The Trace Conjecture – A Counterexample

In the paper 05v8337/xxlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">Geometric K-Theory for Lie Groups and Foliations05v8337/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">, Baum and Connes conjecture in a remark following Corollary 2 of their famous Isomorphism conjecture that for a finitely generated group 05v8337/xxlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0"> with torsion, the trace map tr: K₀(C^*05v8337/xxlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">)05v8337/xxlarge8594.gif" alt="rarr" align="BASELINE" BORDER="0">R maps K₀(C^*05v8337/xxlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">) onto the additive subgroup of Q generated by all rational numbers of the form 1/n where n is the order of a finite subgroup of 05v8337/xxlarge915.gif" alt="Gamma" align="BASELINE" BORDER="0">. We construct a counterexample to this conjecture.     

Testing optimality for quadratic 0-1 problems

Received June 4, 1996 / Revised version received November 19, 1997 Published online November 24, 1998     

The 0-1 Knapsack problem with a single continuous variable

Specifically we investigate the polyhedral structure of the knapsack problem with a single continuous variable, called the mixed 0-1 knapsack problem. First different classes of facet-defining inequalities are derived based on restriction and lifting. The order of lifting, particularly of the continuous variable, plays an important role. Secondly we show that the flow cover inequalities derived for the single node flow set, consisting of arc flows into and out of a single node with binary variable lower and upper bounds on each arc, can be obtained from valid inequalities for the mixed 0-1 knapsack problem. Thus the separation heuristic we derive for mixed knapsack sets can also be used to derive cuts for more general mixed 0-1 constraints. Initial computational results on a variety of problems are presented. Received May 22, 1997 / Revised version received December 22, 1997 Published online November 24, 1998     

A finite‐capacity queue with exhaustive vacation/close‐down/setup times and Markovian arrival processes

We consider a finite045m62p36/xxlarge8208.gif" alt="dash" align="MIDDLE" BORDER="0">capacity single045m62p36/xxlarge8208.gif" alt="dash" align="MIDDLE" BORDER="0">server vacation model with close045m62p36/xxlarge8208.gif" alt="dash" align="MIDDLE" BORDER="0">down/setup times and Markovian arrival processes (MAP). The queueing model has potential applications in classical IP over ATM or IP switching systems, where the close045m62p36/xxlarge8208.gif" alt="dash" align="MIDDLE" BORDER="0">down time corresponds to an inactive timer and the setup time to the time delay to set up a switched virtual connection (SVC) by the signaling protocol. The vacation time may be considered as the time period required to release an SVC or as the time during which the server goes to set up other SVCs. By using the supplementary variable technique, we obtain the queue length distribution at an arbitrary instant, the loss probability, the setup rate, as well as the Laplace013;Stieltjes transforms of both the virtual and actual waiting time distributions.     

Existence and Limiting Behavior of Trajectories Associated with P0-equations

Given a continuous P₀-function F : Rⁿ 08/xxlarge8594.gif" alt="rarr" align="BASELINE" BORDER="0"> Rⁿ, we describe a method of constructing trajectories associated with the P₀-equation F(x) = 0. Various well known equation-based reformulations of the nonlinear complementarity problem and the box variational inequality problem corresponding to a continuous P₀-function lead to P₀-equations. In particular, reformulations via (a) the Fischer function for the NCP, (b) the min function for the NCP, (c) the fixed point map for a BVI, and (d) the normal map for a BVI give raise to P₀-equations when the underlying function is P₀. To generate the trajectories, we perturb the given P₀-function F to a P-function F(x, 08/xxlarge949.gif" alt="epsi" align="BASELINE" BORDER="0">); unique solutions of F(x, 08/xxlarge949.gif" alt="epsi" align="BASELINE" BORDER="0">) = 0 as 08/xxlarge949.gif" alt="epsi" align="BASELINE" BORDER="0"> varies over an interval in (0, 08/xxlarge8734.gif" alt="infin" align="MIDDLE" BORDER="0">) then define the trajectory. We prove general results on the existence and limiting behavior of such trajectories. As special cases we study the interior point trajectory, trajectories based on the fixed point map of a BVI, trajectories based on the normal map of a BVI, and a trajectory based on the aggregate function of a vertical nonlinear complementarity problem.     

Countable Products of Čech-scattered supercomplete spaces

We prove by using well-founded trees that a countable product of supercomplete spaces, scattered with respect to 0238/xxlarge268.gif" alt="Ccaron" align="BASELINE" BORDER="0">ech-complete subsets, is supercomplete. This result extends results given in [1], [5], [6], [19], [26] and its proof improves that given in [19].     

Completely J —positive linear systems of finite order

Completely J — positive linear systems of finite order are introduced as a generalization of completely symmetric linear systems. To any completely J — positive linear system of finite order there is associated a defining measure with respect to which the transfer function has a certain integral representation. It is proved that these systems are asymptotically stable. The observability and reachability operators obey a certain duality rule and the number of negative squares of the Hankel operator is estimated. The Hankel operator is bounded if and only if a certain measure associated with the defining measure is of Carleson type. We prove that a real symmetric operator valued function which is analytic outside the unit disk has a realization with a completely J — symmetric linear space which is reachable, observable and parbalanced. Uniqueness and spectral minimality of the completely J — symmetric realizations are discussed.     

On the dimension of almost -dimensional spaces

Oversteegen and Tymchatyn proved that homeomorphism groups of positive dimensional Menger compacta are 09/S0002-9939-99-04846-7/gif-abstract/img5.gif" HEIGHT=13 WIDTH=6>-dimensional by proving that almost 0$" SRC="http://www.ams.org/proc/1999-127-09/S0002-9939-99-04846-7/gif-abstract/img6.gif" HEIGHT=13 WIDTH=8>-dimensional spaces are at most 09/S0002-9939-99-04846-7/gif-abstract/img7.gif" HEIGHT=13 WIDTH=6>-dimensional. These homeomorphism groups are almost 0$" SRC="http://www.ams.org/proc/1999-127-09/S0002-9939-99-04846-7/gif-abstract/img8.gif" HEIGHT=13 WIDTH=8>-dimensional and at least 09/S0002-9939-99-04846-7/gif-abstract/img9.gif" HEIGHT=13 WIDTH=6>-dimensional by classical results of Brechner and Bestvina. In this note we prove that almost 09/S0002-9939-99-04846-7/gif-abstract/img10.gif" HEIGHT=9 WIDTH=10>-dimensional spaces for 09/S0002-9939-99-04846-7/gif-abstract/img11.gif" HEIGHT=28 WIDTH=46> are 09/S0002-9939-99-04846-7/gif-abstract/img12.gif" HEIGHT=9 WIDTH=10>-dimensional. As a corollary we answer in the affirmative an old question of R. Duda by proving that every hereditarily locally connected, non-degenerate, separable, metric space is 09/S0002-9939-99-04846-7/gif-abstract/img13.gif" HEIGHT=13 WIDTH=6>-dimensional.

     

Non-abelian electrodynamics and the vacuumB (3)

By using an 0(3) gauge group, a non-Abelian theory of vacuum electrodynamics is developed in which the newly discovered longitudinal vacuum fieldsB ⁽³⁾ andi E ⁽³⁾ appear self-consistently with the usual plane wavesB ⁽¹⁾,B ⁽²⁾,E ⁽¹⁾, andE ⁽²⁾ in the circular basis (1), (2), (3), a complex representation of space. Using the charge quantization condition0702_2005_BF02187349_f6.jpg" alt="MediaObjects/10702_2005_BF02187349_f6.jpg"> the vacuum Maxwell equations are given in the non-Abelian representation.