377 research outputs found
The cobordism class of the multiple points of immersions
Let f: M -> N be an even codimensional immersion between smooth manifolds. We
derive an explicit formula for the Pontrjagin numbers and signature of the
multiple point manifolds in terms of singular cohomology of M and N, the maps
induced between these by f and the characteristic classes of the normal bundle.
The main trick is to solve a recursion in cohomology by the use of
(generalized) formal power series.Comment: 17 pages; LaTeX2e document, use of eLaTeX recommended, non-standard
packages: xy, pxfonts, pbox. Accepted by Algebraic Geometry and Topology.
Corrections and clarifications mainly due to the referee of the journa
A polyhedral approach to computing border bases
Border bases can be considered to be the natural extension of Gr\"obner bases
that have several advantages. Unfortunately, to date the classical border basis
algorithm relies on (degree-compatible) term orderings and implicitly on
reduced Gr\"obner bases. We adapt the classical border basis algorithm to allow
for calculating border bases for arbitrary degree-compatible order ideals,
which is \emph{independent} from term orderings. Moreover, the algorithm also
supports calculating degree-compatible order ideals with \emph{preference} on
contained elements, even though finding a preferred order ideal is NP-hard.
Effectively we retain degree-compatibility only to successively extend our
computation degree-by-degree. The adaptation is based on our polyhedral
characterization: order ideals that support a border basis correspond
one-to-one to integral points of the order ideal polytope. This establishes a
crucial connection between the ideal and the combinatorial structure of the
associated factor spaces
The matching polytope does not admit fully-polynomial size relaxation schemes
The groundbreaking work of Rothvo{\ss} [arxiv:1311.2369] established that
every linear program expressing the matching polytope has an exponential number
of inequalities (formally, the matching polytope has exponential extension
complexity). We generalize this result by deriving strong bounds on the
polyhedral inapproximability of the matching polytope: for fixed , every polyhedral -approximation
requires an exponential number of inequalities, where is the number of
vertices. This is sharp given the well-known -approximation of size
provided by the odd-sets of size up to
. Thus matching is the first problem in , whose natural
linear encoding does not admit a fully polynomial-size relaxation scheme (the
polyhedral equivalent of an FPTAS), which provides a sharp separation from the
polynomial-size relaxation scheme obtained e.g., via constant-sized odd-sets
mentioned above.
Our approach reuses ideas from Rothvo{\ss} [arxiv:1311.2369], however the
main lower bounding technique is different. While the original proof is based
on the hyperplane separation bound (also called the rectangle corruption
bound), we employ the information-theoretic notion of common information as
introduced in Braun and Pokutta [http://eccc.hpi-web.de/report/2013/056/],
which allows to analyze perturbations of slack matrices. It turns out that the
high extension complexity for the matching polytope stem from the same source
of hardness as for the correlation polytope: a direct sum structure.Comment: 21 pages, 3 figure
Rigid abelian groups and the probabilistic method
The construction of torsion-free abelian groups with prescribed endomorphism
rings starting with Corner's seminal work is a well-studied subject in the
theory of abelian groups. Usually these construction work by adding elements
from a (topological) completion in order to get rid of (kill) unwanted
homomorphisms. The critical part is to actually prove that every unwanted
homomorphism can be killed by adding a suitable element. We will demonstrate
that some of those constructions can be significantly simplified by choosing
the elements at random. As a result, the endomorphism ring will be almost
surely prescribed, i.e., with probability one.Comment: 12 pages, submitted to the special volume of Contemporary Mathematics
for the proceedings of the conference Group and Model Theory, 201
Lower Bounds on the Oracle Complexity of Nonsmooth Convex Optimization via Information Theory
We present an information-theoretic approach to lower bound the oracle
complexity of nonsmooth black box convex optimization, unifying previous lower
bounding techniques by identifying a combinatorial problem, namely string
guessing, as a single source of hardness. As a measure of complexity we use
distributional oracle complexity, which subsumes randomized oracle complexity
as well as worst-case oracle complexity. We obtain strong lower bounds on
distributional oracle complexity for the box , as well as for the
-ball for (for both low-scale and large-scale regimes),
matching worst-case upper bounds, and hence we close the gap between
distributional complexity, and in particular, randomized complexity, and
worst-case complexity. Furthermore, the bounds remain essentially the same for
high-probability and bounded-error oracle complexity, and even for combination
of the two, i.e., bounded-error high-probability oracle complexity. This
considerably extends the applicability of known bounds
- …