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1.
We point out that if the Hardy–Littlewood maximal operator is bounded on the space L p(t)(ℝ), 1 < ap(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L p(t) (ℝ n ), 1 < ap(t) ≤ b < ∞, t ∈ ℝ n , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.  相似文献   

2.
Let E, F be two Banach spaces, and B(E, F), Φ(E, F), SΦ(E, F) and R(E,F) be the bounded linear, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. In this paper, using the continuity characteristics of generalized inverses of operators under small perturbations, we prove the following result Let ∑ be any one of the following sets {T ∈ Φ(E, F) IndexT =const, and dim N(T) = const.}, {T ∈ SΦ(E, F) either dim N(T) = const. < ∞ or codim R(T) = const.< ∞} and {T ∈ R(E, F) RankT =const.<∞}. Then ∑ is a smooth submanifold of B(E, F) with the tangent space TA∑ = {B ∈ B(E,F) BN(A) (∪) R(A)} for any A ∈ ∑. The result is available for the further application to Thom's famous results on the transversility and the study of the infinite dimensional geometry.  相似文献   

3.
In this note, we show that the number of composite integers n ≤ x such that φ(n)|n - 1 is at most O(x^1/2(loglog x)^1/2), thus improving earlier results by Pomerance and by Shan.  相似文献   

4.
We establish conditions for the oscillation of solutions of the equation y″ + p(t)Ay = 0 in a Banach space, where A is a bounded linear operator and p: ℝ+ → ℝ+ is a continuous function. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 571–576, April, 2007.  相似文献   

5.
We consider a variant of Heilbronn’s triangle problem by investigating for a fixed dimension d≥2 and for integers k≥2 with kd distributions of n points in the d-dimensional unit cube [0,1] d , such that the minimum volume of the simplices, which are determined by (k+1) of these n points is as large as possible. Denoting by Δ k,d (n), the supremum of this minimum volume over all distributions of n points in [0,1] d , we show that c k,d ⋅(log n)1/(dk+1)/n k/(dk+1)Δ k,d (n)≤c k,d ′/n k/d for fixed 2≤kd, and, moreover, for odd integers k≥1, we show the upper bound Δ k,d (n)≤c k,d ″/n k/d+(k−1)/(2d(d−1)), where c k,d ,c k,d ′,c k,d ″>0 are constants. A preliminary version of this paper appeared in COCOON ’05.  相似文献   

6.
In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace τ, and {α t }, a strongly continuous extension to L p (M, τ) of a semigroup of absolute contractions on L 1(M, τ). By means of a non-commutative Banach Principle we prove for a Besicovitch function b and xL p (M, τ), that the averages 1/T0 T b(t)α t (x)dt converge bilateral almost uniformly in L p (M, τ) as T → 0. Communicated by Dénes Petz  相似文献   

7.
Let B H,K = {B H,K (t)} t⩾0 be a bifractional Brownian motion with parameters H ∈ (0, 1) and K ∈ (0, 1]. For a function Φ: [0, ∞) → [0, ∞) and for a partition κ = {t i }n i=0 of an interval [0, T] with T > 0, let {ie418-01}. We prove that, for a suitable Φ depending on H and K, {ie418-02} almost surely. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-16/08  相似文献   

8.
Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, BB(H), define the binary relation A ≤* B by A*A = A*B and AA* = AB*. Then (B(H), “≤*”) is a partially ordered set and the relation “≤*” is called the star order on B(H). Denote by Bs(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B s (H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.  相似文献   

9.
10.
In this paper,we will use the 2r-th Ditzian-Totik modulus of smoothness wp^2r(f,t)p to discuss the direct and inverse theorem of approximation by Left-Bernstein-Durrmeyer quasi-interpolants Mn^[2r-1]f for functions of the space Lp[0,1](1≤p≤ ∞)。  相似文献   

11.
Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on . Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let Aq(K, μ) denote the closure of A(K) in Lq(μ). The aim of this work is to study the structure of the space Aq(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for Aq(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for Pq(μ), the closure of polynomials in Lq(μ), as the special case when K is a closed disk containing the support of μ.  相似文献   

12.
Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) n assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)  相似文献   

