Let X be an infinite set and S be a transformation semigroup on X invariant under conjugations by permutations of X. Such S is termed x-normal. In the paper, we describe elements of a x-normal semigroup S of one-to-one transformations.

The effect of imposing a certain finiteness condition on the group of central automorphisms of a finite-by-nilpotent group is investigated. In particular it is shown that, if each central automorphism of a finite-by-nilpotent group G has finite order, then the factor group G/Z(G) has finite exponent.

Let A and B be C*-algebras and A ⊗ B denote the minimal C*-tensor product of A and B. T. Huruya [1] gave examples of C*-tensor products A ⊗ B with C*-subalgebras A1⊗B2 and A2⊗B2 such that strictly contains , answering a question of S. Wassermann [3, Remark 23]. In this short note, we show that the same situation can occur even if A1 = A2

The method of extracting any root may be understood from the subjoined example of extracting the fifth root of 2,073,071,593.

A proof of Dupin's theorem with some simple illustrations of the method employed.

Before plunging into Dupin's theorem, I think it well to speak of certain infinitesimal rotations which play a part in the proof. By an infinitesimal angle of the first order is meant an angle subtended at the centre of a circle of finite radius by an arc whose length is an infinitesimal of the first order. If we neglect infinitesimals of the second order, equal infinitesimal rotations of the first order about axes which meet and are separated by a small angle of the first order are identical. For instance, if AB and BC be elements of a curve of continuous curvature, an infinitesimal rotation about AB may, if we prefer it, be regarded as taking place about BC; and again, if OA, OB, OC be a set of rectangular axes, small rotations about OA, OB, OC may be regarded as taking place in any order. For if P be a point on a sphere of finite radius, and PQ, PR be the displacements of P due to equal infinitesimal rotations of the first order about two diameters separated by a small angle of the first order, the angle QPR is the angle of separation of the axes, and it follows that QR is an infinitesimal of the second order. Further, if the radius of the sphere is an infinitesimal of the first order, QR is of the third order of small quantities.

The principal problem which we have in view may be stated as follows:

To construct a triangle ABC (Fig. 1) in which are given in magnitude only, the height h = AO from the vertex A, the median m = BI from the vertex B, and the bisector f= CD from the vertex C.

Take first the Addition-Formula of the Tangent :

Assume that 1 – tanx tanO is not zero for all value of x II.

Let G be a group with identity 0 and let be a group of automorphisms of G. The centralizer near-ring determined by G and is the set for all α∈ and f(0)=0}, forming a near-ring under function addition and function composition. This class of nea-rings has been extensively studied (for example see [1], [2], [5] and [6]) and it is known that every finite simple near-ring with identity which is not a ring is isomorphic to C(;G) for a suitable pair (,G) see [6] page 131, Corollary 4.59 and Theorem 4.60.

Two features characterise the treatment of Geometry as presented in Euclid's “Elements”: (1) the propositions are arranged in a definite sequence which cannot be greatly altered without invalidating the proofs; (2) there are no methods of proof applicable to a large number of propositions. If we except the method of reductio ad absurdum, it is scarcely an exaggeration to say, for example, that in Book I no three propositions are proved by the same method.