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

Rigid Sets of Dimension n – 1 in ℝ n

We give conditions allowing an intrinsic isometry on a dense subset to be extended to an isometry of the whole set. This enables us to find examples of (n-1)-dimensional sets rigid in ⁿ .     

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     

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.     

A family of (n−1)! Diagonal polynomial orders ofN n

Letn>0 be an element of the setN of nonnegative integers, and lets(x)=x ₁+...+x _n , forx=(x ₁, ...,x _n ) N _n . Adiagonal polynomial order inN _n is a bijective polynomialp:N _n N (with real coefficients) such that, for allx,y N _n ,p(x)<p(y) whenevers(x)<s(y). Two diagonal polynomial orders areequivalent if a relabeling of variables makes them identical. For eachn, Skolem (1937) found a diagonal polynomial order. Later, Morales and Lew (1992) generalized this polynomial order, obtaining a family of 2 ^n–2 (n>1) inequivalent diagonal polynomial orders. Here we present, for eachn>0, a family of (n – 1)! diagonal polynomial orders, up to equivalence, which contains the Morales and Lew diagonal orders.     

Generalized Latin squares of order n with n 2 − 1 distinct elements

Let n ≥ 3 be a positive integer. We show that the number of equivalence classes of generalized Latin squares of order n with n ² ? 1 distinct elements is 4 if n = 3 and 5 if n ≥ 4. It is also shown that all these squares are embeddable in groups. As an application, we obtain a lower bound for the number of isomorphism classes of certain Eulerian graphs with n ² + 2n ? 1 vertices.     

Maximal subalgebras of rank n −1 of the algebra AP(1, n) and reduction of nonlinear wave equations. I

The concept of canonical decomposition of an arbitrary subalgebra of the algebraAO(1,n) is introduced. With the help of this decomposition all maximal subalgebras L of rankn–1 of the algebraAP(1,n), satisfying the conditionL V=1,...;P _n>, whereV=<P ₀,P ₁,...,P ₁> is the space of translation are described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1552–1559, November, 1990.     

Minima of decomposable forms of degree n in n variables for n ≥ 3

The following theorem is proved: if for all (X0)one has ¦ F(x) ¦ >0, where F(x) is a decomposable form of degree n of n variables, then, for n 3, F(x) is proportional to an integral form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 142–154, 1990.     

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.     

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.     

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.     

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),     

Minimal Covers of Q +(2n + 1, q) by (n − 1)-Dimensional Subspaces

A t-cover of a quadric is a set C of t-dimensional subspaces contained in such that every point of is contained in at least one element of C.We consider (n – 1)-covers of the hyperbolic quadric Q ⁺(2n + 1, q). We show that such a cover must have at least q ^{n + 1} + 2q + 1 elements, give an example of this size for even q and describe what covers of this size should look like.