On projective modules

LetR be a commutative Noetherian ring with identity. The Hermite dimension ofR is defined to be the least integerr such that every stably freeR-module of rank greater thanr is free. In this paper we study ringsR obtained upon inversion of elements of a given ringA. We show that the Hermite dimension ofR does not depend on the Hermite dimension ofA, it depends on the Krull dimension ofA.     

The subalgebra systems of direct powers

A simple characterization of the subalgebra systems of direct powers of finitary universal algebras on a fixed infinite setA is given. For |I|≥|A| such subalgebra system of anI-power is precisely an algebraic closure systemS onA ^I closed under mutations ofI (which encompass both the precomposition by permutations ofI and allowing the values at specified elements ofI to become unrestricted) and such that each function in the intersection ofS is constant. For |I|<|A| the subalgebra systems ofI-powers are obtained as the restrictions toI of such closure systems on someA ^J withJ⊇I and |J|=|A|. Presented by J. D. Monk.     

The condition numbers of the matrix eigenvalue problem

Summary For ann ×n matrixA with distinct eigenvalues explicit expressions are obtained for certain condition numbers associated with the reduction ofA to its Jordan normal form. These condition numbers are also related by inequalities to (i) the departure from normality ofA, (ii) the discriminant of the eigenvalues ofA, (iii) the Gram determinant of the eigenvectors ofA.     

CertainC*-algebras with real rank zero and their corona and multiplier algebras: II

We first determine the homotopy classes of nontrivial projections in a purely infinite simpleC*-algebraA, in the associated multiplier algebraM(A) and the corona algebraM A/A in terms ofK _*(A). Then we describe the generalized Fredholm indices as the group of homotopy classes of non-trivial projections ofA; consequently, we determine theK _*-groups of all hereditaryC*-subalgebras of certain corona algebras. Secondly, we consider a group structure of *-isomorphism classes of hereditaryC*-subalgebras of purely infinite simpleC*-algebras. In addition, we prove that ifA is aC*-algebra of real rank zero, then each unitary ofA, in caseA it unital, each unitary ofM(A) and ofM(A)/A, in caseA is nonunital but -unital, can be factored into a product of a unitary homotopic to the identity and a unitary matrix whose entries are all partial isometries (with respect to a decomposition of the identity).Partially supported by a grant from the National Science Foundation.     

Instability of singular critical points of definitizable operators

LetA be a selfadjoint definitizable operator in a Krein space. It is shown that there exists a finite rank nonnegative perturbation ofA of arbitrarily small norm such that all the singular critical points ofA of finite index disappear.     

Approximation of the spectrum of a non-compact operator given by the magnetohydrodynamic stability of a plasma

Summary The study of the magnetohydrodynamic stability of a plasma leads to a problem of determination of the spectrum of a non-compact selfadjoint operatorA. The spectrum ofA will be approximated by the eigenvalues ofA _h , whereA _h is a linear operator approximatingA in a finite dimensional space (finite element method) andh is a parameter which tends to zero. Generally the spectrum ofA _h pollutes spectrum ofA, i.e. for eachh there exists an eigenvalue h ofA _h which, ash tends to zero, converges to which is not in the spectrum ofA.We present here a sufficient condition on the finite dimensional spaces used, in order to obtain good approximation properties of the spectrum ofA and, especially, the non-pollution property.     

A generalization to the non-separable case of Takesaki's duality theorem forC *-algebras

Takesaki [5] poses the question of how much information about aC ^*-algebraA is contained in its representation theory. He gives it a precise meaning in the following setting: One can furnish the set Rep (A:H) of all representations ofA in a suitable Hilbert spaceH with a topology, with an action of the unitary groupG ofB(H) on it, and with an addition. The setA ^F of operator fields Rep (A:H)B(H) commuting with the action ofG and addition, called the admissible operator fields, turn out to form aW ^*-algebra isomorphic to the bidual ofA with Arens multiplication or with the universal enveloping von Neumann algebra ofA. Takesaki shows in the separable case thatA can be identified inA ^F as the set of continuous admissible operator fields, and leaves the same question open for arbitraryC ^*-algebras. Changing the structures on Rep(A:H) slightly, it is shown here that this result obtains in the general case as well. The proof proceeds along the lines set up in [5] but makes no use of the representation theory of NGCR algebras.     

The normalized matching property from the generalized Macaulay theorem

IfA is a family ofk-element subsets of a finite setM having elements of several different types (i.e., amultiset) and A is the set of all (k–1)-element subsets ofM obtainable by removing a single element ofM from a single member ofA, then, according to the well known normalized matching condition, the density ofA among thek-element subsets ofM never exceeds the density of A among the (k–1)-element subsets ofM. We show that this useful fact can be regarded as yet another corollary of the generalized Macaulay theorem.     

On Tate-Shafarevich groups over galois extensions

LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L) ^G ]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(A ^x/K)] whereA x is the twist ofA by the non-trivial characterχ ofG.     

The Hilbert series of prime PI rings

We study the Hilbert series of finitely generated prime PI algebras. We show that given such an algebraA there exists some finite dimensional subspaceV ofA which contains 1 _A and generatesA as an algebra such that the Hilbert series ofA with respect to the vector spaceV is a rational function.     

