Linking (n − 2)-dimensional panels inn-space I: (k − 1,k)-graphs and (k − 1,k)-frames

A (k – 1,k)-graph is a multi-graph satisfyinge (k – 1)v – k for every non-empty subset ofe edges onv vertices, with equality whene = |E(G)|. A (k – 1,k)-frame is a structure generalizing an (n – 2, 2)-framework inn-space, a structure consisting of a set of (n – 2)-dimensional bodies inn-space and a set of rigid bars each joining a pair of bodies using ball joints. We prove that a graph is the graph of a minimally rigid (with respect to edges) (k – 1,k)-frame if and only if it is a (k – 1,k)-graph. Rigidity here means infinitesimal rigidity or equivalently statical rigidity.     

(-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)     

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.     

Minimal coverings and maximal packings of (k−1)-subsets by k-subsets

This paper studies the asymptotic behavior of functions M(n, k, k–1, ) and m(n, k, k–1, ), equal to the respective cardinalities of the minimal -covering and maximal -packing of all (k–1)-subsets of the n-element set of its k-subsets. It is shown that, if sequence k=k(n) is such that k(n)/n 0 as then m(n, k, k–1, ).( _k–1 ⁿ ).k¹, and if as n , thenM(n,k,k–1,).( _k–1 ⁿ ).k^–1. A consequence of these results is the validity of the Erdös-Hanani conjecture concerning the asymptotic behavior of functions M(n, k, k–1, 1) and m(n, k, k–1, 1).Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 565–571, April, 1977.In conclusion, the author wishes to thank A. A. Sapozhenko under whose direction the present work was achieved.     

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.     

Singular (k,n − k) boundary value problems between conjugate and right focal

For 1 ⩽k⩽n − 1 and 0 ⩽q⩽k − 1, solutions are obtained for the boundary value problem, (−1)^n−k = f(x,y), y⁽ⁱ⁾=0, 0⩽i⩽k − 1, and y⁽ⁱ⁾ = 0, q⩽j⩽n − k + q − 1, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.     

Lokal-reguläre Bogen ohne (n−2,k)-Sekanten im projektivenP n (n≦k)

Ohne Zusammenfassung     

Propertyf −1(S) =g −1(S) for entire and meromorphicP-adic functions

LetW be an algebraically closed filed of characteristic zero, letK be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value, and letA(K) (resp. ℳ(K)) be the set of entire (resp. meromorphic) functions inK. For everyn≥7, we show that the setS _n(b) of zeros of the polynomialx ⁿ−b (b≠0) is such that, iff, g ∈W[x] or iff, g ∈A(K), satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), thenf ⁿ=g ⁿ. For everyn≥14, we show thatS _n(b) is such that iff, g ∈W({tx}) or iff, g ∈ ℳ(K) satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), then eitherf ⁿ=g ⁿ, orfg is a constant. Analogous properties are true for complex entire and meromorphic functions withn≥8 andn≥15, respectively. For everyn≥9, we show that the setY _n(c) of zeros of the polynomial , (withc≠0 and 1) is an ursim ofn points forW[x], and forA(K). For everyn≥16, we show thatY _n(c) is an ursim ofn points forW(x), and for ℳ(K). We follow a method based on thep-adic Nevanlinna Theory and use certain improvement of a lemma obtained by Frank and Reinders.     

GLn(k)和SLn(k)的阶

本文利用矩阵行的初等变换 ,采用递推的方法 ,求出了有限域 k上 n次一般线性群 GLn(k)和 n次特殊线性群 SLn(k)的阶 .     

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),     

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.     

Über einige Grundeigenschaften der Bogen ohne (n−2,k)-Sekanten im projektivenP n (n≦k)

Ohne Zusammenfassung     

不等式k~(1/2)+1/(k+1)~(1/2)>(k+1)~(1/2)的几种证法

为什么要证明不等式k~(1/2)+1/(k+1)~(1/2)>(k+1)~(1/2)下面通过实例来说明,高中数学第三册P.147.3(4)题:求证1/1~(1/2)+1/2~(1/2)+…+1/n~(1/2)>n~(1/2)(n>1)。我们用数学归纳法来证明。 (1)当n=2时不等式左边=1/1~(1/2)+1/2~(1/2)=(2+2~(1/2))/2右边=2~(1/2)=(2~(1/2)+2~(1/2))/2,显然不等式成立。 (2)假设当n=k(k>1)时不等式成立,