Mowaffaq Hajja

Means and their inequalities

The theory of means has its roots in the work of the Pythagoreans who introduced the harmonic, geometric, and arithmetic means with reference to their theories of music and arithmetic. Later, Pappus introduced seven other means and gave the well-known elegant geometric proof of the celebrated inequalities among the harmonic, geometric, and arithmetic means. Nowadays, the families and types of means that are being investigated by researchers and the variety of questions that are being asked about them are beyond the scope of any single survey, with the voluminous book Handbook of Means and Their Inequalities by P. S. Bullen being the best such reference in this direction. The theory of means has grown to occupy a prominent place in mathematics with hundreds of papers on the subject appearing every year. The strong relations and interactions of the theory of means with the theories of inequalities, functional equations, and probability and statistics add greatly to its importance.

Orthocentric simplices and their centers

A simplex is said to be orthocentric if its altitudes intersect in a common point, called its orthocenter. In this paper it is proved that if any two of the traditional centers of an orthocentric simplex (in any dimension) coincide, then the simplex is regular. Along the way orthocentric simplices in which all facets have the same circumradius are characterized, and the possible barycentric coordinates of the orthocenter are described precisely. In particular these barycentric coordinates are used to parametrize the shapes of orthocentric simplices. The substantial, but widespread, literature on orthocentric simplices is briefly surveyed in order to place the new results in their proper context, and some of the previously known results are given with new proofs from the present perspective.

Finite Group Actions on Rational Function Fields

is an invertible n x n matrix with integer entries and where a,(a) E K\(O). If a,(c) = 1 Vi, then CJ is called purely monomial. It is proved in [6, Theorem] that if G is any finite group of monomial K-automorphisms of K(x,, x2) then its fixed held K(x,, x2)’ is rational ( =purely transcendental) (over K). This does not generalize to the three variable case since the fixed field of the (order 4 cyclic group generated by the) monomial Q-automorphism o defined on Q(x,, x2, x3) by is not rational [S, last paragraph]. However, we prove here that the fixed field of every finite abeliun group of purely monomial K-automorphisms of

Rationality of Finite Groups of Monomial Automorphisms of k(x, y)

Abstract It has been shown ( Hajja, J. Algebra 85 (1983) , 243–250) that every finite cyclic group of monomial k-automorphisms of k(x, y) has a purely transcendental fixed field. Here, the same result is established with the assumption that the group is cyclic dropped.

A note on monomial automorphisms

Abstract Previous results (Hajja, J. Algebra73 (1981), 30–36) on monomial automorphisms are strengthened and the rationality of all monomial automorphisms on k(x,y) is established. The relation of certain steps in the proofs to a special case of a problem of Zariski on whether quasi-rationality implies rationality is also noted.

The measure of solid angles in n-dimensional Euclidean space

A formula in terms of a definite integral for the measure of a polygonal solid angle in a Euclidean space of arbitrary dimension is proved. The formula is applied to the study of the geometry of n-simplices.

A note on affine automorphisms

Let be a purely transcendental extension of the field kof transcendence degree d. A k-automorphism of K is said to be affine (resp. linear) if it acts on the k-subspace of K generated by . In this article, we give a simple canonical form for affine automorphisms (modulo linear ones) and we extend to affine automorphisms (a stronger form of) the negligibility theorem proved for linear ones in [3,Theorem 1.5], We also discuss the question of how much of an affine automorphism (of finite order) is determined by its order and we show that in certain cases an affine automorphism is completely characterized by its order.

Triangle centres: some questions in Euclidean geometry

The circumcentre E of a triangle ABC is defined, as in figure 1, by the two relations EA = EB EB = EC The other centres (such as the incentre, the centroid, etc.) can be defined by two similar relations. This note is an elaboration on the simple fact that if two centres of a triangle coincide then it is equilateral. We take a certain centre of a given triangle and investigate what can be deduced from the assumption that it satisfies one of the two defining relations of another centre. This is done for each pair of, what one may think of as, the seven most natural centres.

On the Fermat-Torricelli points of tetrahedra and of higher dimensional simplexes

Introduction It is well known (see, e.g., [6, pp. 24-34], [2, p. 22], [5]) that if ABC is a non-collinear triangle, then there exists a unique point P in the triangle that minimizes the quantity XA + XB + XC. If any of the vertices of ABC holds an angle greater than or equal to 2-7T/3 = 120?, then P coincides with that vertex. Otherwise, P lies inside the triangle and satisfies the equiangular property

Quasi-linearity of Cyclic Monomial Automorphisms*

LetK1,K2 be purely transcendental extensions ofk of finite transcendence degrees and lets1,s2 bek-automorphisms ofK1,K2 of finite orders. In Theorem 1.5, it is shown that ifs1 acts linearly (on some base ofK1) and if order(s1) divides order(s2), thens1 ⊛ s2 is (quasi-) equivalent toI ⊛ s2, whereI is the identity automorphism ofK1 and wheres1 ⊛ s2 is thek-automorphism induced bys1 ands2 on the quotient fieldK1 ⊛ K2 ofK1 ⊗k K2. This fact and results from [1] are then used to prove that every cyclic monomial automorphism is quasilinearizable (Theorem 2.5).