A numerical procedure for computing the Moore-Penrose inverse

An algorithm for computing the Moore-Penrose inverse of an arbitraryn×m real matrixA is presented which uses a Gram-Schmidt like procedure to form anA-orthogonal set of vectors which span the subspace perpendicular to the kernel ofA. This one procedure will work for any value ofn andm, and for any value of rank (A).     

Modular annihilator algebras

The elements of minimal left (right) ideals in a semi-prime modular annihilator algebraA completely characterized by the property of being singles not in radA. An elements ofA is calledsingle if wheneverasb=0 for somea,b inA then at least one ofas,sb is zero.     

On the generalized Lie structure of associative algebras

We study the structure of Lie algebras in the category ^H MA ofH-comodules for a cotriangular bialgebra (H, 〈|〉) and in particular theH-Lie structure of an algebraA in ^H MA. We show that ifA is a sum of twoH-commutative subrings, then theH-commutator ideal ofA is nilpotent; thus ifA is also semiprime,A isH-commutative. We show an analogous result for arbitraryH-Lie algebras whenH is cocommutative. We next discuss theH-Lie ideal structure ofA. We show that ifA isH-simple andH is cocommutative, then any non-commutativeH-Lie idealU ofA must contain [A, A]. IfU is commutative andH is a group algebra, we show thatU is in the graded center ifA is a graded domain. Dedicated to the memory of S. A. Amitsur Supported by a Fulbright grant. Supported by NSF grant DMS-9203375.     

On eigenvalues of symmetric (+1, −1) matrices

To every symmetric matrixA with entries ±1, we associate a graph G(A), and ask (for two different definitions of distance) for the distance ofG(A) to the nearest complete bipartite graph (cbg). Letλ ₁(A),λ ¹ (A) be respectively the algebraically largest and least eigenvalues ofA. The Frobenius distance (see Section 4) to the nearest cbg is bounded above and below by functions ofn −λ ₁ (A), wheren=ord A. The ordinary distance (see Section 1) to the nearest cbg is shown to be bounded above and below by functions ofλ ¹ (A). A curious corollary is: there exists a functionf (independent ofn, and given by (1.1)), such that |λ _i (A) | ≦f(λ ¹(A), whereλ _i (A) is any eigenvalue ofA other thanλ _i (A). This work was supported (in part) by the U.S. Army under contract #DAHC04-C-0023.     

Complex powers of differential operators on manifolds with conical singularities

We construct the complex powersA _z for an elliptic cone (or Fuchs type) differential operatorA on a manifold with boundary. We show thatA _z exists as an entire family ofb-pseudodifferential operators. We also examine the analytic structure of the Schwartz kernel ofA _z , both on and off the diagonal. Finally, we study the meromorphic behavior of the zeta function Tr(A _z ). Supported by a Ford Foundation Fellowship administered by the National Research Council.     

Connected components of attractors and other stable sets

Summary We study the action of a continuous mapf on a locally connected metric space, with an emphasis on the action induced byf on the space of connected components of an attractorA or a stable setY off. The main result for attractors is that, ifA is an attractor in the sense of C. Conley, thenf permutes the components ofA and that the permutation is cyclic if and only if the attractor is indecomposable. This is related to similar results obtained recently by Buescu and Stewart. In the case of an indecomposable stable invariant setY we show that either the action off on the components ofY is as described above for the action on the attractor, or elseY has infinitely many components, the action off on them is a generalized adding machine, so that there are no periodic points inY. Results along these lines are due to Buescu and Stewart and to Melbourne, Dellnitz and Golubitsky.     

Graded rings and essential ideals

LetG be a group andA aG-graded ring. A (graded) idealI ofA is (graded) essential ifI⊃J≠0 wheneverJ is a nonzero (graded) ideal ofA. In this paper we study the relationship between graded essential ideals ofA, essential ideals of the identity componentA _e and essential ideals of the smash productA#G ^*. We apply our results to prime essential rings, irredundant subdirect sums and essentially nilpotent rings.     

Steiner polygons in the Steiner problem

A polygon, whose vertices are points in a given setA ofn points, is defined to be a Steiner polygon ofA if all Steiner minimal trees forA lie in it. Cockayne first found that a Steiner polygon can be obtained by repeatedly deleting triangles from the boundary of the convex hull ofA. We generalize this concept and give a method to construct Steiner polygons by repeatedly deletingk-gons,k n. We also prove the uniqueness of Steiner polygons obtained by our method.     

Homeomorphisms of universal dendrites

Homeomorphisms of some special universal dendrites are considered in the paper. In particular, dendrites are characterized for which the action onX of the group of autohomeomorphisms ofX has exactlyn≥3 orbits, and for each orbitB ofX and for each areA ⊂X the intersectionA∪B is a dense subset ofA. Some of the results generalize earlier ones from the author’s paper [2].     

Trace polynomials and invariant theory

LetA be a finite-dimensional simple (non-associative) algebra over an algebraically closed fieldF of characteristic 0. LetG be the group of its automorphisms which acts onkA, the direct sum ofk copies ofA. SupposeA is generated byk elements. In this paper, generators of the field of rational invariantF(kA) ^G are described in terms of operations of the algebraA.