36,517 research outputs found
Equal Entries in Totally Positive Matrices
We show that the maximal number of equal entries in a totally positive (resp.
totally nonsingular) matrix is (resp.
)). Relationships with point-line incidences in the plane,
Bruhat order of permutations, and completability are also presented. We
also examine the number and positionings of equal minors in a
matrix, and give a relationship between the location of
equal minors and outerplanar graphs.Comment: 15 page
Counting Euler Tours in Undirected Bounded Treewidth Graphs
We show that counting Euler tours in undirected bounded tree-width graphs is
tractable even in parallel - by proving a upper bound. This is in
stark contrast to #P-completeness of the same problem in general graphs.
Our main technical contribution is to show how (an instance of) dynamic
programming on bounded \emph{clique-width} graphs can be performed efficiently
in parallel. Thus we show that the sequential result of Espelage, Gurski and
Wanke for efficiently computing Hamiltonian paths in bounded clique-width
graphs can be adapted in the parallel setting to count the number of
Hamiltonian paths which in turn is a tool for counting the number of Euler
tours in bounded tree-width graphs. Our technique also yields parallel
algorithms for counting longest paths and bipartite perfect matchings in
bounded-clique width graphs.
While establishing that counting Euler tours in bounded tree-width graphs can
be computed by non-uniform monotone arithmetic circuits of polynomial degree
(which characterize ) is relatively easy, establishing a uniform
bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound
claimed in the previous version which has been subsequently remove
Shared-object System Equilibria: Delay and Throughput Analysis
We consider shared-object systems that require their threads to fulfill the
system jobs by first acquiring sequentially the objects needed for the jobs and
then holding on to them until the job completion. Such systems are in the core
of a variety of shared-resource allocation and synchronization systems. This
work opens a new perspective to study the expected job delay and throughput
analytically, given the possible set of jobs that may join the system
dynamically.
We identify the system dependencies that cause contention among the threads
as they try to acquire the job objects. We use these observations to define the
shared-object system equilibria. We note that the system is in equilibrium
whenever the rate in which jobs arrive at the system matches the job completion
rate. These equilibria consider not only the job delay but also the job
throughput, as well as the time in which each thread blocks other threads in
order to complete its job. We then further study in detail the thread work
cycles and, by using a graph representation of the problem, we are able to
propose procedures for finding and estimating equilibria, i.e., discovering the
job delay and throughput, as well as the blocking time.
To the best of our knowledge, this is a new perspective, that can provide
better analytical tools for the problem, in order to estimate performance
measures similar to ones that can be acquired through experimentation on
working systems and simulations, e.g., as job delay and throughput in
(distributed) shared-object systems
Distinguishing between exotic symplectic structures
We investigate the uniqueness of so-called exotic structures on certain exact
symplectic manifolds by looking at how their symplectic properties change under
small nonexact deformations of the symplectic form. This allows us to
distinguish between two examples based on those found in
\cite{maydanskiy,maydanskiyseidel}, even though their classical symplectic
invariants such as symplectic cohomology vanish. We also exhibit, for any ,
an exact symplectic manifold with distinct but exotic symplectic
structures, which again cannot be distinguished by symplectic cohomology.Comment: 33 pages, 6 figures. Final version, accepted by Journal of Topolog
Stability conditions and quantum dilogarithm identities for Dynkin quivers
We study fundamental group of the exchange graphs for the bounded derived
category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category
D(\Gamma_N Q) of the Calabi-Yau-N Ginzburg algebra associated to Q. In the case
of D(Q), we prove that its space of stability conditions (in the sense of
Bridgeland) is simply connected; as applications, we show that its
Donanldson-Thomas invariant can be calculated via a quantum dilogarithm
function on exchange graphs. In the case of D(\Gamma_N Q), we show that
faithfulness of the Seidel-Thomas braid group action (which is known for Q of
type A or N = 2) implies the simply connectedness of its space of stability
conditions.Comment: Journal (almost) equivalent versio
Integration over Tropical Plane Curves and Ultradiscretization
In this article we study holomorphic integrals on tropical plane curves in
view of ultradiscretization. We prove that the lattice integrals over tropical
curves can be obtained as a certain limit of complex integrals over Riemannian
surfaces.Comment: 32pages, 12figure
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