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1.
For an irrational number x and n ≥ 1, we denote by k n (x) the exact number of partial quotients in the continued fraction expansion of x given by the first n decimals of x. G. Lochs proved that for almost all x, with respect to the Lebesgue measure In this paper, we prove that an iterated logarithm law for {k n (x): n ≥ 1}, more precisely, for almost all x, for some constant σ > 0. Author’s address: Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China  相似文献   

2.
Let K be either the rational number field or an imaginary quadratic field. We give irrationality results for the number , where q (∣q∣ > 1) is an integer in K, rK × (∣r∣ < ∣q∣), and with q n r l (n ≥ 1). Authors’ addresses: Kenji Amano, NS solutions Corporation, 2-27-1 Shinkawa, Chuo-ku, Tokyo 104-0033, Japan; Yohei Tachiya, Department of Mathematics, Keio University, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan  相似文献   

3.
In this paper we consider special elements of the Fock space #x2131; n . That is the space of entire functionsf:ℂ: n →ℂ, such that the followingL 2- condition is satisfied: . Here we show that there exists an entire functiong:ℂ n →ℂ such that for every one-dimensional subspace Π⊂ℂ n and for all 0<∈<2 we have , but in the limit case ∈=0 we have . This result is analogue to a result from [1]. There holomorphic functions on the unit-ball are investigated. Furthermore the proof — as the one in [1] — uses a theorem from [2]. Therefore we give another application of the results from [2] — namely for spaces of entire functions.  相似文献   

4.
In [2], it was shown that if a and b are multiplicatively independent integers and ɛ > 0, then the inequality gcd (an − 1,bn − 1) < exp(ɛn) holds for all but finitely many positive integers n. Here, we generalize the above result. In particular, we show that if f(x),f1(x),g(x),g1(x) are non-zero polynomials with integer coefficients, then for every ɛ > 0, the inequality holds for all but finitely many positive integers n.  相似文献   

5.
Let be the modular curve associated to a congruence subgroup Γ of level N with , and let be its canonical model over . The main aim of this paper is to show that the endomorphism algebra of its Jacobian is generated by the Hecke operators T p , with , together with the “degeneracy operators” D M,d , D t M,d , for . This uses the fundamental results of Ribet on the structure of together with a basic result on the classification of the irreducible modules of the algebra generated by these operators. Received: 18 December 2007  相似文献   

6.
Let τ(n) be the Ramanujan τ-function, x ≥ 10 be an integer parameter. We prove that
We also show that
where ω(n) is the number of distinct prime divisors of n and p denotes prime numbers. These estimates improve several results from [6, 9]. Received: 23 November 2006  相似文献   

7.
In this note we are interested in the graded modulesM k=I(k)/Ik and , whereI is a saturated ideal in the homogeneous coordinate ringS=K[x0,…,xn] of ℙn,I (k) is the symbolic power and is the saturation of the ordinary power. Very little is known about these modules, and we provide a bound on their diameters, we compute the Hilbert functions and we study some characteristic submodules in the special case ofn+1 general points in ℙn.
Sunto In questa nota siamo interessati ai moduli graduatiM k=I(k)/Ik e , doveI è un ideale saturato nell'anello delle coordinate omogeneeS:=K[x0,…,xn] di ℙn,I (k) è la potenza simbolica e è la saturazione della potenza ordinaria. Poco è noto su questi moduli e qui viene fornito un limite superiore ai loro diametri. Ne calcoliamo inoltre le funzioni di Hilbert e studiamo alcuni sottomoduli caratteristici nel caso speciale din+1 punti in posizione generale, in ℙn.
  相似文献   

8.
A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, yX. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (xn) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (xn) be aT-martingale taking on values inH. If then x n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (xn) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X * of norm 1 such that If, in addition, the spaceX is strictly convex, x n /n converges weakly; and if the norm ofX * is Fréchet differentiable (away from zero), x n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093  相似文献   

9.
If
denotes the error term in the classical Rankin-Selberg problem, then it is proved that
where Δ1(x) = ∫ x 0 Δ(u)du. The latter bound is, up to ‘ɛ’, best possible. Received: 8 February 2007  相似文献   

10.
Let an≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα n ≡0, then Bn (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q n (k) (α n ,f(t))=d k /dx k B n+k (α n ,F k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα n ≡0, k=1, then Qn (1) (0, f(t), x) are Kantorovič polynomials Pn(f). Ifα n =0, k=2, then Qn (2), (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈LP [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define[1–2],[8–10]: . As usual, for the space Lp [a,b](1≤p<∞), we have and L[a, b]=l1[a, b]. Letα n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials [3] [4]. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.  相似文献   

