共查询到20条相似文献,搜索用时 15 毫秒
1.
Dongho Byeon 《Journal of Number Theory》2011,131(8):1513-1529
Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x2+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for and consecutive integers x=x0,x0+1,…,x0+s−1, |fm(x)| is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials fm(x). 相似文献
2.
《Journal of Computational and Applied Mathematics》1998,88(1):103-118
We are concerned with the discrete focal boundary value problem Δ3x(t − k) = f(x(t)), x(a) = Δx(t2) = Δ2x(b + 1) = 0. Under various assumptions on f and the integers a, t2, and b we prove the existence of three positive solutions of this boundary value problem. To prove our results we use fixed point theorems concerning cones in a Banach space. 相似文献
3.
Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x2+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for and consecutive integers is either 1 or prime. In Byeon (J. Number Theory 94 (2002) 177), we showed that there are only finitely many Rabinowitsch polynomials fm(x) such that 1+4m is square free. In this note, we shall remove the condition that 1+4m is square free. 相似文献
4.
C. Aistleitner 《Acta Mathematica Hungarica》2010,129(1-2):1-23
Let (n k ) k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying n k+1/n k > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝ 0 1 f(x) dx = 0. Then the probabilistic behavior of the system (f(n k x)) k≧1 is very similar to the behavior of sequences of i.i.d. random variables. For example, Erd?s and Gál proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos 2πx and lacunary $ (n_k )_{k \geqq 1} $ : (1) $$ \mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2 $$ for almost all x ∈ (0, 1), where ‖f‖2 = (∝ 0 1 f(x)2 dx)1/2 is the standard deviation of the random variables f(n k x). If (n k ) k≧1 has certain number-theoretic properties (e.g. n k+1/n k → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always ‖f‖2. For general lacunary (n k ) k≧1 this is not necessarily true: Erd?s and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (n k ) k≧1, such that the lim sup in the LIL (1) is not equal to ‖f‖2 and not even a constant a.e. In this paper we show that the class of possible functions on the right-hand side of (1) can be very large: we give an example of a trigonometric polynomial f such that for any function g(x) with sufficiently small Fourier coefficients there exists a lacunary sequence (n k ) k≧1 such that (1) holds with √‖f‖ 2 2 + g(x) instead of ‖f‖2 on the right-hand side. 相似文献
5.
W.R Lawrence 《Journal of Number Theory》1980,12(2):201-209
Let f(x, y) be an indefinite binary quadratic form, d(f) its discriminant, m(f) the infimum of |f(x, y)| over all integers x, y not both zero, and put . In this paper we prove the existence of countably many disjoint open intervals Ij contained in such that there is no f with μ(f) in Ij (j = 1, 2,…) and such that for any interval I containing two intervals Ij, Ik there is an f with μ(f) in I. 相似文献
6.
R.S. Singh 《Journal of multivariate analysis》1976,6(1):111-122
On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators fn(p) of f(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on Rm, p = (p1,…,pm), pj ≥ 0, fixed integers, and for x = (x1,…,xm) in Rm, f(p)(x) = ?p1+…+pm f(x)/(?p1x1 … ?pmxm). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of fn(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of fn(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out. 相似文献
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8.
M. Eshaghi Gordji H. Khodaei 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(11):5629-5643
In this paper, we achieve the general solution and the generalized Hyers–Ulam–Rassias stability of the following functional equation
f(x+ky)+f(x−ky)=k2f(x+y)+k2f(x−y)+2(1−k2)f(x)