共查询到20条相似文献，搜索用时 875 毫秒
1.
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
2.
Aleksandar Ivić 《Archiv der Mathematik》2008,90(5):412-419
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
3.
Xianling Fan Shao-Gao Deng 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(2):255-271
We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving
the p(x)-Laplacian of the form
where Ω is a bounded smooth domain in , and p(x) > 1 for with and φ ≢ 0 on ∂Ω. Using the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that, there exists λ* > 0 such that the problem has at least two positive solutions if λ = λ*, has at least one positive solution if λ = λ*, and has no positive solution if λ = λ*. To prove the result we establish a special strong comparison principle for the Neumann problems.
The research was supported by the National Natural Science Foundation of China 10371052,10671084). 相似文献
4.
Filip Saidak 《Archiv der Mathematik》2005,85(4):345-361
Assuming a quasi Generalized Riemann Hypothesis (quasi-GRH for short) for Dedekind zeta functions over Kummer fields of the
type
we prove the following prime analogue of a conjecture of Erd?s & Pomerance (1985) concerning the exponent function fa(p) (defined to be the minimal exponent e for which ae ≡ 1 modulo p):
where
The main result is obtained by computing all the higher moments corresponding to ω(fa(p)), and by comparing them, via the Fréchet-Shohat theorem, with estimates due to Halberstam concerning the moments of ω(p − 1).
Received: 25 October 2004; revised: 12 February 2005 相似文献
| ((‡)) |
5.
We give the exact closed form solution of the following ordinary differential equation:
which is a modified logistic one, wherein x(t) is the population of a homogeneous species x at time t. Other than integrating the above nonlinear differential equation by means of Mathieu functions of the first kind, we also
provide a condition of a couple of inequalities involving a, b, c, h and x
0 whose fulfillment is sufficient to ensure that a bounded solution for x(t) there exists.
相似文献
6.
In this paper we prove that any residue class λ modulo a large prime number p can be represented in the form
for some positive integers m1, n1,... ,m5, n5 of the size O(p27/28). This improves one of the results from [6] on representability of λ modulo p in the form
with
. We also prove that any residue class modulo p can be represented in the form
with
. This improves the result of [7].
Received: 27 March 2006 相似文献
7.
Aleksandar Ivić 《Monatshefte für Mathematik》2005,145(1):35-46
Let P(n) denote the largest prime factor of an integer n (2), and let N (x) denote the number of natural numbers n such that 2 n x, and n does not divide P(n)!. We prove that
where (u) is the Dickman-de Bruijn function. In terms of elementary functions we have
thereby sharpening and correcting recent results of K. Ford and J.-M. De Koninck and N. Doyon. 相似文献
8.
The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations 总被引：1，自引：0，他引：1
Kentaro Hirata 《Mathematische Annalen》2008,340(3):625-645
We investigate the boundary growth of positive superharmonic functions u on a bounded domain Ω in , n ≥ 3, satisfying the nonlinear elliptic inequality
where c > 0, α ≥ 0 and p > 0 are constants, and is the distance from x to the boundary of Ω. The result is applied to show a Harnack inequality for such superharmonic functions. Also, we study
the existence of positive solutions, with singularity on the boundary, of the nonlinear elliptic equation
where V and f are Borel measurable functions conditioned by the generalized Kato class. 相似文献
9.
Zhian Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(3):399-418
We derive the optimal decay rates of solution to the Cauchy problem for a set of nonlinear evolution equations with ellipticity
and dissipative effects
with initial data
where α and ν are positive constants such that α < 1, ν < α(1 − α), which is a special case of (1.1). We show that the solution
to the system decays with the same rate to that of its associated homogenous linearized system. The main results are obtained
by the use of Fourier analysis and interpolation inequality under some suitable restrictions on coefficients α and ν. Moreover,
we discuss the asymptotic behavior of the solution to general system (1.1) at the end.
The research was supported by the F. S. Chia Scholarship of the University of Alberta.
Received: January 27, 2005; revised: April 27, 2005 相似文献
10.
We consider the Cauchy problem for the Perona–Malik equation
in a bounded open set , with Neumann boundary conditions.