11.
We are studying the Diophantine exponent μ n,l defined for integers 1≤l<n and a vector α∈ℝ n by letting
where is the scalar product, denotes the distance to the nearest integer and is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the exponent takes all values in the interval [l+1,∞), with the value n attained for almost all α. We calculate the Hausdorff dimension of the set of vectors α with μ n,l (α)=μ for μ≥n. Finally, letting w n denote the exponent obtained by removing the restrictions on , we show that there are vectors α for which the gaps in the increasing sequence μ n,1(α)≤...≤μ n,n-1(α)≤w n (α) can be chosen to be arbitrary.  相似文献   

12.
For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L2(ℝ), such that the generating function g∈L2(ℝ) satisfies the condition:∫|g(x)|2(1+|x| A )/log d (2+|x|)dx < ∞ and .  相似文献   

13.
A Robin boundary problem with Hardy potential and critical nonlinearities   总被引:1,自引:0,他引:1  
Let Ω be a bounded domain with a smooth C 2 boundary in ℝn (n ≥ 3), 0 ∈ , and υ denote the unit outward normal to ∂Ω. In this paper, we are concerned with the following class of boundary value problems:
(*)
where 2* = 2n/(n − 2) is the limiting exponent for the embedding of H 1(Ω) into L p (Ω), 2 < p < 2*, , η ≥ 0 and λ ∈ ℝ1 are parameters, and α(x) ∈ C(∂Ω), α(x) ≥ 0. Through a compactness analysis of the functional corresponding to the problem (*), we obtain the existence of positive solutions for this problem under various assumptions on the parameters μ, λ and the fact that 0 ∈ Ω or 0 ∈ ∂Ω. The research was supported by NSFC(10471052, 10571069, 10631030) and the Key Project of Chinese Ministry of Education(107081) and NCET-07-0350.  相似文献   

14.
In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.  相似文献   

15.
Let G be a simple graph with n vertices. For any , let , and , and and u and v has distance 2 in E(G)}. Let l ≥ 1 be an integer. A graph G on nl vertices is [l, n]-pan-connected if for any , and any integer m with lmn, G has a (u, v)-path of length m. In 1998, Wei and Zhu (Graphs Combinatorics 14:263–274, 1998) proved that for a three-connected graph on n ≥ 7 vertices, if NC(G) ≥ n − δ(G) + 1, then G is [6, n]-pan-connected. They conjectured that such graphs should be [5, n]-pan-connected. In this paper, we prove that for a three-connected graph on n ≥ 7 vertices, if NC 2(G) ≥ n − δ(G) + 1, then G is [5, n]-pan-connected. Consequently, the conjecture of Wei and Zhu is proved as NC 2(G) ≥ NC(G). Furthermore, we show that the lower bound is best possible and characterize all 2-connected graphs with NC 2(G) ≥ n − δ(G) + 1 which are not [4, n]-pan-connected.   相似文献   

16.
In this paper, we prove that if a sequence of homeomorphisms , with bounded planar domains, of Sobolev space has uniformly equibounded distortions in EXP(Ω) and weakly converges to f in then the matrices A(x, f j ) of the corresponding Laplace-Beltrami operators Γ-converge in the Orlicz–Sobolev space , where Q(t) = t 2log(e + t), to the matrix A(x, f) of the Laplace-Beltrami operator associated to f.   相似文献   

17.
Let F be a subfield of a commutative field extending ℝ. Let We say thatf : preserves distanced ≥ 0 if for eachx,y ∈ ℝ ∣x- y∣= d implies ϕ2(f(x),f(y)) = d2 . We prove that each unit-distance preserving mappingf : has a formI o (ρ,ρ), where is a field homomorphism and is an affine mapping with orthogonal linear part.  相似文献   

18.
The problem (where B is a unit ball in R n )
, with , is known to have a curve of positive solutions bifurcating from infinity at λ = λ1, the principal eigenvalue. It turns out that a similar situation may occur, when g(u) is oscillatory for large u, instead of being small. In case n = 1, we can also prove existence of infinitely many solutions at λ = λ1 on this curve. Similarly, we consider oscillatory bifurcation from zero.   相似文献   

19.
Book Reviews     
For , let E*, λ*) be the set It has been proved in [1] and [3] that E*, λ*) is an uncountable set. In the present paper, we strengthen this result by showing that where dim denotes the Hausdorff dimension.  相似文献   

20.
In this paper we consider , in one case that fx 0 (t) is a ΛBMV function on [0, ∞], and in another case thatfεL 1 m-1(Rn) and when |k|=m−1 and f(x)=0 when |x−x0|<δ for some δ>0. Our theormes improve the results of Pan Wenjie ([1]).  相似文献   

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