Global Behavior of Solutions of x n + 1 = max {x n , A}/x n x n – 1

We investigate the asymptotic behavior, the oscillatory character, and theperiodic nature of solutions of the difference equation

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.     

Boundedness properties of the difference equation x n+1=(α+βx n +δx n−2)/(x n−1)

Consider the third-order difference equation x _n+1 = (α+βx _n +δx _n ? 2)/(x _n ? 1) with α ∈ [0,∞) and β,δ ∈ (0,∞). It is shown that this difference equation has unbounded solutions if and only if δ>β.     

On Globally Periodic Solutions of the Difference Equation x n + 1 = f ( x n )/ x n − 1

We study the second-order difference equation x n +1 = f ( x n ) x n m 1 where f ] C 1 ([0, X ),[0, X )) and x n ] (0, X ) for all n ] Z . For the cases p h 5, we find necessary and sufficient conditions on f for all solutions to be periodic with period p . We answer some questions and conjectures of Kulenovi ' and Ladas.     

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.     

Zu Fragen der Analysis im Zusammenhang mit der Gleichungx 1 2 +...+x n 2 —ax 1...x n =b

x ₁ ² +...+x _n ² —ax ₁...x _n =b. First we describe a combinatorial presentation of a group of automorphisms of this equation, ifn=3, then we getPGL (2, ) as such a group of automorphisms of this equation. This gives analytical applications becausePGL (2, ) acts discontinously on the set {(x ₁,x ₂,x ₃)0<x ₁,x ₂,x ₃ andx ₁ ² +x ₂ ² +x ₃ ² –x ₁ x ₂ x ₃=b0}³. Further we ask for fundamental solutions of this equation. Finally, letx ₁,x ₂,x ₃ withx ¹ x ₂ ² +x ₃ ² ––x ₁ x ₂ x ₃=0 Then there areA, BSL(2, ) with trA=x ₁, trB=x ₂ and trA B=x ₃, and the group (A, B) is a discrete free group of rank two. In analysis we are interested in the question whether there are evenA, BSL(2, ) with trA=x ₁, trB=x ₂ and trA B=x ₃. We give necessary and sufficient conditions for that and remark that this question is connected with the ternary quadratic formk1p ²+k2q ²–r ²,k ¹=x ₁ ² ,k ₂=16(x ₂ ² +x ₁ ² +x ₃ ² –x ₁ x ₂ x ₃–4), which has some invariant properties.     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).     

K 0 of hypersurfaces defined by x 1 2 + ... + x n 2 = ±1

Let k be a field of characteristic ≠ 2 and let Q _n,m (x ₁, ..., x _n , y ₁, ..., y _m ) = x ₁ ² +...+x _n ² ? (y ₁ ² +...+y _m ² ) be a quadratic form over k. Let R(Q _n,m ) = R _n,m = k[x ₁, ..., x _n , y ₁, ..., y _m ]/(Q _n,m ? 1). In this note we will calculate $\tilde K_0 \left( {R_{n,m} } \right)$ for every n,m ≥ 0. We will also calculate CH ₀(R _n,m ) and the Euler class group of R _n,m when k = ?.     

关于满足I(x,n(x))=n(x)的D-蕴涵的刻画

研究I(x,n(x))=n(x),其中I为由连续三角模T、连续三角余模S和强否定n生成的D-蕴涵,即I(x,y)=S(T(n(x),n(y)),y),给出了满足I(x,n(x))=n(x)的充要条件。     

从(n 1)/n×(n 1)=(n 1)/n (n 1)谈起

一问题的提出本刊2003年第5期刊载了《运用发现法解题》(以下简称《解题》)一文,文章在谈到“归纳发现法”时,提到这样一个例子: 1.观察下列各式     

Zolotarev's first problem — the best approximation by polynomials of degree ≤n − 2 tox n − nσx n−1 in [−1, 1]

Chebyshev determined $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n + a_1 x^{n - 1} + \cdots + a_n |$$ as 2^1?n , which is attained when the polynomial is 2^1?n T _n(x), whereT _n(x) = cos(n arc cosx). Zolotarev's First Problem is to determine $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n - n\sigma x^{n - 1} + a_2 x^{n - 2} + \cdots + a_n |$$ as a function ofn and the parameter σ and to find the extremal polynomials. He solved this in 1878. Another discussion was given by Achieser in 1928, and another by Erdös and Szegö in 1942. The case when 0≤|σ|≤ tan²(π/2n) is quite simple, but that for |σ|> tan²(π/2n) is quite different and very complicated. We give two new versions of the proof and discuss the change in character of the solution. Both make use of the Equal Ripple Theorem.     

A note on the diophantine equation (a n − 1)(b n − 1) = x 2

Let a, b be fixed positive integers such that a ≠ b, min(a, b) > 1, ν(a?1) and ν(b ? 1) have opposite parity, where ν(a ? 1) and ν(b ? 1) denote the highest powers of 2 dividing a ? 1 and b ? 1 respectively. In this paper, all positive integer solutions (x, n) of the equation (a ⁿ ? 1)(b ⁿ ? 1) = x ² are determined.     

A Note on the Difference Equation x n +1 = ∑ i =0 k α i x n − i p i

In this note we improve Theorem 2 in Ref. [3] , about the difference equation x n +1 = ~ i =0 k f i x n m i p i , n =0,1,2,..., where k is a positive integer, f i , p i ] (0, X ) for i =0,..., k , and the initial conditions x m k , x m k +1 ,..., x 0 are arbitrary positive numbers.     

差分方程xn+1=xn+xαn/nβ解的渐近性

我们研究差分方程(1)的解的渐近性和全局吸引性.     

方程x1＋x2＋……＋xm=n的正整数解的个数及其应用

在近几年的高考试题中，出现了可化为求方程x1＋x2＋…＋xm=n（m，n∈N^＋，m≤n）的正整数解的个数的问题，下面就这个问题谈几点看法，供大家参考。     

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 .     

直线方程x=my n的妙用

解析几何是用代数方法研究几何问题的一门数学学科.由于它开创了数、形结合的研究方法,因此,它给数学注入了新的活力.直线是最常见、最基本的简单几何图形之一,它的方程有多种不同的形式,在使用直线方程的各种形式时,要注意它们各自的限制条件,如:点斜式的使用条件是直线必须存在斜率;截距式的使用条件是两截距都存在且不为零;两点式的使用条件为直线不与坐标轴垂直;等等.在使用直线的方程时,通常我们都应该根据直线满足的几何条件,选择合适的方程形式.但是有时会出现这样的问题,不知道直线的斜率是否存在.这时,通常的做法是分类讨论,即根据…     

关于方程x+μsinx+x=0在|x|≤(n+1)π上恰好存在n个极限环的定理

本文证明了微分方程x+μsinx+x=0(1)的等价方程组 d_x/dt=v,dv/dt=-x-μsinv (2)在相平面(x,v)的带域|v|≤(n+1)π上恰好存在n个极限环(n=1,2,…).本文对比(1)式更广泛的一类Lienard型方程证明了此结论.