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.     

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.     

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 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.     

On improving the accuracy of Horner’s and Goertzel’s algorithms

It is known that Goertzels algorithm is much less numerically accurate than the Fast Fourier Transform (FFT) (cf. [2]). In order to improve accuracy we propose modifications of both Goertzels and Horners algorithms based on the divide-and-conquer techniques. The proof of the numerical stability of these two modified algorithms is given. The numerical tests in Matlab demonstrate the computational advantages of the proposed modifications. The appendix contains the proof of numerical stability of Goertzels algorithm of polynomial evaluation. AMS subject classification 65F35, 65G50