37 research outputs found
Generalized Bell Inequality Experiments and Computation
We consider general settings of Bell inequality experiments with many
parties, where each party chooses from a finite number of measurement settings
each with a finite number of outcomes. We investigate the constraints that Bell
inequalities place upon the correlations possible in a local hidden variable
theories using a geometrical picture of correlations. We show that local hidden
variable theories can be characterized in terms of limited computational
expressiveness, which allows us to characterize families of Bell inequalities.
The limited computational expressiveness for many settings (each with many
outcomes) generalizes previous results about the many-party situation each with
a choice of two possible measurements (each with two outcomes). Using this
computational picture we present generalizations of the Popescu-Rohrlich
non-local box for many parties and non-binary inputs and outputs at each site.
Finally, we comment on the effect of pre-processing on measurement data in our
generalized setting and show that it becomes problematic outside of the binary
setting, in that it allows local hidden variable theories to simulate maximally
non-local correlations such as those of these generalised Popescu-Rohrlich
non-local boxes.Comment: 16 pages, 2 figures, supplemental material available upon request.
Typos corrected and references adde
Minimal half-spaces and external representation of tropical polyhedra
We give a characterization of the minimal tropical half-spaces containing a
given tropical polyhedron, from which we derive a counter example showing that
the number of such minimal half-spaces can be infinite, contradicting some
statements which appeared in the tropical literature, and disproving a
conjecture of F. Block and J. Yu. We also establish an analogue of the
Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently
represented internally (in terms of extreme points and rays) or externally (in
terms of half-spaces containing it). A canonical external representation of a
polyhedron turns out to be provided by the extreme elements of its tropical
polar. We characterize these extreme elements, showing in particular that they
are determined by support vectors.Comment: 19 pages, 4 figures, example added with a new figure, figures
improved, references update
Hamiltonian submanifolds of regular polytopes
We investigate polyhedral -manifolds as subcomplexes of the boundary
complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian}
if it contains the full -skeleton of the polytope. Since the case of the
cube is well known and since the case of a simplex was also previously studied
(these are so-called {\it super-neighborly triangulations}) we focus on the
case of the cross polytope and the sporadic regular 4-polytopes. By our results
the existence of 1-Hamiltonian surfaces is now decided for all regular
polytopes.
Furthermore we investigate 2-Hamiltonian 4-manifolds in the -dimensional
cross polytope. These are the "regular cases" satisfying equality in Sparla's
inequality. In particular, we present a new example with 16 vertices which is
highly symmetric with an automorphism group of order 128. Topologically it is
homeomorphic to a connected sum of 7 copies of . By this
example all regular cases of vertices with or, equivalently, all
cases of regular -polytopes with are now decided.Comment: 26 pages, 4 figure
Computing the vertices of tropical polyhedra using directed hypergraphs
We establish a characterization of the vertices of a tropical polyhedron
defined as the intersection of finitely many half-spaces. We show that a point
is a vertex if, and only if, a directed hypergraph, constructed from the
subdifferentials of the active constraints at this point, admits a unique
strongly connected component that is maximal with respect to the reachability
relation (all the other strongly connected components have access to it). This
property can be checked in almost linear-time. This allows us to develop a
tropical analogue of the classical double description method, which computes a
minimal internal representation (in terms of vertices) of a polyhedron defined
externally (by half-spaces or hyperplanes). We provide theoretical worst case
complexity bounds and report extensive experimental tests performed using the
library TPLib, showing that this method outperforms the other existing
approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section
5 (using directed hypergraphs), detailed appendix; v3: major revision of the
article (adding tropical hyperplanes, alternative method by arrangements,
etc); v4: minor revisio
Few smooth d-polytopes with n lattice points
We prove that, for fixed n there exist only finitely many embeddings of
Q-factorial toric varieties X into P^n that are induced by a complete linear
system. The proof is based on a combinatorial result that for fixed nonnegative
integers d and n, there are only finitely many smooth d-polytopes with n
lattice points. We also enumerate all smooth 3-polytopes with at most 12
lattice points. In fact, it is sufficient to bound the singularities and the
number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result
Flanker performance in female college students with ADHD: a diffusion model analysis
Attention-deficit hyperactivity disorder (ADHD) is characterized by poor adaptation to environmental demands, which leads to various everyday life problems. The present study had four aims: (1) to compare performance in a flanker task in female college students with and without ADHD (N = 39) in a classical analyses of reaction time and error rate and studying the underlying processes using a diffusion model, (2) to compare the amount of focused attention, (3) to explore the adaptation of focused attention, and (4) to relate adaptation to psychological functioning. The study followed a 2-between (group: ADHD vs. control) × 2-within (flanker conflict: incongruent vs. congruent) × 2-within (conflict frequency: 20 vs. 80 %) design. Compared to a control group, the ADHD group displayed prolonged response times accompanied by fewer errors in a flanker task. Results from the diffusion model analyses revealed that the members of the ADHD group showed deficits in non-decisional processes (i.e., higher non-decision time) and leaned more toward accuracy than participants without ADHD (i.e., setting higher boundaries). The ADHD group showed a more focused attention and less adaptation to the task conditions which is related to psychological functioning. Deficient non-decisional processes and poor adaptation are in line with theories of ADHD and presumably typical for the ADHD population, although this has not been shown using a diffusion model. However, we assume that the cautious strategy of trading speed of for accuracy is specific to the subgroup of female college students with ADHD and might be interpreted as a compensation mechanism