If n = 1, we prove some a priori estimates on u and u
x
. Then we consider the semi-discrete scheme obtained by replacing the space derivatives by finite differences. Extending the
previous estimates to the discrete setting we prove a compactness result for this scheme and we characterize the possible
limits in some cases. Finally, for n > 1 we give examples to show that the corresponding estimates on are in general false. 相似文献
11.
Xiaojing Yang 《Archiv der Mathematik》2005,85(5):460-469
In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator
is discussed, where α, β are positive constants satisfying
for some ω ∈R+ /Q, h(t) ∈L∞ [0, 2π ] is 2π-periodic, x±=max {±x, 0 }.
Received: 23 September 2004 相似文献
12.
13.
We establish a new bound for the exponential sum
where λ is an element of the residue ring modulo a large prime number
and
are arbitrary subsets of the residue ring modulo p − 1 and γ (n) are any complex numbers with | γ (n)| ≦ 1.
Received: 15 June 2005 相似文献
14.
Wen-Bin Zhang 《Mathematische Zeitschrift》2009,261(1):233-234
Let be the generalized integers n
j
associated with a set of generalized primes p
i
in Beurling’s sense. On the basis of the general mean-value theorems, established in our previous work, for multiplicative
function f(n
j
) defined on , we prove extensions, in functional form and in mean-value form, of the Elliott–Daboussi theorem to high order mean-values.
For the main result, let α,ρ, and τ be positive real constants such that α > 1,ρ≥1 and . Then a multiplicative function f satisfies the following conditions, with some constant , (1) All four series
converge and (2)
if and only if the order τρ mean-value
exists with and the limit
exists with . The proof is deduced from an intrinsic connection between m
f
and .
An erratum to this article can be found at 相似文献
15.
We consider the following semilinear elliptic equation with singular nonlinearity:
where
and Ω is an open subset in
. Let u be a non-negative finite energy stationary solution and
be the rupture set of u. We show that the Hausdorff dimension of Σ is less than or equal to [(n−2) α+(n+2)]/(α +1). 相似文献
16.
Jérôme Droniou Juan-Luis Vázquez 《Calculus of Variations and Partial Differential Equations》2009,34(4):413-434
We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation
posed in a bounded domain , with pure Neumann boundary conditions
Under the assumption that with p = N if N ≥ 3 (resp. p > 2 if N = 2), we prove that the problem has a solution if ∫Ω
f
dx = 0, and also that the kernel is generated by a function , unique up to a multiplicative constant, which satisfies a.e. on Ω. We also prove that the equation
has a unique solution for all ν > 0 and the map is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation
The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure
data and to parabolic problems. 相似文献
17.
Zhaoli Liu Jiabao Su Zhi-Qiang Wang 《Calculus of Variations and Partial Differential Equations》2009,35(4):463-480
In this paper, we study existence of nontrivial solutions to the elliptic equation
and to the elliptic system
where Ω is a bounded domain in with smooth boundary ∂Ω, , f (x, 0) = 0, with m ≥ 2 and . Nontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, for and , and for and , where I
m
is the m × m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity
on the asymptotic behaviors of the nonlinearity f and .
Z. Liu was supported by NSFC(10825106, 10831005). J. Su was supported by NSFC(10831005), NSFB(1082004), BJJW-Project(KZ200810028013)
and the Doctoral Programme Foundation of NEM of China (20070028004). 相似文献
18.
We consider one-dimensional difference Schr?dinger equations with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V
0(x) of degree k
0, and assume positive Lyapunov exponents and Diophantine ω. We prove that the integrated density of states is H?lder continuous for any k > 0. Moreover, we show that is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem
on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval
, does not exceed , k > 0. Moreover, for , this averaged number does not exceed exp , for any . For the integrated density of states of the problem this implies that for any . To investigate the distribution of the Dirichlet eigenvalues of on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus and show also that no more than 2k
0 of them fall into any disk of radius exp. In addition, we obtain the lower bound (with δ > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies.
Received: February 2006, Accepted: December 2007 相似文献
19.
In this paper we establish an asymptotic formula for the sum
when y is large compared to x1/2 log x.
Received: 27 January 2005 相似文献