Mathematics Commons^™

The Fine Structure Of The Kasparov Groups I: Continuity Of The Kk-Pairing, Claude Schochet

Mathematics Faculty Research Publications

In this paper it is demonstrated that the Kasparov pairing is continuous with respect to the natural topology on the Kasparov groups, so that a KK-equivalence is an isomorphism of topological groups. In addition, we demonstrate that the groups have a natural pseudopolonais structure, and we prove that various KK-structural maps are continuous.

Ultraconvergence Of Zz Patch Recovery At Mesh Symmetry Points, Zhimin Zhang, Runchang Lin

Mathematics Research Reports

Ultraconvergence property of the Zienkiewicz-Zhu gradient patch recovery technique based on local discrete least squares fitting is established for a large class of even-order finite elements. The result is valid at all rectangular mesh symmetry points. Different smoothing strategies are discussed. Superconvergence recovery for the Q8 element is proved and ultraconvergence numerical examples are demonstrated.

Geometric Realization And K-Theoretic Decomposition Of C*-Algebras, Claude Schochet

Mathematics Faculty Research Publications

Suppose that A is a separable C*-algebra and that G∗ is a (graded) subgroup of the ℤ/2-graded group K∗(A). Then there is a natural short exact sequence

0 → G∗ → K∗(A) → K∗(A)/G∗ → 0.

In this note we demonstrate how to geometrically realize this sequence at the level of C*-algebras. As a result, we KK-theoretically decompose A as

0 → A ⊗ [cursive]K → A_ƒ → SA_t → 0

where K∗(A_t) is the torsion subgroup of …

Extended Powers Of Manifolds And The Adams Spectral Sequence, Robert R. Bruner

Mathematics Faculty Research Publications

The extended power construction can be used to create new framed manifolds out of old. We show here how to compute the effect of such operations in the Adams spectral sequence, extending partial results of Milgram and the author. This gives the simplest method of proving that Jones’ 30-manifold has Kervaire invariant one, and allows the construction of manifolds representing Mahowald’s classes η4 and η5, among others.

Ossa's Theorem And Adams Covers, Robert R. Bruner

Mathematics Faculty Research Publications

We show that Ossa’s theorem splitting ku ∧ BV for elementary abelian groups V follows from general facts about ku ∧ BZ/2 and Adams covers. For completeness, we also provide the analogous results for ko ∧ BV .

Invariant Measure For Diffusions With Jumps, Jose-Luis Menaldi, Maurice Robin

Mathematics Faculty Research Publications

Our purpose is to study an ergodic linear equation associated to diffusion processes with jumps in the whole space. This integro-differential equation plays a fundamental role in ergodic control problems of second order Markov processes. The key result is to prove the existence and uniqueness of an invariant density function for a jump diffusion, whose lower order coefficients are only Borel measurable. Based on this invariant probability, existence and uniqueness (up to an additive constant) of solutions to the ergodic linear equation are established.

The Topological Snake Lemma And Corona Algebras, Claude Schochet

Mathematics Faculty Research Publications

We establish versions of the Snake Lemma from homological algebra in the context of topological groups, Banach spaces, and operator algebras. We apply this tool to demonstrate that if ƒ : B → B′ is a quasi-unital C^*-map of separable C^*-algebras, so that it induces a map of Corona algebras ƒ̄ : QB → QB′, and if ƒ is mono, then the induced map ƒ̄ is also mono.

Some Root Invariants And Steenrod Operations In Ext_A(F2,F2), Robert R. Bruner

Mathematics Faculty Research Publications

We give the results of computations of root invariants in Ext over the Steenrod algebra through the 25-stem, with partial information through the 45-stem. This allows the computation of some new Steenrod operations as well.

Some Remarks On The Root Invariant, Robert R. Bruner

Mathematics Faculty Research Publications

We show how the root invariant of a product depends upon the product of the root invariants, give some examples of the equivariant definition of the root invariant, and verify a weakened form of the algebraic Bredon-Löffler conjecture.

A Yoneda Description Of The Steenrod Operations, Robert R. Bruner

Mathematics Faculty Research Publications

No abstract provided.

Infinite-Dimensional Hamilton-Jacobi-Bellman Equations In Gauss-Sobolev Spaces, Pao-Liu Chow, Jose-Luis Menaldi

Mathematics Faculty Research Publications

We consider the strong solution of a semi linear HJB equation associated with a stochastic optimal control in a Hilbert space H: By strong solution we mean a solution in a L²(μ,H)-Sobolev space setting. Within this framework, the present problem can be treated in a similar fashion to that of a finite-dimensional case. Of independent interest, a related linear problem with unbounded coefficient is studied and an application to the stochastic control of a reaction-diffusion equation will be given.

On Stable Homotopy Equivalences, Robert R. Bruner, F. R. Cohen, C. A. Mcgibbon

Mathematics Faculty Research Publications

No abstract provided.

Regularity Of The Free-Boundary In Singular Stochastic Control, S. A. Williams, P. L. Chow, J. L. Menaldi

Mathematics Faculty Research Publications

No abstract provided.

Ext In The Nineties, Robert R. Bruner

Mathematics Faculty Research Publications

We describe a package of programs to calculate minimal resolutions, chain maps, and null homotopies in the category of modules over a connected algebra overe Z_2 and in the category of unstable modules over the mod 2 Steenrod algebra. They are available for free distribution and intended for use as an Adams spectral sequence 'pocket calculator'. We provide a sample of the results obtained from them.

