Characterization (mathematics)

Abstract: This note is concerned, for matrices with elements in an algebraically closed field of arbitrary characteristic p, with pencils generated by pairs of matrices with property L. A pair of n by n matrices is said to have property L if for a special ordering of the characteristic roots a< of A and Bi of B, the characteristic roots of \A+ptB are Xa<+p,/3,- for all values of X and pt. (See [1-5].) In §§1-5 another characterization of pairs of matrices with property L is given for a large class of such pairs. The method employed for this purpose is used in §6 for the study of pencils (not necessarily with property L) of diagonable matrices, i.e., matrices which are similar to a diagonal matrix. (These matrices are also called nondefective.) It is shown that for p=0, as well as for n^p, such pencils are always generated by commutative matrices. In §7 the significance of this result for general pencils of commutative matrices is investigated. 1. The ^-discriminant. The new characterization of pairs A, B of matrices with property L is obtained by considering those ratios X/p. for which \A +p,B has a multiple characteristic root. We see as follows that this is the case either for at most n(n — l) ratios or for every X/p.

Abstract: The goals of this paper are to provide a characterization of dihedral angles of convex ideal (those with all vertices on the sphere at infinity) polyhedra in H3, and also of those convex polyhedra with some vertices on the sphere at infinity and some in the finite part of H3. These characterizations are given in, respectively, Theorems 0.1 and 10.5. The first theorem is proved in detail, while the proof of the second (which is similar) is only outlined. The results of this paper grow out of the general framework of the author's doctoral dissertation [13], as published in [20]. A lot of the language, and some of the auxilary results come from there as well, so familiarity with the latter reference is very helpful. The necessity of the conditions postulated in Theorem 0.1 has been shown in [14]; hence only the sufficiency direction will be shown in the current paper. In 1832, Jakob Steiner asked for a combinatorial characterization of convex polyhedra inscribed in the sphere. This was considered intractable it took almost a hundred years to find a single example of an "uninscribable" combinatorial type (by Steinitz [24]). However, Theorem 0.1 gives such a characterization. Section 11 is dedicated to a (very brief) historical survey and some purely graph-theoretic and computational-geometric consequences of Theorem 0.1. In order to state this theorem, suppose that a convex ideal polyhedron P in H3 is given. Let P* denote the Poincar' dual of P, and assign to each edge e* of P* a weight w(e*) equal to the exterior dihedral angle at the corresponding edge e of P. Then the following result holds: