6,068 research outputs found
On the Geometric Interpretation of the Nonnegative Rank
The nonnegative rank of a nonnegative matrix is the minimum number of
nonnegative rank-one factors needed to reconstruct it exactly. The problem of
determining this rank and computing the corresponding nonnegative factors is
difficult; however it has many potential applications, e.g., in data mining,
graph theory and computational geometry. In particular, it can be used to
characterize the minimal size of any extended reformulation of a given
combinatorial optimization program. In this paper, we introduce and study a
related quantity, called the restricted nonnegative rank. We show that
computing this quantity is equivalent to a problem in polyhedral combinatorics,
and fully characterize its computational complexity. This in turn sheds new
light on the nonnegative rank problem, and in particular allows us to provide
new improved lower bounds based on its geometric interpretation. We apply these
results to slack matrices and linear Euclidean distance matrices and obtain
counter-examples to two conjectures of Beasly and Laffey, namely we show that
the nonnegative rank of linear Euclidean distance matrices is not necessarily
equal to their dimension, and that the rank of a matrix is not always greater
than the nonnegative rank of its square
Approximation of the Euclidean ball by polytopes with a restricted number of facets
We prove that there is an absolute constant such that for every and there exists a polytope with at most facets that satisfies
and
where is the -dimensional Euclidean unit ball.
This result closes gaps from several papers of Hoehner, Ludwig, Sch\"utt and
Werner. The upper bounds are optimal up to absolute constants. This result
shows that a polytope with an exponential number of facets (in the dimension)
can approximate the -dimensional Euclidean ball with respect to the
aforementioned distances
Stochastic model for the 3D microstructure of pristine and cyclically aged cathodes in Li-ion batteries
It is well-known that the microstructure of electrodes in lithium-ion
batteries strongly affects their performance. Vice versa, the microstructure
can exhibit strong changes during the usage of the battery due to aging
effects. For a better understanding of these effects, mathematical analysis and
modeling has turned out to be of great help. In particular, stochastic 3D
microstructure models have proven to be a powerful and very flexible tool to
generate various kinds of particle-based structures. Recently, such models have
been proposed for the microstructure of anodes in lithium-ion energy and power
cells. In the present paper, we describe a stochastic modeling approach for the
3D microstructure of cathodes in a lithium-ion energy cell, which differs
significantly from the one observed in anodes. The model for the cathode data
enhances the ideas of the anode models, which have been developed so far. It is
calibrated using 3D tomographic image data from pristine as well as two aged
cathodes. A validation based on morphological image characteristics shows that
the model is able to realistically describe both, the microstructure of
pristine and aged cathodes. Thus, we conclude that the model is suitable to
generate virtual, but realistic microstructures of lithium-ion cathodes
Elliptic and K-theoretic stable envelopes and Newton polytopes
In this paper we consider the cotangent bundles of partial flag varieties. We
construct the -theoretic stable envelopes for them and also define a version
of the elliptic stable envelopes. We expect that our elliptic stable envelopes
coincide with the elliptic stable envelopes defined by M. Aganagic and A.
Okounkov. We give formulas for the -theoretic stable envelopes and our
elliptic stable envelopes. We show that the -theoretic stable envelopes are
suitable limits of our elliptic stable envelopes. That phenomenon was predicted
by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms
of the elliptic and trigonometric weight functions which originally appeared in
the theory of integral representations of solutions of qKZ equations twenty
years ago. (More precisely, the elliptic weight functions had appeared earlier
only for the case.) We prove new properties of the trigonometric
weight functions. Namely, we consider certain evaluations of the trigonometric
weight functions, which are multivariable Laurent polynomials, and show that
the Newton polytopes of the evaluations are embedded in the Newton polytopes of
the corresponding diagonal evaluations. That property implies the fact that the
trigonometric weight functions project to the -theoretic stable envelopes.Comment: Latex, 37 pages; v.2: Appendix and Figure 1 added; v.3: missing shift
in Theorem 2.9 added and a proof of Theorem 2.9 adde
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