Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K _1,2n (A, ) = G _2n (A, )/E _2n (A, ), n 3, where G _2n (A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E _2n (A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G _2n (A, ) G _2n ⁰(A, ) ; G _2n ¹(A, ) ... E _2n (A, ) of the general quadratic group G _2n (A, ) such that G _2n (A, )/G _2n ⁰(A, ) is Abelian, G _2n ⁰(A, ) G _2n ¹(A, ) ... is a descending central series, and G _2n ^d(A)(A, ) = E _2n (A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K _1,2n (A, ) is solvable when d(A) < .     

K1 of Some Non-Commutative Completed Group Rings

We compute K₁ of completed group rings of some two dimensional p-adic Lie groups. Dedicated to Professor Spencer Bloch on his sixtieth birthday     

On the K0 groups of nest algebras

In this paper we compute the K ₀ group of any nest subalgebra of ().     

The relationship between the boundedly controlled K1 and Whitehead groups

Let Z be a boundedness control space and p: X Z be a continuous map. The boundedly controlled Whitehead group Wh ^bc (X, p) is defined to be a quotient of the boundedly controlled K ₁-group K ₁ ^bc (X, p) by a certain subgroup whose generators are explicitly given. In general, little is known about this subgroup and it is even possible that it vanishes; i.e. that the boundedly controlled K ₁ and Whitehead groups are identical. This paper examines the structure of this subgroup in the case when p is the open cone on a PL map between compact polyhedra. As a byproduct, it calculates Wh ^bc (X, p) in some of these cases.Partially supported by the NSF under grant number DMS-8803149.     

The decomposition of Kv into K2×K5''''s

The decomposition of the complete graph Kv into Kr×Kc's, the products of Kr and Kc,is originated from the use of DNA library screening. In this paper, we consider the case where r=2 and c = 5, and show that such a decomposition exists if and only if v ≡ 1 (mod 25).     

Double Short Exact Sequences and K1 of an Exact Category

We introduce a modification of the Bass group K ₁ ^det . Our group is defined by generators and relations, in terms of double short exact Sequences instead of automorphisms. We establish its relation to Quillen's K₁ by dealing with loops in the G-construction.     

K2 of a Ring via the G-Construction

We interpret the Steinberg symbols x_i,j(a) as homotopies contracting the elementary matrices e_i,j(a), the latters being represented by certain arcs in a simplicial model of the K-theory. We further prove the Steinberg relations for these homotopies. This provides an explicit map from K₂ of a ring, defined classically as ker(St(R) → GL(R)), to π₂ of the G-construction assigned to R. Though the two groups are known to be isomorphic, a certain work is to be done to prove that this explicit map is an isomorphism. Mathematics Subject Classification 1991: Primary 19B99, 19D99; secondary 18E10, 18F25.     

ComplexK-theory ofBSL3(Z)

ComplexK-theory ofBSL₃(Z) andBSt₃(Z) are computed. We first study a Brown-Peterson (BP) version of Soulé's arguments. Then we consider complexK-theory by using a Conner-Floyd type theorem.     

On the structure of the K2 of the ring of integers in a number field

For any prime p, the p-primary part of the tame and the wild kernel of a number field F is described in terms of ideal class groups of p-power cyclotomic extensions of F.     

The Lower Algebraic K -theory of Γ3

We explicitly compute the lower algebraic K-theory of Γ₃ a discrete subgroup of the group of isometries of hyperbolic 3-space. An erratum to this article is available at .     

L1 regression estimate and its bootstrap

We study the asymptotic distribution of the L ₁ regression estimator under general conditions with matrix norming and possibly non i.i.d. errors. We then introduce an appropriate bootstrap procedure to estimate the distribution of this estimator and study its asymptotic properties. It is shown that this bootstrap is consistent under suitable conditions and in other situations the bootstrap limit is a random distribution. This work was supported by J.C. Bose National Fellowship, Government of India     

SU(2)-equivariantk-theory of a class ofC*-algebras and nonhomogeneous random walks

The SU(2) equivariantK ₀-Theory of a class ofC*-algebras is studied. These algebras arise from nonhomogeneous actions of SU(2) or SO(3) on uniformly hyperfinite algebras. The problems are shown to be equivalent to studying nonhomogeneous random walks associated with infinite products of characters — in particular, properties related to unimodality and positivity. Concrete sufficient conditions are developed for reducing the problem to the maximal torus action, for which extensive results are known; these become necessary and sufficient when there is a bound on the degree of the characters, or a strong unimodality assumption is made. When the degrees are unbounded, examples are constructed to indicate the range of behavior possible when reduction to the maximal torus is impossible. Finally, limit ratio theorems are obtained for the distribution of irreducibles in products of characters having a unimodality property.Supported in part by an operating grant from NSERC, Canada.