-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     

ℒ1 subspaces ofl 1

It is proved that every subspace ofl ₁ which is an Λ_{1, λ} space with λ close enough to 1 is isomorphic tol ₁.     

On almost 1-1 extensions

We show that a broad class of extensions of measure preserving systems in the context of ergodic theory can be realized by topological models for which the extension is “almost one-one”.     

BestL 1-approximations of classesW 1 r by Splines fromW 1 r

We obtain the exact values of the bestL ₁-approximations of the classesW ₁ ^r of periodic functions by periodic polynomial splines of degreer and defect 1 with equidistant knots that belong to the classW ₁ ^r .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1410–1413, October, 1994.     

Non-linear oscillations of a satellite at 1:1:1 resonance

The motion of a satellite about its centre of mass in a central Newtonian gravitational field is investigated. The satellite is considered to be a dynamically symmetrical rigid body. It is assumed that the ratio of the polar and equatorial principal central moments of inertia of the satellite is 4/3, or close to this. The orbit of the centre of mass is elliptic, and the orbit eccentricity is assumed to be small. In the limit case, when the orbit of the centre of mass is circular, a steady motion exists (corresponding to relative equilibrium of the satellite in the orbital system of coordinates) in which the axis of dynamic symmetry is directed along the velocity vector of the centre of mass of the satellite; here, the frequencies of the small linear oscillations of the axis of symmetry are equal or close to one another. But in an elliptic orbit of small eccentricity, multiple 1:1:1 resonance occurs in this case, as the oscillation frequencies mentioned are equal or close to the frequency of motion of the centre of mass of the satellite in orbit. The non-linear problem of the existence, bifurcations and stability of periodic motions of the satellite with a period equal to the rotation period of its centre of mass in orbit is investigated.     

关于Goormaghtigh方程(x~3-1)/(x-1)=(y~n-1)/(y-1)

本文证明了：Ｇｏｏｒｍａｇｈｔｉｇｈ方程适合的例外解（ｘ，ｙ，ｎ），满足以及     

Binary-(−1, 1)-nilalgebras

Translated from Matematicheskie Zametki, Vol. 48, No. 1, pp. 103–109, July, 1990.     

The 1-median and 1-highway problem

In this paper we study a facility location problem in the plane in which a single point (median) and a rapid transit line (highway) are simultaneously located in order to minimize the total travel time of the clients to the facility, using the L₁ or Manhattan metric. The highway is an alternative transportation system that can be used by the clients to reduce their travel time to the facility. We represent the highway by a line segment with fixed length and arbitrary orientation. This problem was introduced in [Computers & Operations Research 38(2) (2011) 525–538]. They gave both a characterization of the optimal solutions and an algorithm running in O(n³logn) time, where n represents the number of clients. In this paper we show that the previous characterization does not work in general. Moreover, we provide a complete characterization of the solutions and give an algorithm solving the problem in O(n³) time.     

On wQN1 and wQN1 spaces

Several properties of a topological space related to the behavior of open coverings or to the behavior of sequences of continuous real-valued functions defined on the space were recently studied. We modify some notions replacing continuous functions by lower or upper semicontinuous ones and we show that some of so obtained notions coincide with other (continuous case) notions.     

关于函数方程f1n1+af1m1f2m2+f2n2=1

对于函数方程f1n1+af1m1f2m2+f2n2=1,其中a∈C/{0},n1,n2,m1,m2∈N,给出存在非常数亚纯函数解和整函数解的必要条件.     

1|1 parallel transport and connections

A vector bundle with connection over a supermanifold leads naturally to a notion of parallel transport along superpaths. In this note we show that every such parallel transport along superpaths comes form a vector bundle with connection, at least when the base supermanifold is a manifold.     

Unraveling Π 1 1 sets, revisited

We present a method of unraveling Π ₁ ¹ sets which greatly simplifies the construction in Neeman [6]. Apart from adding elegance, this method is also useful for proofs of determinacy involving long games; see Neeman [5]. This material is based upon work supported by the National Science Foundation under Grant No. DMS-0094174.