3,959 research outputs found
The Hyperdeterminant and Triangulations of the 4-Cube
The hyperdeterminant of format 2 x 2 x 2 x 2 is a polynomial of degree 24 in
16 unknowns which has 2894276 terms. We compute the Newton polytope of this
polynomial and the secondary polytope of the 4-cube. The 87959448 regular
triangulations of the 4-cube are classified into 25448 D-equivalence classes,
one for each vertex of the Newton polytope. The 4-cube has 80876 coarsest
regular subdivisions, one for each facet of the secondary polytope, but only
268 of them come from the hyperdeterminant.Comment: 30 pages, 6 figures; An author's name changed, typos fixe
The polytopal structure of the tight-span of a totally split-decomposable metric
The tight-span of a finite metric space is a polytopal complex that has appeared in several areas of mathematics. In this paper we determine the polytopal structure of the tight-span of a totally split-decomposable (finite) metric. These metrics are a generalization of tree-metrics and have importance within phylogenetics. In previous work, we showed that the cells of the tight-span of such a metric are zonotopes that are polytope isomorphic to either hypercubes or rhombic dodecahedra. Here, we extend these results and show that the tight-span of a totally split-decomposable metric can be broken up into a canonical collection of polytopal complexes whose polytopal structures can be directly determined from the metric. This allows us to also completely determine the polytopal structure of the tight-span of a totally split-decomposable metric. We anticipate that our improved understanding of this structure may lead to improved techniques for phylogenetic inference
Transmission and Spectral Aspects of Tight Binding Hamiltonians for the Counting Quantum Turing Machine
It was recently shown that a generalization of quantum Turing machines
(QTMs), in which potentials are associated with elementary steps or transitions
of the computation, generates potential distributions along computation paths
of states in some basis B. The distributions are computable and are thus
periodic or have deterministic disorder. These generalized machines (GQTMs) can
be used to investigate the effect of potentials in causing reflections and
reducing the completion probability of computations. This work is extended here
by determination of the spectral and transmission properties of an example GQTM
which enumerates the integers as binary strings. A potential is associated with
just one type of step. For many computation paths the potential distributions
are initial segments of a quasiperiodic distribution that corresponds to a
substitution sequence. The energy band spectra and Landauer Resistance (LR) are
calculated for energies below the barrier height by use of transfer matrices.
The LR fluctuates rapidly with momentum with minima close to or at band-gap
edges. For several values of the parameters, there is good transmission over
some momentum regions.Comment: 22 pages Latex, 13 postscript figures, Submitted to Phys. Rev.
- …