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1.

Let τ(

We also show that

where ω(

*n*) be the Ramanujan τ-function,*x*≥ 10 be an integer parameter. We prove that*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.

N. G. Kuznetsov 《Journal of Mathematical Sciences》2008,150(1):1856-1859

The combinatorial identity

is established with the help of the differentiation of the convolution of some function with the sine function. Bibliography:
5 titles.
__________
Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 65–67. 相似文献

3.

4.

Fa-en WU~ 《中国科学A辑(英文版)》2007,50(8):1078-1086

Let D be a bounded domain in an n-dimensional Euclidean space Rn. Assume that 0 ＜ λ1 ≤λ2 ≤ … ≤ λκ ≤ … are the eigenvalues of the Dirichlet Laplacian operator with any order l{(-△)lu=λu, in D u=(δ)u/(δ)(→n)=…(δ)l-1u/(δ)(→n)l-1=0,on (δ)D.Then we obtain an upper bound of the (k 1)-th eigenvalue λκ 1 in terms of the first k eigenvalues.k∑i=1(λκ 1-λi) ≤ 1/n[4l(n 2l-2)]1/2{k∑i=1(λκ 1-λi)1/2λil-1/l k∑i=1(λκ 1-λi)1/2λ1/li}1/2.This ineguality is independent of the domain D. Furthermore, for any l ≥ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang. 相似文献

5.

M. N. Yakovlev 《Journal of Mathematical Sciences》2007,141(6):1710-1722

It is proved that the boundary-value problem

, has a solution, provided that the following conditions are fulfilled:

, and, for ϕ(u) ≡ 0, the Galerkin method converges in the norm of the space H

^{1}(a, b; a). Several theorems of a similar kind are presented. Bibliography: 4 titles. __________ Translated from*Zapiski Nauchnykh Seminarov POMI*, Vol. 334, 2006, pp. 246–266. 相似文献6.

We develop structural formulas satisfied by some families of orthogonal matrix polynomials of size 2 × 2 satisfying second-order
differential equations with polynomial coefficients. We consider here three one-parametric families of weight matrices, namely,

and

and their corresponding orthogonal polynomials. We also show that the orthogonal polynomials with respect to the second family
are eigenfunctions of two linearly independent second-order differential operators. 相似文献

7.

In the present paper, the following Dirichlet problem and Neumann problem involving the

and

are studied and some new multiplicity results of solutions for systems (1.λ) and (2.λ) are obtained. Moreover, by using the
KKM principle we give also two new existence results of solutions for systems (1.1) and (2.1).
This Work is supported in part by the National Natural Science Foundation of China (10561011). 相似文献

*p*-Laplacian((1.λ)) |

((2.λ)) |

8.

9.

Wang Lei Pan Ting Dept. of Math. Zhejiang Univ. Hangzhou China. Univ. of International Relation Hangzhou China. 《高校应用数学学报(英文版)》2004,19(2):212-222

Ibαf ( x) =∫R ∏mj=1( bj( x) - bj( y) ) 1| x - y| n-αf ( y) dyare considered.The following priori estimates are proved.For 1

01Φ1t| {y∈Rn:| Ibαf( y) | >t}| 1q ≤csupt>01Φ1t| {y∈Rn:ML( log L) 1r ,α(‖b‖f ) ( y) >t}| 1q,where‖b‖=∏mj=1‖bj‖Oscexp Lrj,Φ( t) =t( 1 + log+t) 1r,1r =1r1+ ...+ 1rm,ML(… 相似文献

10.

V. V. Vysotsky 《Journal of Mathematical Sciences》2007,147(4):6873-6883

Let S

for 0 ≤ t ≤ 1. We also show that

, where the U

_{i}be a random walk with standard exponential increments. The sum ∑_{i=1}^{k}S_{i}is called the k-step area of the walk. The random variable ∑_{i=1}^{k}S_{i}plays an important role in the study of the so-called one-dimensional sticky particles model. We find the distribution of this variable and prove that_{i,n}are order statistics of n i.i.d. random variables uniformly distributed on*[0, 1]*. Bibliography:*6*titles. __________ Translated from*Zapiski Nauchnykh Seminarov POMI*, Vol. 341, 2007, pp. 48–67. 相似文献