On the non-commutative neutrix product (x + r lnx +) ox − −s

The non-commutative neutrix product of the distributionsx ₊ ^r lnx ₊ andx _– ^–s is evaluated forr=0, 1, 2, ands=1, 2, . Further neutrix products are then deduced.     

On the neutrix convolution product ofx − r Inx − andx + −3

The existence of the neutrix convolution product of distributionx _? ^r Inx _? andx ₊ ^?3 is proved and some convolution products are evaluated.     

On the composition subalgebras of Bridgeland’s Hall algebras

Let A be a finite dimensional hereditary algebra over a finite field k and 𝒫 the category consisting of finite dimensional projective (left) A-modules. In this paper, we consider the composition subalgebra of Bridgeland’s Hall algebra of the category 𝒞_m(𝒫) of m-cyclic complexes for any positive integer m≥2 and determine its generating relations.     

On the gi/m/∞ service system with batch arrivals and different types of service distributions

We study the infinite-server system with batch arrivals ands different types of customers. With probabilityp _i an arriving customer is of typei (i=1,..., s) and requires an exponentially distributed service time with parameter _i (G ^GI /M ₁ ...M _s /). For theGI ^GI /M ₁...M _s / system it is shown that the binomial moments of thes-variate distribution of the number of type-i customers in the system at batch arrival epochs are determined by a recurrence relation and, in steady state, can be computed recursively. Furthermore, forG ^GI /M ₁...M _s /, relations between the distributions (and their binomial moments) of the system size vector at batch arrival and random epochs are given. Thus, earlier results by Takács [14], Gastwirth [9], Holman et al. [11], Brandt et al. [3] and Franken [6] are generalized.     

公式∫dx/x=lnx C与∫dx/x=ln|x| C

莫绍揆教授于 1 985年在文 [1 ]中谈起了积分公式∫dxx =lnx C (1 )　与 ∫dxx =ln| x| C (2 )　讨论了这两个公式的写法引起的争论 ,提出了有意义的建议 ,我很赞同 .由于至今一些高校教师和大学生中仍有争论或困惑 ,今特介绍该文提出的分析和见解 ,供大家参考 .指责 (1 )式的人说 ,该式右边的 x必须为正 ,而左边的 x却可正可负 .于是就会得出下面无意义的式子 :∫- 2- 1dxx =lnx - 2- 1=ln(-2 ) -ln(-1 ) .赞成 (1 )式的人辩解说 ,该式可写成∫- 2- 1dxx =lnx - 2- 1=ln-2-1 =ln2 .但这只能说是人为的临时约定 .因为按牛顿 -莱布尼兹…     

On the trigonometric polynomials of Fejér and Young

The trigonometric polynomials of Fejér and Young are defined by $S_n (x) = \sum\nolimits_{k = 1}^n {\tfrac{{\sin (kx)}} {k}}$S_n (x) = \sum\nolimits_{k = 1}^n {\tfrac{{\sin (kx)}} {k}} and $C_n (x) = 1 + \sum\nolimits_{k = 1}^n {\tfrac{{\cos (kx)}} {k}}$C_n (x) = 1 + \sum\nolimits_{k = 1}^n {\tfrac{{\cos (kx)}} {k}}, respectively. We prove that the inequality $\left( {{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}} \right)\sqrt {15} \leqslant {{C_n \left( x \right)} \mathord{\left/ {\vphantom {{C_n \left( x \right)} {S_n \left( x \right)}}} \right. \kern-\nulldelimiterspace} {S_n \left( x \right)}}$\left( {{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}} \right)\sqrt {15} \leqslant {{C_n \left( x \right)} \mathord{\left/ {\vphantom {{C_n \left( x \right)} {S_n \left( x \right)}}} \right. \kern-\nulldelimiterspace} {S_n \left( x \right)}} holds for all n ≥ 2 and x ∈ (0, π). The lower bound is sharp.     

On the regularity index of s fat points not on a linear (r−1)-space,s≤r+3

We will give formulas to compute the regularity index of s fat points in general position on a linear r-space, s≤r+3 (Theorem 3.1). Then we will show formulas to compute the regularity index of s equimultiple fat points not on a linear (r?1)-space, s≤r+3 (Theorem 4.6). Our results show that the Segre’s bound is attained in these cases.     

On the Periodic Structure of Delayed Difference Equations of the Form x n = f ( x n − k ) on I and S 1

The aim of this paper is to give an account of some results recently obtained in Combinatorial Dynamics and apply them to get for k S 2 the periodic structure of delayed difference equations of the form x n = f ( x n m k ) on I and S 1 .     

On the Behavior of Solutions of x n+1 = p+(x n−1/xn )

Our goal in this article is to complete the study of the behavior of solutions of the equation in the title when the parameter p is positive and the initial conditions are arbitrary positive numbers. Our main focus is the case 0 < p < 1. We will show that in this case, all solutions which do not monotonically converge to the equilibrium have a subsequence which converges to p and a subsequence which diverges to infinity. For the sake of completeness, we will also present the results (which were previously known) with alternative proofs for the case p = 1 and the case p > 1.     

On the diophantine equation x 2 − Dy 2 = n

We propose a method to determine the solvability of the diophantine equation x2-Dy2=n for the following two cases:(1) D = pq,where p,q ≡ 1 mod 4 are distinct primes with(q/p)=1 and(p/q)4(q/p)4=-1.(2) D=2p1p2 ··· pm,where pi ≡ 1 mod 8,1≤i≤m are distinct primes and D=r2+s2 with r,s ≡±3 mod 8.     

On the elliptic curve y 2 = x 3 −2rDx and factoring integers

Let D=pq be the product of two distinct odd primes.Assuming the parity conjecture,we construct infinitely many r≥1 such that E2rD:y2=x3-2rDx has conjectural rank one and vp(x([k]Q))≠vq(x([k]Q))for any odd integer k,where Q is the generator of the free part of E(Q).Furthermore,under the generalized Riemann hypothesis,the minimal value of r is less than c log4 D for some absolute constant c.As a corollary,one can factor D by computing the generator Q.     

Regularity and explicit representation of (0,1,...,m−2,m) interpolation on the zeros of (1−x 2)P n −2/(α,β) (x)

Necessary and sufficient conditions for the regularity andq-regularity of (0,1,...,m–2,m) interpolation on the zeros of (1–x ²)P _n ^–2/(,) (x) (,>–1) in a manageable form are established, whereP _n ^–2/(,) (x) stands for the (n–2)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, we show that under a mild assumption if the problem of (0,1,...,m–2,m) interpolation has an infinity of solutions then the general form of the solutions isf ₀(x)+C f(x) with an arbitrary constantC.This work is supported by the National Natural Science Foundation of China.