The extension of the D(?k 2)-pair {k 2,k 2 + 1}

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k ², k ²+1, c, d} is a D(?k ²)-quadruple with c < d, then c = 1 and d = 4k ²+1. This extends the work of the first author [20] and that of Dujella [4].     

On the Normal Order of ϕk+1(n)/ϕk(n), where ϕk is the k-Fold Iterate of Euler's Function

Let be the j-fold iterated function of . Let and > 0 be fixed, Q be a prime, and let N _k(Q|x) denote the number of those nx for which Q . We give the asymptotics of N _k(Q|x) in the range .     

On the k—smoothness and k—strong smoothness

5l.IntroductionFrompal,er[11,L2],to[3j,tI1efol1owingresultsareobtained:1)LetXbeaBanachspace,thenxes(X)={xeX;llxIl=1}isG-differetiableifandonIyif,foreachyeX,letXI=span(x,y).2)LetXbeaBanachspace,thenxeS(X)isF-differetiableifandonlyifIn[4j,NanchaoxunandWangjianhuaintroducedk-smoothnessandk-strongsmooth-ness.lnthisPaPer,wegive:Theorem1LetXbeaBanachspace,thcnthefollowingstatementsareequivalent.(i)xo6S(X)isk-smooth.(ii)IfX*cyXisk-dimensionalsubspaceandx,eX*,thenwhereU(x,,r)={xeX;l1x-xo…     

A Note on the Recursive Sequence x n + 1 = p k x n + p k − 1 x n − 1 +...+ p 1 x n − k + 1

We present some comments on the behavior of solutions of the difference equation where p _i 0, i = 1,..., k, k N, and x _–k ,..., x _–1 R.     

Evaluation of Series Involving the Product of the Tail of $${\zeta(k)}$$ and $${\zeta(k+1)}$$ζ(k+1)

We aim at evaluating the following class of series involving the product of the tail of two consecutive zeta function values

$$\sum\limits_{n=1}^{\infty}\left(\zeta(k)-1-\frac{1}{2^k}-\cdots-\frac{1}{n^k}\right)\left(\zeta(k+1)-1-\frac{1}{2^{k+1}}-\cdots-\frac{1}{n^{k+1}}\right),$$

where \({k\geq 2}\) is an integer. We show that the series can be expressed in terms of Riemann zeta function values and a special integral involving a polylogarithm function.

     

$$L^p$$ Christoffel–Minkowski problem: the case $$1< p<k+1$$1<p<k+1

We consider a fully nonlinear partial differential equation associated to the intermediate \(L^p\) Christoffel–Minkowski problem in the case \(1<p<k+1\). We establish the existence of convex body with prescribed k-th even p-area measure on \(\mathbb S^n\), under an appropriate assumption on the prescribed function. We construct examples to indicate certain geometric condition on the prescribed function is needed for the existence of smooth strictly convex body. We also obtain \(C^{1,1}\) regularity estimates for admissible solutions of the equation when \( p\ge \frac{k+1}{2}\).     

(-1)~(k 1)=1(k∈Z)

解方程 sin5x=sin4x。解法一:因为与a有相同的正弦值的弧度数x的集合是{x|x=kπ (-1)~ka,k∈Z},所以原方程可以化成 5x=kπ (-1)4x (k∈Z) 解之得:x=kπ/5 (-1)~(k 1) 所以原方程的解集是{x|2= 解法二:原方程等价为sin4x=sin5x,m同解法一得:4x=kπ (-1)~k5x 解之得:x=kπ/4 (-1)~(k 1)5(k∈z)     

从k 1→k反推 助你归纳法成功

使用数学归纳法证明与自然数有关的命题,最为关键的一处是发现由k→k l的关系。具体实施数学归纳法时,思维习惯总是遵循由k→k l去探索。笔者常常在解题教学中,教导学生在正向探求受阻时,就超越传统思维习惯的束缚,打破常规,由k l→k反推,透视问题的表象,从更深层次去揭示问题的本质。这种反推的思维对象,对数学归纳法的成功与失败,具有特别的意义。这里选例几个范例,共尝从k l→k反推之乐趣。     

On High G k−1,k -Defective Varieties

We give a rough characterization for n-dimensional varieties with G_k?1,k-defect equal to a > 0 if k ≥ n. Then we apply this in the case that a ≥ n ? 2 to become a fine classification.     

