New Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Abstract For relatively prime positive integers u0 and r, and for 0 〈 k ≤ n, define uk ：= u0 ＋ kr. Let Ln ：= 1cm（u0,u1,... ,un） and let a,l≥2 be any integers. In this paper, the authors show that, for integers α≥ a, r ≥max（a,l - 1） and n ≥lατ, the following inequality holds Ln≥u0r^（l-1）α＋a-l（r＋1）^n.Particularly, letting l = 3 yields an improvement on the best previous lower bound on Ln obtained by Hong and Kominers in 2010.     

F_q[x]上的素数定理和Dirichlet定理及最小范数不可约多项式

讨论了F_q[x]上的zeta函数和L函数的解析性质,并在不假定黎曼猜想的情况下,导出了F_q[x]上的多项式环及其算术级数中不可约多项式的分布.然后,通过一系列的技术性处理,给出了算术级数中不可约多项式的最小范数的估计.成功地把素数定理及Dirichlet定理推广到了F_q[x]中,最重要的是,对应于最小素数问题,得到的最小范数的估计值本质上要比有理整数环上假定黎曼猜想情况下所推得的结果还好.     

On the Stability Radius of Matrix Polynomials

The stability radius of a matrix polynomial P ( λ) relative to an open region Ωof the complex plane and its relation to the numerical range of P ( λ) are investigated. Using an expression of the stability radius in terms of λon the boundary of Ωand ‖P ( λ) -1 ‖2 , a lower bound is obtained. This bound for the stability radius involves the distances of Ωto the connected components of the numerical range of P ( λ) and can be applied in conjunction with polygonal approximations of the numerical range. The special case of hyperbolic matrix polynomials is also considered.     

On Sums of Coefficients of Products of Polynomials

We study the nilpotency of the sums of all coefficients of some sorts of products of polynomials over reversible, IFP, and NI rings, and introduce an SCN ring as a generalization. We characterize SCN rings in relation with related ring properties, and also provide several useful properties and ring extensions of SCN rings.     

On Reproducing Kernel Hilbert Spaces of Polynomials

Let (X,Y) be a minimal compactifiurtion of ?². In this paper, we determine the structure of such a (X,Y) in the case where X is a normal hypersurface of degree d ≤ 4 in P³.     

Non-Unique Factorization of Polynomials Over Residue Class Rings of the Integers

We investigate non-unique factorization of polynomials in ? _{p ⁿ} [x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, ? _{p ⁿ} [x] is atomic. We reduce the question of factoring arbitrary nonzero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of ? _{p ⁿ} [x] is a direct sum of monoids corresponding to irreducible polynomials in ? _p [x], and we show that each of these monoids has infinite elasticity. Moreover, for every m ∈ ?, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles.     

On the probability that n and [n c] are coprime

We prove that for any positive real number which is not an integer, the density of the integers which are coprime to , a result conjectured by Moser, Lambek and Erd Hs. This revised version was published online in June 2006 with corrections to the Cover Date.     

Least Common Multiple of Path, Star with Cartesian Product of Some Graphs

A graph G without isolated vertices is a least common multiple of two graphs H₁ and H₂ if G is a smallest graph,in terms of number of edges,such that there exists a decomposition of G into edge disjoint copies of H₁ and H₂.The collection of all least common multiples of H₁ and H₂ is denoted by LCM(H₁,H₂) and the size of a least common multiple of H₁ and H₂ is denoted by lcm(H₁...     

On Commuting Graphs of Finite Matrix Rings

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R and two distinct vertices are joint by an edge whenever they commute. It is conjectured that if R is a ring with identity such that Γ(R) ≈ Γ(M _n (F)), for a finite field F and n ≥ 2, then R ≈ M _n (F). Here we prove this conjecture when n = 2.     

Coordinates of R[x,y]: Constructions and Classifications

Let R be a PID. We construct and classify all coordinates of R[x, y] of the form p ₂ y + Q ₂(p ₁ x + Q ₁(y)) with p ₁, p ₂ ∈ qt(R) and Q ₁, Q ₂ ∈ qt(R)[y]. From this construction (with R = K[z]) we obtain nontame automorphisms σ of K[x, y, z] (where K is a field of characteristic 0) such that the subgroup generated by σ and the affine automorphisms contains all tame automorphisms.     

Some Simple Derivations of k[x,y]

It is shown that the derivation y ^r ? _x  + (xy ^s  + g)? _y , where 0 ≤ r < s are integers, is a simple derivation of k[x, y], the polynomial ring in two variables over a field k of characteristic zero.     

一元多项式学习中易犯的几个错误

结合一元多项式中的一些重要概念,如多项式的最大公因式、多项式的重根及不可约多项式等,分析一元多项式学习中易犯的错误,并强调运用定理时要注意其适用的条件和前提．     

Z[3]上无限维矩阵广义逆

研究整环Z[3]上无限维矩阵V关于无限维矩阵构造下的逆、M-P逆和群逆,给出V的不同的逆、M-P逆等,推广了Saranya和Sivakumar的结果,并且得到Z[3]上无限维矩阵广义逆更广泛的性质.     

Simplicity of Some Derivations of k[x,y]

If k is a field of characteristic zero, c ∈ k?0, and 1 ≤ s ≤ r are integers such that either r ? s + 1 divides s or s divides r ? s + 1, then it is shown that the derivation y ^r ? _x  + (xy ^s  + c)? _y of the polynomial ring k[x, y] is simple.     

A Class of Simple Derivations of k[x,y]

If k is a field of characteristic zero, c ∈ k?{0}, s, t ≥ 1, and r ≥ 0 are integers, then it is shown that the k-derivation y ^r ? _x  + (y ^s x ^t  + c)? _y of the polynomial algebra k[x, y] is simple.     

环Z/p~kZ上矩阵广义逆的拓展

设R=Z/pkZ(其中k>1,p是一个奇素数),A是R上一个给定的可相似对角化的n阶矩阵.利用组合方法和有限局部环上的矩阵方法,讨论了矩阵A的拓展广义逆,得到了矩阵A的拓展广义逆存在的充要条件和一些的计数定理.