The operators ϕ, L, L* and their inverses are not necessarily everywhere defined. Our principle result is a nonlinear extension of an earlier theorem
of v. Neumann for A=L*L.Theorem: Suppose that either (L*)−1 is bounded or that both L−1 is bounded and, D(ϕ) υ N (L*). The L*ϕ·L, is maximal cyclically monotone. Maximality of sums
is also considered, and the theory is applied to concrete differential operators of the form
, with monotone functions f1 and various boundary conditions.
相似文献
8.
Letf(x) ∈L p[0,1], 1?p? ∞. We shall say that functionf(x)∈Δk (integerk?1) if for anyh ∈ [0, 1/k] andx ∈ [0,1?kh], we have Δ h k f(x)?0. Denote by ∏ n the space of algebraic polynomials of degree not exceedingn and define $$E_{n,k} (f)_p : = \mathop {\inf }\limits_{\mathop {P_n \in \prod _n }\limits_{P_n^{(\lambda )} \geqslant 0} } \parallel f(x) - P_n (x)\parallel _{L_p [0,1]} .$$ We prove that for any positive integerk, iff(x) ∈ Δ k ∩ L p[0, 1], 1?p?∞, then we have $$E_{n,k} (f)_p \leqslant C\omega _2 \left( {f,\frac{1}{n}} \right)_p ,$$ whereC is a constant only depending onk. 相似文献
9.
Joseph Rosenblatt 《Mathematische Annalen》1977,230(3):245-272
For a mean zero norm one sequence (f
n
)L
2[0, 1], the sequence (f
n
{nx+y}) is an orthonormal sequence inL
2([0, 1]2); so if
, then
converges for a.e. (x, y)[0, 1]2 and has a maximal function inL
2([0, 1]2). But for a mean zerofL
2[0, 1], it is harder to give necessary and sufficient conditions for theL
2-norm convergence or a.e. convergence of
. Ifc
n
0 and
, then this series will not converge inL
2-norm on a denseG
subset of the mean zero functions inL
2[0, 1]. Also, there are mean zerofL[0, 1] such that
never converges and there is a mean zero continuous functionf with
a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c
n
| = 0(n
–) for >1/2, then
converges a.e. and unconditionally inL
2[0, 1]. In addition, for any mean zerof of bounded variation, the series
has its maximal function in allL
p[0, 1] with 1p<. Finally, if (f
n
)L
[0, 1] is a uniformly bounded mean zero sequence, then
is a necessary and sufficient condition for
to converge for a.e.y and a.e. (x
n
)[0, 1]. Moreover, iffL
[0, 1] is mean zero and
, then for a.e. (x
n
)[0, 1],
converges for a.e.y and in allL
p
[0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one. 相似文献
10.
For f L
n
(T
d
) and
, the modulus of smoothness
is shown to be equivalent to
where T
n is the best trigonometric polynomial approximant of degree n to f in L
p and is the Laplacian. The above result is shown to be incorrect for 0 < p
. 相似文献
11.
Dale Umbach 《Annals of the Institute of Statistical Mathematics》1981,33(1):135-140
Summary LetF be a distribution function over the real line. DefineR
p(y)=∫|x−y|pdF(x) forp≧1. Forp>1 there is a unique minimizer ofR
p(·), sayγ
p. Under mild conditions onF it is shown that
exists and that the limit is a median. Thus, one could use this limit as a definition of a unique median. Also it is shown
that
whereR is the right extremity ofF andL is the left extremity ofF provided that −∞<L≦R<∞. A similar result is available ifL=−∞,R=∞, yetF has symmetric tails. 相似文献
12.
Complete moment and integral convergence for sums of negatively associated random variables 总被引:2,自引:0,他引:2
For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE(max1≤k≤n|Sk|^1/q-∈bn^1/qp)^+〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 (log n) ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence. 相似文献
13.
Let (Ω,f,P) be a probability space and letT be a measure-preserving weak mixing transformation. We define a large class of sequences of integers calledp-sequences, such that iff∈L
1 there exists a set Ω′⊂Ω of probability one and for ω∈Ω′ we have
for everyp-sequence {kn}. 相似文献
14.
Jugal Ghorai 《Annals of the Institute of Statistical Mathematics》1980,32(1):341-350
LetX
1,...,X
n
be i.i.d. random variable with a common densityf. Let
be an estimate off(x) based on a complete orthonormal basis {φ
k
:k≧0} ofL
2[a, b]. A Martingale central limit theorem is used to show that
, where
and
. 相似文献
15.
I. K. Matsak 《Ukrainian Mathematical Journal》1998,50(9):1405-1415
We prove that
where X is a normal random element in the space C [0,1], MX = 0, σ = {(M|X(t)|2)1/2
t∈[0,1}, (X
n
) are independent copies of X, and . Under additional restrictions on the random element X, this equality can be strengthened.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1227–1235, September, 1998. 相似文献
16.
CHEN Qionglei & ZHANG Zhifei Department of Mathematics Zhejiang University Hangzhou China Academy of Mathematics System Sciences Chinese Academy of Sciences Beijing China 《中国科学A辑(英文版)》2004,47(6):842-853
In this paper we give the (Lα p, Lp) boundedness of the maximal operator of a class of super singular integrals defined bywhich improves and extends the known result. Moreover, by applying an off-Diagonal T1 Theorem, we also obtain the (Lp, Lq) boundedness of the commutator defined by 相似文献
17.
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
18.
Nikolay Moshchevitin 《Czechoslovak Mathematical Journal》2012,62(1):127-137
Let Θ = (θ
1,θ
2,θ
3) ∈ ℝ3. Suppose that 1, θ
1, θ
2, θ
3 are linearly independent over ℤ. For Diophantine exponents
$\begin{gathered}
\alpha (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\sup }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\}, \hfill \\
\beta (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\inf }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\} \hfill \\
\end{gathered}$\begin{gathered}
\alpha (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\sup }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\}, \hfill \\
\beta (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\inf }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\} \hfill \\
\end{gathered} 相似文献
19.
T. O. Kononovych 《Ukrainian Mathematical Journal》2004,56(9):1403-1416
We obtain upper bounds in terms of Fourier coefficients for the best approximation by an angle and for norms in the metric of L
p
for functions of two variables defined by trigonometric series with coefficients such that
as l
1 + l
2 and
20.
Uri Fixman 《Integral Equations and Operator Theory》2000,37(1):9-19
LetA be the linear operator inL
p
(0, 1), 1<p<∞,p≠2, defined by
,x∈L
p
(0, 1),s∈[0,1]. We show that the real values of numbers in the numerical range ofA have maximum
, whereq=p/(p−1). This amounts to an inequality between integrals, for which we determine the case of equality. 相似文献
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