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1.
In this paper,we consider sets of points with some restricts on the digits of theirα-Lroth expansions.More precisely,for any countable partitionα={An,n∈N}of the unit interval I,we completely determine the Hausdorf dimensions of the sets F(α,φ)=x=[l1(x),l2(x),...]α∈I:ln(x)φ(n),n 1,whereφis an arbitrary positive function defined on N satisfyingφ(n)→∞as n→∞.  相似文献   

2.
Пустьl 1 иl 2 — неотрицательные убывающие функции на (0, ∞). Допустим, что $$\int\limits_0^\infty {S^{n_i - 1} l_i (S)\left( {1 + \log + \frac{1}{{S^{n_i } l_i (S)}}} \right)dS}< \infty ,$$ , гдеn 1 иn 2 — натуральные числа. Тогда для каждой функции \(f \in L^1 (R^{n_1 + n_2 } )\) при почти всех (x0, у0) мы имеем $$\mathop {\lim }\limits_{\lambda \to \infty } \lambda ^{n_1 + n_2 } \int\limits_{R^{n_1 } } {\int\limits_{R^{n_2 } } {l_1 } } (\lambda |x|)l_2 (\lambda |y|)f(x_0 - x,y_0 - y)dx dy = f(x_0 ,y_0 )\int\limits_{R^{n_1 } } {\int\limits_{R^{n_2 } } {l_i (|x|)l_2 } } (|y|)dx dy.$$   相似文献   

3.
4.
A probability measureμ on a locally compactσ — compact amenable Hausdorff groupG is called mixing by convolutions if for every pair of probabilitiesν 1,ν 2 onG we have: $$\mathop {\lim }\limits_{n \to \infty } \left\| {\left( {\nu _1 - \nu _2 } \right) \star \mu ^{ \star n} } \right\| = \mathop {\lim }\limits_{n \to \infty } \left\| {\left( {\nu _1 - \nu _2 } \right) \star \mu ^{ \star n} } \right\| = 0.$$ . It is proved that the set of all mixing by convolutions probabilities is a norm (variation) dense subset of the setP(G) of all probabilities onG. IfG is additionally second countable the mixing measures are residual inP(G).  相似文献   

5.
Let F be a cubic cyclic field with t(2)ramified primes.For a finite abelian group G,let r3(G)be the 3-rank of G.If 3 does not ramify in F,then it is proved that t-1 r3(K2O F)2t.Furthermore,if t is fixed,for any s satisfying t-1 s 2t-1,there is always a cubic cyclic field F with exactly t ramified primes such that r3(K2O F)=s.It is also proved that the densities for 3-ranks of tame kernels of cyclic cubic number fields satisfy a Cohen-Lenstra type formula d∞,r=3-r2∞k=1(1-3-k)r k=1(1-3-k)2.This suggests that the Cohen-Lenstra conjecture for ideal class groups can be extended to the tame kernels of cyclic cubic number fields.  相似文献   

6.
7.
Supposef(x1,..., xn) is a polynomial of even degree d having coefficients in the finite field k=[q] and satisfying certain natural conditions, and let χ be the quadratic character of k. Then $$\left| {\sum {x_1 , \ldots ,} x_n \in k\chi (f(x_1 , \ldots ,x_n ))} \right| \leqslant Cq^{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} $$ where the constant C depends only on d and n.  相似文献   

8.
LetP κ,n (λ,β) be the class of functions \(g(z) = 1 + \sum\nolimits_{v = n}^\infty {c_\gamma z^v }\) , regular in ¦z¦<1 and satisfying the condition $$\int_0^{2\pi } {\left| {\operatorname{Re} \left[ {e^{i\lambda } g(z) - \beta \cos \lambda } \right]} \right|} /\left( {1 - \beta } \right)\cos \lambda \left| {d\theta \leqslant \kappa \pi ,} \right.z = re^{i\theta } ,$$ , 0 < r < 1 (κ?2,n?1, 0?Β<1, -π<λ<π/2;M κ,n (λ,β,α),n?2, is the class of functions \(f(z) = z + \sum\nolimits_{v = n}^\infty {a_v z^v }\) , regular in¦z¦<1 and such thatF α(z)∈P κ,n?1(λ,β), where \(F_\alpha (z) = (1 - \alpha )\frac{{zf'(z)}}{{f(z)}} + \alpha (1 + \frac{{zf'(z)}}{{f'(z)}})\) (0?α?1). Onr considers the problem regarding the range of the system {g (v?1)(z?)/(v?1)!}, ?=1,2,...,m,v=1,2,...,N ?, on the classP κ,1(λ,β). On the classesP κ,n (λ,β),M κ,n (λ,β,α) one finds the ranges of Cv, v?n, am, n?m?2n-2, and ofg(?),F ?(?), 0<¦ξ¦<1, ξ is fixed.  相似文献   

9.
For an m × n matrix B = (b ij ) m×n with nonnegative entries b ij , let B(k, l) denote the set of all k × l submatrices of B. For each AB(k, l), let a A and g A denote the arithmetic mean and geometric mean of elements of A respectively. It is proved that if k is an integer in ( $\tfrac{m} {2}$ ,m] and l is an integer in ( $\tfrac{n} {2}$ , n] respectively, then $$\left( {\prod\limits_{A \in B\left( {k,l} \right)} {a_A } } \right)^{\tfrac{1} {{\left( {_k^m } \right)\left( {_l^n } \right)}}} \geqslant \frac{1} {{\left( {_k^m } \right)\left( {_l^n } \right)}}\left( {\sum\limits_{A \in B\left( {k,l} \right)} {g_A } } \right),$$ with equality if and only if b ij is a constant for every i, j.  相似文献   

10.
Устанавливается лиш шщивость и односторо нняя дифференцируемость метрической проекции $$P_H^t :x \to P_H^t \left( x \right) = \left\{ {y \in H: \parallel x - y\parallel \leqq t + \varrho \left( {x, H} \right)} \right\}, t \geqq 0$$ на класс из пространстваB(Q) огр аниченных на множест веQ функцийx c нормой ∥x∥=sup {∣x(q)∣:qQ}, гдеΩ—метрика наQ. В частн ости, дляy i B(Q),t i ≧0, метрикΩ i и \(P_i = P_{H(\Omega _i )}^{t_i } \left( {y_i } \right), i = 1, 2\) доказано неравенство $$h\left( {P_1 , P_2 } \right) \leqq \frac{3}{2}\mathop {\sup }\limits_{q_1 , q_2 \in Q} \left| {\Omega _1 \left( {q_1 , q_2 } \right) - \Omega _2 \left( {q_1 , q_2 } \right)} \right| + 2\parallel y_1 - y_2 \parallel + 1\left| {t_1 - t_2 } \right|,$$ гдеh — расстояние Хау сдорфа. Константы 3/2, 2, 1 в э том неравенстве неулучш аемы.  相似文献   

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