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1.
Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the are known for . Upper bounds for were first given by Jaeschke, and those for were then sharpened by the first author in his previous paper (Math. Comp. 70 (2001), 863-872).

In this paper, we first follow the first author's previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes (spsp's) to the first five or six prime bases, which have the form with odd primes and ; then we tabulate all Carmichael numbers , to the first six prime bases up to 13, which have the form with each prime factor . There are in total 36 such Carmichael numbers, 12 numbers of which are also spsp's to base 17; 5 numbers are spsp's to bases 17 and 19; one number is an spsp to the first 11 prime bases up to 31. As a result the upper bounds for and are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit number:

We conjecture that

and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor and propose necessary conditions on to be a strong pseudoprime to the first prime bases. Comparisons of effectiveness with Arnault's, Bleichenbacher's, Jaeschke's, and Pinch's methods for finding (Carmichael) numbers with three prime factors, which are strong pseudoprimes to the first several prime bases, are given.
2.

3.
For ann x n real matrixX, let ϕ(X)=X ο (X −1) T , where ο stands for the Hadamard (entrywise) product. SupposeA, B, C andD aren x n real nonsingular matrices, and among them there are at least one solutions to the equation ϕ(X)=1/nJ n . An equivalent condition which enable $M = \left( {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right)$ become a real solution to the equation ϕ(X)=1/2nJ 2n , is given. As applications, we get new real solutions to the matrix equation ϕ(X)-1/2nJ 2n by applying the results of Zhang, Yang and Cao [SIAM. J. Matrix Anal. Appl, 21 (1999), pp: 642–645] and Chen [SIAM. J. Matrix Anal. Appl, 22 (2001), pp:965–970]. At the same time, all solutions of the matrix equation ϕ(X)=1/4J 4 are also given.  相似文献
4.
Based on the double penalized estimation method,a new variable selection procedure is proposed for partially linear models with longitudinal data.The proposed procedure can avoid the effects of the nonparametric estimator on the variable selection for the parameters components.Under some regularity conditions,the rate of convergence and asymptotic normality of the resulting estimators are established.In addition,to improve efficiency for regression coefficients,the estimation of the working covariance matrix is involved in the proposed iterative algorithm.Some simulation studies are carried out to demonstrate that the proposed method performs well.  相似文献
5.
Let k≥2 be an integer,and let σ(n) denote the sum of the positive divisors of an integer n.We call n a quasi-multiperfect number if σ(n)=kn+1.In this paper,we give some necessary properties of them.  相似文献
6.
The loop-Witt algebra is the Lie algebra of the tensor product of the Witt algebra and the Laurent polynomial algebra. In this paper we study the universal central extension, derivations and automorphism group for the loop-Witt algebra.  相似文献
7.
In this note,one kind of insurance risk models with the policies having multiple validity times are investigated.Explicit expressions for the ruin probabilities are obtained by using the martingale method.As a consequence,the obtained probability serves as an upper bound for the ruin probability of a newly developed entrance processes based risk model.  相似文献
8.

9.
Min Tang   《Discrete Mathematics》2009,309(21):6288-6293
Let A={a1,a2,…}(a1<a2<) be an infinite sequence of nonnegative integers, let k≥2 be a fixed integer and denote by rk(A,n) the number of solutions of ai1+ai2++aikn. Montgomery and Vaughan proved that r2(A,n)=cn+o(n1/4) cannot hold for any constant c>0. In this paper, we extend this result to k>2.  相似文献
10.
Min Tang  Yong-Gao Chen   《Discrete Mathematics》2009,309(21):6294-6298
Let A={a1,a2,…}(a1<a2<) be an infinite sequence of nonnegative integers. Let k≥2 be a fixed integer and for , let Rk(A,n) be the number of solutions of ai1++aik=n,ai1,…,aikA, and let and denote the number of solutions with the additional restrictions ai1<<aik, and ai1≤≤aik respectively. Recently, Horváth proved that if d>0 is an integer, then there does not exist n0 such that for n>n0. In this paper, we obtain the analogous results for Rk(A,n), and .  相似文献