825 research outputs found
Embedding Stacked Polytopes on a Polynomial-Size Grid
A stacking operation adds a -simplex on top of a facet of a simplicial
-polytope while maintaining the convexity of the polytope. A stacked
-polytope is a polytope that is obtained from a -simplex and a series of
stacking operations. We show that for a fixed every stacked -polytope
with vertices can be realized with nonnegative integer coordinates. The
coordinates are bounded by , except for one axis, where the
coordinates are bounded by . The described realization can be
computed with an easy algorithm.
The realization of the polytopes is obtained with a lifting technique which
produces an embedding on a large grid. We establish a rounding scheme that
places the vertices on a sparser grid, while maintaining the convexity of the
embedding.Comment: 22 pages, 10 Figure
Compatibility of quantum measurements and inclusion constants for the matrix jewel
In this work, we establish the connection between the study of free
spectrahedra and the compatibility of quantum measurements with an arbitrary
number of outcomes. This generalizes previous results by the authors for
measurements with two outcomes. Free spectrahedra arise from matricial
relaxations of linear matrix inequalities. A particular free spectrahedron
which we define in this work is the matrix jewel. We find that the
compatibility of arbitrary measurements corresponds to the inclusion of the
matrix jewel into a free spectrahedron defined by the effect operators of the
measurements under study. We subsequently use this connection to bound the set
of (asymmetric) inclusion constants for the matrix jewel using results from
quantum information theory and symmetrization. The latter translate to new
lower bounds on the compatibility of quantum measurements. Among the techniques
we employ are approximate quantum cloning and mutually unbiased bases.Comment: v5: section 3.3 has been expanded significantly to incorporate the
generalization of the Cartesian product and the direct sum to matrix convex
sets. Many other minor modifications. Closed to the published versio
Computing the bounded subcomplex of an unbounded polyhedron
We study efficient combinatorial algorithms to produce the Hasse diagram of
the poset of bounded faces of an unbounded polyhedron, given vertex-facet
incidences. We also discuss the special case of simple polyhedra and present
computational results.Comment: 16 page
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
Threshold functions and Poisson convergence for systems of equations in random sets
We present a unified framework to study threshold functions for the existence
of solutions to linear systems of equations in random sets which includes
arithmetic progressions, sum-free sets, -sets and Hilbert cubes. In
particular, we show that there exists a threshold function for the property
" contains a non-trivial solution of
", where is a random set and each of
its elements is chosen independently with the same probability from the
interval of integers . Our study contains a formal definition of
trivial solutions for any combinatorial structure, extending a previous
definition by Ruzsa when dealing with a single equation.
Furthermore, we study the behaviour of the distribution of the number of
non-trivial solutions at the threshold scale. We show that it converges to a
Poisson distribution whose parameter depends on the volumes of certain convex
polytopes arising from the linear system under study as well as the symmetry
inherent in the structures, which we formally define and characterize.Comment: New version with minor corrections and changes in notation. 24 Page
Virtual polytopes
Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes
represent a natural algebraic generalization of convex polytopes. Introduced as the Grothendick group associated to the semigroup of convex
polytopes, they admit a variety of geometrizations. A selection of applications demonstrates their versatility
Virtual Polytopes
Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as elements of the Grothendieck group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. The present survey connects the theory of virtual polytopes with other geometrical subjects, describes a series of geometrizations together with relations between them, and gives a selection of applications
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