372,697 research outputs found
The -problem for Gabor systems
A Gabor system generated by a window function and a rectangular
lattice is given by One of
fundamental problems in Gabor analysis is to identify window functions
and time-frequency shift lattices such that the corresponding
Gabor system is a Gabor frame for
, the space of all square-integrable functions on the real line .
In this paper, we provide a full classification of triples for which
the Gabor system generated by the ideal
window function on an interval of length is a Gabor frame for
. For the classification of such triples (i.e., the
-problem for Gabor systems), we introduce maximal invariant sets of some
piecewise linear transformations and establish the equivalence between Gabor
frame property and triviality of maximal invariant sets. We then study dynamic
system associated with the piecewise linear transformations and explore various
properties of their maximal invariant sets. By performing holes-removal surgery
for maximal invariant sets to shrink and augmentation operation for a line with
marks to expand, we finally parameterize those triples for which
maximal invariant sets are trivial. The novel techniques involving
non-ergodicity of dynamical systems associated with some novel non-contractive
and non-measure-preserving transformations lead to our arduous answer to the
-problem for Gabor systems
Ball and Spindle Convexity with respect to a Convex Body
Let be a convex body. We introduce two notions of
convexity associated to C. A set is -ball convex if it is the
intersection of translates of , or it is either , or . The -ball convex hull of two points is called a -spindle. is
-spindle convex if it contains the -spindle of any pair of its points. We
investigate how some fundamental properties of conventional convex sets can be
adapted to -spindle convex and -ball convex sets. We study separation
properties and Carath\'eodory numbers of these two convexity structures. We
investigate the basic properties of arc-distance, a quantity defined by a
centrally symmetric planar disc , which is the length of an arc of a
translate of , measured in the -norm, that connects two points. Then we
characterize those -dimensional convex bodies for which every -ball
convex set is the -ball convex hull of finitely many points. Finally, we
obtain a stability result concerning covering numbers of some -ball convex
sets, and diametrically maximal sets in -dimensional Minkowski spaces.Comment: 27 pages, 5 figure
Two ideals connected with strong right upper porosity at a point
Let be the set of upper strongly porous at subsets of and let be the intersection of maximal ideals . Some characteristic properties of sets are obtained. It
is shown that the ideal generated by the so-called completely strongly porous
at subsets of is a proper subideal of Comment: 18 page
Maximum Resilience of Artificial Neural Networks
The deployment of Artificial Neural Networks (ANNs) in safety-critical
applications poses a number of new verification and certification challenges.
In particular, for ANN-enabled self-driving vehicles it is important to
establish properties about the resilience of ANNs to noisy or even maliciously
manipulated sensory input. We are addressing these challenges by defining
resilience properties of ANN-based classifiers as the maximal amount of input
or sensor perturbation which is still tolerated. This problem of computing
maximal perturbation bounds for ANNs is then reduced to solving mixed integer
optimization problems (MIP). A number of MIP encoding heuristics are developed
for drastically reducing MIP-solver runtimes, and using parallelization of
MIP-solvers results in an almost linear speed-up in the number (up to a certain
limit) of computing cores in our experiments. We demonstrate the effectiveness
and scalability of our approach by means of computing maximal resilience bounds
for a number of ANN benchmark sets ranging from typical image recognition
scenarios to the autonomous maneuvering of robots.Comment: Timestamp research work conducted in the project. version 2: fix some
typos, rephrase the definition, and add some more existing wor
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