1-1 1-1 1-1 1-…=?

思考求算式1-1 1-1 1-1 1-…的代数和。解法一设x=1-1 1-1 1-1 1-…。 x=1-(1-1 1-1 1-1 1-…), 所以 x=1-x,x=1/2。一个似乎没有任何漏洞的运算。看完了下面的几种解法,你恐怕就不这样认为了。解法二设x=(1-1) (1-1) (1-1) …。 x=0 0 0 0 …,所以x=0。解法三设x=1-(1-1)-(1-1)-(1-1)-…。 x=1-0-0-0-0-…,所以x=1。解法四设x=1-1 1-1 1-1 1-…。将原算式第一项与第二项,第三项和第四项,第五项和第六项……交换位置有     

[1 1/(1/2) … 1/(1/(1999))]=?

陈宽国同志在《数学通讯》98年第5期中遗留的一个问题．本文我们肯定由于显然，占的值越大，精确度越高．在②、③中分别令k一2，则令n—1999，则左边。87．61，右边。87．98，于是得①．[1 1/(1/2) … 1/(1/(1999))]=?@吴勤文$新疆昌吉州教育局教研室!831100     

-1=1?

推证 :- 1 =i2 =( i4) 12 =1 .显然结论错误 ,但究竟错在哪里呢 ?诡辩揭密 :关于复数指数幂运算律 ,详见人教版原高二代数教材 P1 91有 :　　　　 ( zm) n =zmn,　　　　 zm . zn =zm+ n,　　　　 zmzn =zm-n,其中 m、n∈ N.实质上 ,m,n∈ Z也成立 .可见 ,i2 ≠ ( i4) 12 ,即 - 1≠ 1 .-1=1?!712000$陕西省咸阳市天王中学@曹万宏     

1/x_3=1/x_1+1/x_2

春耕一片田地,甲队单独进行a天可以完成,乙队单独进行b天可以完成。甲、乙两队合耕几天可以完成? 设甲、乙两队合耕x天可以完成,那末 1/x=1/a+1/b 两根导线的电阻分别是R_1和R_2,则其总电阻R满足等式 1/R=1/R_1+1/R_2。物体、象和焦点到透镜的距离这三个量之间有下列关系: 1/f+1/q=1/h。以上三个实际问题反映到数学中来,就是研究x_1,x_2,x_3这三个正量之间的下列关系: 1/x_3=1/x_1+1/x_2。(*) 在等式(*)中,如果x_1,x_2两个量已知,那么我们可以用解分式方程的方法,解得 x=x_1x_2/(x_1+x_2)。(*′)     

Non-singularSaffman-taylorFinger1

Ｎｏｎ－ｓｉｎｇｕｌａｒＳａｆｆｍａｎ－ｔａｙｌｏｒＦｉｎｇｅｒ１ＫａｉＨｕａＺＨＡＯＨｕｉｄａｎＹＵ（Ｄｅｐａｒｔｍｅｎｔｏｆｐｈｙｓｉｃｓ，ＣｅｎｔｅｒｆｏｒＮｏｎｌｉｎｅａｒＳｃｉｅｎｃｅ，ＰｅｋｉｎｇＵｎｉｖｅｒｓｉｔｙ，Ｂｅｉｊｉｎｇ１００...     

ThePropertiesofConcentrationsinaPolymerBlend1

ＴｈｅＰｒｏｐｅｒｔｉｅｓｏｆＣｏｎｃｅｎｔｒａｔｉｏｎｓｉｎａＰｏｌｙｍｅｒＢｌｅｎｄ１ＹｕｎＨＵＡＮＧ，ＤｅｌｕＺＨＡＯ（ＤｅｐａｒｔｍｅｎｔｏｆＰｈｙｓｉｃｓ，ＰｅｋｉｎｇＵｎｉｖｅｒｓｉｔｙ，Ｂｅｉｊｉｎｇ１００８７１，Ｃｈｉｎａ）（Ｉｎｓｔ...     

Averaging operators for exponential splittings

Averaging operations are considered in connection with exponential splitting methods. Toeplitz plus Hankel related matrices are resplit by applying appropriate averaging operators leading to a hierarchy of structured matrices. With the resulting parts, the option of using exponential splitting methods becomes available. A related, seemingly important group of unitary unipotents is looked at. Based on a formula due to Lenard, a very fast iterative method to find the nearest Toeplitz plus Hankel matrix in the Frobenius norm is devised.     

On two particular cases of solving the normal Hankel problem

The normal Hankel problem is one of characterizing all the complex matrices that are normal and Hankel at the same time. The matrix classes that can contain normal Hankel matrices admit a parameterization by real 2 × 2 matrices with determinant one. Here, the normal Hankel problem is solved in the case where the characteristic matrix of a given class is an order two Jordan block for the eigenvalue 1 or ?1.     

Tangential Nevanlinna-Pick Interpolation and Its Connection with Hamburger Matrix Moment Problem

This paper is concerned with the solution of a certain tangential Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called block Hankel vector method to establish two intrinsic connections between the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the truncated Hamburger matrix moment problem associated with the block Hankel vector under consideration: one is a congruent relationship between their information matrices, and the other is a divisor-remainder connection between their solutions. These investigations generalize our previous work on the Nevanlinna-Pick interpolation and power matrix moment problem.     

Hankel operators in analytic spaces

A connection between solutions of an operator equation with Hankel matrices is established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 998–1000, July, 1990.     

On the skew-symmetric part of the Toeplitz component in the real normal (<Emphasis Type="Italic">T</Emphasis> + <Emphasis Type="Italic">H</Emphasis>)-problem

The real normal Toeplitz-plus-Hankel problem is to characterize the matrices that can be represented as sums of two real matrices of which one is Toeplitz and the other Hankel. For a matrix of this type, relations are found between the skew-symmetric part of the Toeplitz component and the matrix obtained by reversing the order of columns in the Hankel component.     

On inversion of square matrices partitioned into non-square blocks

Inversion theorems for structured block matrices with non-square blocks are presented. The considered classes contain Toeplitz, Toeplitz plus Hankel and Van der Monde type matrices.     

由特征值唯一确定的实3×3Hankel矩阵(英文)

本文研究了由特征值唯一确定的3×3实Hankel矩阵.借助于M.Fielder[1]的结论并经过细致的讨论,得到3×3实Hankel矩阵由其特征值唯一确定的充分必要条件,刻画了3×3实Hankel矩阵的一种特征值性质.     

Type II Matrices and Their Bose-Mesner Algebras

Type II matrices were introduced in connection with spin models for link invariants. It is known that a pair of Bose-Mesner algebras (called a dual pair) of commutative association schemes are naturally associated with each type II matrix. In this paper, we show that type II matrices whose Bose-Mesner algebras are imprimitive are expressed as so-called generalized tensor products of some type II matrices of smaller sizes. As an application, we give a classification of type II matrices of size at most 10 except 9 by using the classification of commutative association schemes.     

Riordan arrays and the LDU decomposition of symmetric Toeplitz plus Hankel matrices

We examine a result of Basor and Ehrhardt concerning Hankel and Toeplitz plus Hankel matrices, within the context of the Riordan group of lower-triangular matrices. This allows us to determine the LDU decomposition of certain symmetric Toeplitz plus Hankel matrices. We also determine the generating functions and Hankel transforms of associated sequences.     

Structure theory and fast inversion of Hankel striped matrices. I

A structure theory for Hankel striped matrices is developed that generalizes the corresponding theory for Hankel matrices [4]. This leads, in particular, to an inverslon algorithm for Hankel and Toeplitz striped matrices working without additional assumptions.