13.
We investigate the behaviour of the logarithmic small deviation probability of a sequence (σ n θ n ) in l p , 0<p≤∞, where (θ n ) are i.i.d. random variables and (σ n ) is a decreasing sequence of positive numbers. In particular, the example σ n n μ (1+log n)ν is studied thoroughly. Contrary to the existing results in the literature, the rate function and the small deviation constant are expressed expli- citly in the present treatment. The restrictions on the distribution of θ 1 are kept to an absolute minimum. In particular, the usual variance assumption is removed. As an example, the results are applied to stable and Gamma-distributed random variables.  相似文献   

14.
Let Tn be the full transformation semigroup on the n-element set Xn. For an arbitrary integer r such that 2 ≤ r ≤ n-1, we completely describe the maximal subsemigroups of the semigroup K(n, r) = {α∈Tn : |im α| ≤ r}. We also formulate the cardinal number of such subsemigroups which is an answer to Problem 46 of Tetrad in 1969, concerning the number of subsemigroups of Tn.  相似文献   

15.
In this paper we prove that an elementary Abelian p-group of rank 4p−2 is not a CI(2)-group, i.e. there exists a 2-closed transitive permutation group containing two non-conjugate regular elementary Abelian p-subgroups of rank 4p−2, see Hirasaka and Muzychuk (J. Comb. Theory Ser. A 94(2), 339–362, 2001). It was shown in Hirasaka and Muzychuk (loc cit) and Muzychuk (Discrete Math. 264(1–3), 167–185, 2003) that this is related to the problem of determining whether an elementary Abelian p-group of rank n is a CI-group. As a strengthening of this result we prove that an elementary Abelian p-group E of rank greater or equal to 4p−2 is not a CI-group, i.e. there exist two isomorphic Cayley digraphs over E whose corresponding connection sets are not conjugate in Aut E. This research was supported by a fellowship from the Pacific Institute for the Mathematical Sciences.  相似文献   

16.
Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. The finite groups for which ν(G) ≤ 2 were determined by Dedekind and by Schmidt in the early times of group theory. On the other hand, if G is a finite p-group, La Haye and Rhemtulla have proved that either ν(G) ≤ 1 or ν(G) ≥ p. In this note, we determine all finite p-groups satisfying ν(G) = p for p > 2.  相似文献   

17.
For an integer k > 0, a graph G is k-triangular if every edge of G lies in at least k distinct 3-cycles of G. In (J Graph Theory 11:399–407 (1987)), Broersma and Veldman proposed an open problem: for a given positive integer k, determine the value s for which the statement “Let G be a k-triangular graph. Then L(G), the line graph of G, is s-hamiltonian if and only L(G) is (s + 2)-connected” is valid. Broersma and Veldman proved in 1987 that the statement above holds for 0 ≤ sk and asked, specifically, if the statement holds when s = 2k. In this paper, we prove that the statement above holds for 0 ≤ s ≤ max{2k, 6k − 16}.  相似文献   

18.
New subgeometry partitions of PG(n − 1, q m ) by subgeometries isomorphic to PG(n − 1, q) are constructed.   相似文献   

19.
Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all i ≥ 0: (i) exp Ωi(G) ≤ p i for odd p, and expΩi(G) ≤ 2 i+1 for p = 2; (ii) the index |G: G p i| coincides with the number of elements of G of order at most p i. Supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the University of the Basque Country, grant UPV05/99.  相似文献   

20.
Let D = (V, E) be a primitive digraph. The vertex exponent of D at a vertex v∈ V, denoted by expD(v), is the least integer p such that there is a v →u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that exPD(V1) ≤ exPD(V2) …≤ exPD(Vn). Then exPD(Vk) is called the k- point exponent of D and is denoted by exPD (k), 1≤ k ≤ n. In this paper we define e(n, k) := max{expD (k) | D ∈ PD(n, 2)} and E(n, k) := {exPD(k)| D ∈ PD(n, 2)}, where PD(n, 2) is the set of all primitive digraphs of order n with girth 2. We completely determine e(n, k) and E(n, k) for all n, k with n ≥ 3 and 1 ≤ k ≤ n.  相似文献   

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