一类k步k—1阶格式的研究

1.引言 基于数值微分的方法(或称Gear法)是当前处理Stiff问题的主导方法。该方法具有如下的形式: k y_(n 1)=sum from i=1 to k α_iy_(n 1-i) hβ_0f_(n 1)。 (1) 众所周知,当k≤6时,该方法是Stiff稳定的。积分过程所形成的隐式方程组必须用Newton-Raphson法求解,因为一般的简单迭代收敛性条件     

Hamiltonian[k,k+1]-因子

本文考虑n/2-临界图中Hamiltonian[k,k+1]-因子的存在性。Hamiltonian[k,k+1]-因子是指包含Hamiltonian圈的[k,k+1]-因子;给定阶数为n的简单图G,若δ(G)≥n/2而δ(G\e)相似文献   

巧用1/(n(n k))=1/k((1/n)-(1/(n k)))解竞赛题

一、用来解方程例1(1999年河北省竞赛题)方程1/(x(x-1)) 1/(x(x 1)) … 1/((x 1997)(x 1998)) =(1999)/(2000)的根为()．(A)1999 (B)-2 (C)-1999或2 (D)1999或-2解根据公式原方程化为1/(x-1)-1/x 1/x-1/(x 1) … 1/(1/(x 1997))-1/(x 1998)=(1999)/(2000),1/(x-1)-1/(x 1998)=(1999)/(2000),(x 1998)-(x-1)=(1999)/(2000)(x-1)(x 1998),1999=(1999)/(2000)(x-1)(x 1998),     

一类k步k＋1阶SDBDF型混杂法

本文在ＳＤＢＤＦ方法的基础上构造了一类ｋ步ｋ＋１阶的ＳＤＢＤＦ型混杂法，讨论了该方法的稳定性质，证明了ｋ＝１，２，３时，方法是Ａ稳定的，ｋ＝４～１０时方法是ｓｔｉｆ稳定的，同ＢＤＦ型和Ａｄａｍｓ型混杂法相比，该方法的ｓｔｉｆ稳定阶大有提高，这两类方法的ｓｔｉｆ稳定阶只能分别达到８阶和９阶〔２，３，５〕．在实现Ｎｅｗｔｏｎ迭代计算时，我们的方法要优于原ＳＤＢＤＦ方法，因此对于求解ｓｔｉｆ问题，我们的这类方法具有一定的优势，最后给出了数值实例     

从k到k 1思路受阻分析及转化策略

用数学归纳法证明不等式，特别是数列不等式，是一个行之有效的方法，也是中学数学中的一个基本方法，近些年高考试题中几次出现这类问题．运用这种方法时，往往有好多学生在证ｋ到（ｋ＋１）的过程中，卡了壳、断了思路，这是一种普遍现象．现就笔者在讲授这部分内容时学生暴露的问题，分析一下思路受阻的几种原因及转化策略．１　从ｋ到（ｋ＋１）添项不足在从ｋ到（ｋ＋１）的证明过程中，如果分析不透命题结构，就会造成添项不足，证明夭折．例１　已知Ｓｎ＝１＋１２＋１３＋…＋１ｎ，（ｎ∈Ｎ）．用数学归纳法证明Ｓ２ｎ＞１＋ｎ２　…     

一类隐式的k+1阶线性k步法

本文首先给出了一类比Adams-Moulton方法的绝对稳定区间大的隐式k+1阶线性k步法基本公式.求出了3-9步新公式的分数形式的精确系数,阶数,局部截断误差主项系数和绝对稳定区间,然后构造了由4阶隐式新公式和同阶显式Nyström公式组合而成的预估-校正方法,比著名的Adams-Bashforth-Moulton和Nyström-Adams-Moulton预估校正方法的绝对稳定区间大,最后用对比数值试验对结果进行了验证.     

On codimensions k immersions of m-manifolds for k = 1 and k = m − 2

Let us consider M a closed smooth connected m-manifold, N a smooth (2m − 2)-manifold and a continuous map, with . We prove that if is injective, then f is homotopic to an immersion. Also we give conditions to a map between manifolds of codimension one to be homotopic to an immersion. This work complements some results of Biasi et al. (Manu. Math. 104, 97–110, 2001; Koschorke in The singularity method and immersions of m-manifolds into manifolds of dimensions 2m − 2, 2m − 3 and 2m − 4. Lecture Notes in Mathematics, vol. 1350. Springer, Heidelberg, 1988; Li and Li in Math. Proc. Camb. Phil. Soc. 112, 281–285, 1992).     

关于∞∑k=1f(k)xk-1计算的矩阵方法

本文将幂级数∞∑k=1f(k)xk-1的计算转化为求解矩阵方程Aβ=α,使计算非常简单和有效.