Generalized Lame-Clapeyron Solution For A One-Phase Source Stefan Problem, José Luis Menaldi, Domingo Alberto Tarzia

Mathematics Faculty Research Publications

In this paper we obtain a generalized Lamé-Clapeyron solution for a one-phase Stefan problem with a particular type of sources. Necessary and sufficient conditions are given in order to characterize the source term which provides a unique solution. Some estimates on the free boundary and the temperature are presented. In particular, asymptotic expansions are given for small Stefan number and source.

Regularizing Effect For Integro-Differential Parabolic Equations, Maria Giovanna Garroni, José Luis Menaldi

Mathematics Faculty Research Publications

No abstract provided.

Remarks On Estimates For The Green Function, Jose Luis Menaldi

Mathematics Faculty Research Publications

No abstract provided.

Mat 751 Algebraic Topology I - Fall '89, David Handel

Mathematics Faculty Research Publications

A collection of notes for the course Mat 751, Algebraic Topology I, prepared by Professor David Handel of the Wayne State University Mathematics Department. The notes include examples, exercises, and additional lecture notes on related concepts.

Math 752 Algebraic Topology Ii - Winter '84, David Handel

Mathematics Faculty Research Publications

A collection of notes for the course MAT 752, Algebraic Topology II, prepared by Professor David Handel of the Wayne State University Mathematics Department. This course builds on MAT 751, Algebraic Topology I, and the notes include examples, exercises, and suggestions for further reading.

Two Generalizations Of The Adams Spectral Sequence, Robert R. Bruner

Mathematics Faculty Research Publications

No abstract provided.

The Classification Of Extensions Of C*-Algebras, Jonathan Rosenberg, Claude Schochet

Mathematics Faculty Research Publications

Our objective in this note is to outline a number of results concerning the Kasparov groups Ext(A,B) which are analogous to known information about the Brown, Douglass, and Fillmore (BDF) theory regarding groups which classify extensions of the form 0 → B ⊗ K → E → A → 0. These results enable one to compute the groups with relatively mild restrictions on the C*-algebras A and B. This in turn should make it possible to analyze the way in which a wide variety of C*-algebra extensions are put together, at least stably.

Homogeneous Extensions Of C*-Algebras And K-Theory. I, Claude Schochet

Mathematics Faculty Research Publications

No abstract provided.

K-Theory And Steenrod Homology: Applications To The Brown-Douglas-Fillmore Theory Of Operator Algebras, Jerome Kaminker, Claude Schochet

Mathematics Faculty Research Publications

The remarkable work of L. G. Brown, R. Douglas and P. Fillmore on operators with compact self-commutators once again ties together algebraic topology and operator theory. This paper gives a comprehensive treatment of certain aspects of that connection and some adjacent topics. In anticipation that both operator theorists and topologists may be interested in this work, additional background material is included to facilitate access.

K₁ Of The Compact Operators Is Zero, L. G. Brown, Claude Schochet

Mathematics Faculty Research Publications

We prove that K₁ of the compact operators is zero. This theorem has the following operator-theoretic formulation: any invertible operator of the form (identity) + (compact) is the product of (at most eight) multiplicative commutators (A_jB_jA_j⁻¹B_j⁻¹)^±1, where each B_j is of the form (identity) + (compact). The proof uses results of L. G. Brown, R. G. Douglas, and P. A. Fillmore on essentially normal operators and a theorem of A. Brown and C. Pearcy on multiplicative …

Steenrod Homology And Operator Algebras, Jerome Kaminker, Claude Schochet

Mathematics Faculty Research Publications

The recent work of Larry Brown, R. G. Douglas, and Peter Fillmore on operator algebras has created a new bridge between functional analysis and algebraic topology. This note constitutes an effort to make that bridge more concrete.

On The Bordism Ring Of Complex Projective Space, Claude Schochet

Mathematics Faculty Research Publications

The bordism ring MU∗(CP^∞) is central to the theory of formal groups as applied by D. Quillen, J. F. Adams, and others recently to complex cobordism. In the present paper, rings E∗(CP^∞) are considered, where E is an oriented ring spectrum, R=π∗(E), and pR=0 for a prime p. It is known that E∗(CP^∞) is freely generated as an R-module by elements {β_τ|r≧0}. The ring structure, however, is not known. It is shown that the elements {β …

Radicals And Torsion Theories In Locally Compact Groups, Robert R. Bruner

Mathematics Faculty Research Publications

In this paper we will study the properties of locally compact Abelian Hausdorff topological groups (hereafter known as LCA groups) by means of their mapping properties. The results contained herein are an outgrowth of work done by Professor Armacost [Al] on "sufficiency classes" of LCA groups. The sufficiency class S\textunderscore(H) of an LCA group H is the class of all LCA groups G such that there are sufficiently many continuous homomorphisms from G to H to separate the points of G. This condition is easily seen to be equivalent to the requirement that ∩ker(f)=0, where f ranges over all elements …

A Two-Stage Postnikov System Where E₂ ≠ E∞ In The Eilenberg-Moore Spectral Sequence, Claude Schochet

Mathematics Faculty Research Publications

Let ΩB → PB → B be the path fibration over the simply-connected space B, let ΩB → E → X be the induced fibration via the map ƒ : X → B, and let X and B be generalized Eilenberg-Mac Lane spaces. G. Hirsch has conjectured that H*E is additively isomorphic to Τοτ_H*B(Z₂,H*X), where cohomology is with Z₂ coefficients. Since the Elienberg-Moore spectral sequence which converges to H*E has E₂ = Τοτ_H*B(Z₂,H …