62,759 research outputs found
Convexity properties of the condition number II
In our previous paper [SIMAX 31 n.3 1491-1506(2010)], we studied the
condition metric in the space of maximal rank matrices. Here, we show that this
condition metric induces a Lipschitz-Riemann structure on that space. After
investigating geodesics in such a nonsmooth structure, we show that the inverse
of the smallest singular value of a matrix is a log-convex function along
geodesics (Theorem 1).
We also show that a similar result holds for the solution variety of linear
systems (Theorem 31).
Some of our intermediate results, such as Theorem 12, on the second covariant
derivative or Hessian of a function with symmetries on a manifold, and Theorem
29 on piecewise self-convex functions, are of independent interest.
Those results were motivated by our investigations on the com- plexity of
path-following algorithms for solving polynomial systems.Comment: Revised versio
Classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems
Necessary and sufficient conditions for convexity and strong convexity,
respectively, of sublevel sets that are defined by finitely many real-valued
-maps are presented. A novel characterization of strongly convex sets
in terms of the so-called local quadratic support is proved. The results
concerning strong convexity are used to derive sufficient conditions for
attainable sets of continuous-time nonlinear systems to be strongly convex. An
application of these conditions is a novel method to over-approximate
attainable sets when strong convexity is present.Comment: 20 pages, 3 figure
Symbol Error Rates of Maximum-Likelihood Detector: Convex/Concave Behavior and Applications
Convexity/concavity properties of symbol error rates (SER) of the maximum
likelihood detector operating in the AWGN channel (non-fading and fading) are
studied. Generic conditions are identified under which the SER is a
convex/concave function of the SNR. Universal bounds for the SER 1st and 2nd
derivatives are obtained, which hold for arbitrary constellations and are tight
for some of them. Applications of the results are discussed, which include
optimum power allocation in spatial multiplexing systems, optimum power/time
sharing to decrease or increase (jamming problem) error rate, and implication
for fading channels.Comment: To appear in 2007 IEEE International Symposium on Information Theory
(ISIT 2007), Nice, June 200
A combinatorial non-positive curvature I: weak systolicity
We introduce the notion of weakly systolic complexes and groups, and initiate
regular studies of them. Those are simplicial complexes with
nonpositive-curvature-like properties and groups acting on them geometrically.
We characterize weakly systolic complexes as simply connected simplicial
complexes satisfying some local combinatorial conditions. We provide several
classes of examples --- in particular systolic groups and CAT(-1) cubical
groups are weakly systolic. We present applications of the theory, concerning
Gromov hyperbolic groups, Coxeter groups and systolic groups.Comment: 35 pages, 1 figur
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