Divisibility Properties of Power Matrices Associated with Arithmetic Functions on a Divisor Chain

Let a,b and n be positive integers withn≥2,f be an integer-valued arithmetic function,and the set S={x_(1),…,x_(n)}of n distinct positive integers be a divisor chain such that x_(1)|x_(2)|⋯|x_(n).We first show that the matrix(f_(a)(S))having f evaluated at the ath power(x_(i),x_(j))^(a) of the greatest common divisor of x_(i) and x_(j) as its i,j-entry divides the GCD matrix(f^(b)(S))in the ring M_(n)(Z)of n×n matrices over integers if and only if f^(b−a)(x_(1))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Consequently,we show that the matrix(f^(a)[S])having f evaluated at the ath power[x_(i),x_(j)]^(a) of the least common multiple of x_(i) and x_(j) as its i,j-entry divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(b−a)(x_(n))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with2≤i≤n.Finally,we prove that the matrix(f^(a)(S))divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(a)(x_(1))|f^(b)(x_(i))and(f^(a)(x_(i))−f^(a)(x_(i−1)))|(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Our results extend and strengthen the theorems of Hong obtained in 2008.

Characterization of Arithmetic Functions that Preserve the Sum-of-squares Operation

We characterize all functions f ： N → C such that f（m^2 ＋ n^2） = f（m）^2 ＋ f（n）^2 for all m, n ∈ N. It turns out that all such functions can be grouped into three families, namely f ≡ 0, f（n） = ±n （subject to some restrictions on when the choice of the sign is possible） and f（n） =±l/2（again subject to some restrictions on when the choice of the sign is possible）.

Boundedness and Spectrum of Multiplicative Convolution Operators Induced by Arithmetic Functions

In this paper, we consider a multiplicative convolution operator Mf acting on a Hilbert spaces l^2(N,ω;). In particular, we focus on the operators M1 and Mμ, where μ, is the Mobius function. We investigate conditions on the weight ω under which the operators M1 and Mμ are bounded. We show that for a positive and completely multiplicative function f,M1 is bounded on l^2(N, f^2) if and only if ||f||1 <∞, in which case ||M1||2,ω=||f||1, where ωn = f^2(n). Analogously, we show that Mμ is bounded on l^2(N, 1/n^2α) with ||M1||2,ω=ζ(α)/ζ(2α),where ωn= 1 /n^2α,α> 1. As an application, we obtain some results on the spectrum of M1^*M1 and M^*μMμ. Moreover, von Neumann algebra generated by a certain family of bounded operators is also considered.

NOTES ON ERDOS' CONJECTURE

Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The authors investigate a conjecture of Erdos in probabilistic number theory and show that in order for the sequence FN to be weakly convergent, it is both sufficient and necessary that there exist three numbers X0 and X1 < X2 such that limsup(FN(X2) - FN(X1)) > 0 holds, and Lo = N→ ∞ lim FN(X0) exists. Moreover, the authors point out that they can also obtain the same result in the weakened case of lim inf P(Xn = 0) > 0.

Powers of General Digital Sums

Let m0,m1,m2,…be positive integers with mi〉 2 for all i. It is well known that each nonnegative integer n can be uniquely represented as n= a0 ＋ a1m0＋a2m0m1＋…＋atm0m1m2…mt-1,where 0≤ai≤mi-1 for all i and at≠0.let each fi be a function defined on {0,1,2…,mi-1} with fi（0）=0.write S（n）=i=0∑tfi（ai）.In this paper, we give the asymptotic formula for x^-1∑n≤xS（n）^k,where k is a positive integer.

数论函数Ω_k的动力系统(英文)

In this paper, we generalize the results of Goldring W. in 2006 and study dynamics of the arithmetic function Ωk.

Engineering DNA logic systems with non-canonical DNA-nanostructures:basic principles,recent developments and bio-applications

Engineering DNA logic systems is considered as one of the most promising strategies for next-generation molecular computers.Owing to the inherent features of DNA,such as low cost,easy synthesis,and controllable hybridization,various DNA logic devices with different functions have been developed in the recent decade.Besides,a variety of logic-programmed biological applications are also explored,which initiates a new chapter for DNA logic computing.Although this field has gained rapid developments,a systematical review that could not only elaborate the logic principles of diverse DNA logic devices but also outline recent representative works is urgently needed.In this review,we first elaborate the general classification and logical principle of diverse DNA logic devices,in which the operating strategy of these devices and representative examples are selectively presented.Then,we review state-of-the-art advancements in DNA computing based on different non-canonical DNA-nanostructures during the past decade,in which some classical works are summarized.After that,the innovative applications of DNA computing to logic-controlled bioanalysis,cell imaging,and drug load/delivery are selectively presented.Finally,we analyze current obstacles and suggest appropriate prospects for this area.

On the Equation n1n2= n3n4 Restricted to Factor Closed Sets

We study the number of solutions N（B,F） of the diophantine equation n1n2 = n3n4,where 1 ≤ n1≤B, 1≤ n3 ≤B, n2, n4 ∈ F and F C [1, B] is a factor closed set. We study more particularly the case when F = {m = p1^ε1…pk^εk ,εj∈ {0, 1}, 1≤ j ≤ k}, p1,… ,pk being distinct prime numbers.

An Integral Representation for the Weighted Geometric Mean and Its Applications

By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.