192 research outputs found
Separability problem for multipartite states of rank at most four
One of the most important problems in quantum information is the separability
problem, which asks whether a given quantum state is separable. We investigate
multipartite states of rank at most four which are PPT (i.e., all their partial
transposes are positive semidefinite). We show that any PPT state of rank two
or three is separable and has length at most four. For separable states of rank
four, we show that they have length at most six. It is six only for some
qubit-qutrit or multiqubit states. It turns out that any PPT entangled state of
rank four is necessarily supported on a 3x3 or a 2x2x2 subsystem. We obtain a
very simple criterion for the separability problem of the PPT states of rank at
most four: such a state is entangled if and only if its range contains no
product vectors. This criterion can be easily applied since a four-dimensional
subspace in the 3x3 or 2x2x2 system contains a product vector if and only if
its Pluecker coordinates satisfy a homogeneous polynomial equation (the Chow
form of the corresponding Segre variety). We have computed an explicit
determinantal expression for the Chow form in the former case, while such
expression was already known in the latter case.Comment: 19 page
Two-Center Black Holes Duality-Invariants for stu Model and its lower-rank Descendants
We classify 2-center extremal black hole charge configurations through
duality-invariant homogeneous polynomials, which are the generalization of the
unique invariant quartic polynomial for single-center black holes based on
homogeneous symmetric cubic special Kaehler geometries. A crucial role is
played by an horizontal SL(p,R) symmetry group, which classifies invariants for
p-center black holes. For p = 2, a (spin 2) quintet of quartic invariants
emerge. We provide the minimal set of independent invariants for the rank-3 N =
2, d = 4 stu model, and for its lower-rank descendants, namely the rank-2 st^2
and rank-1 t^3 models; these models respectively exhibit seven, six and five
independent invariants. We also derive the polynomial relations among these and
other duality invariants. In particular, the symplectic product of two charge
vectors is not independent from the quartic quintet in the t^3 model, but
rather it satisfies a degree-16 relation, corresponding to a quartic equation
for the square of the symplectic product itself.Comment: 1+31 pages; v2: amendments in Sec. 9, App. C added, other minor
refinements, Refs. added; v3: Ref. added, typos fixed. To appear on
J.Math.Phy
Interpretations of Presburger Arithmetic in Itself
Presburger arithmetic PrA is the true theory of natural numbers with
addition. We study interpretations of PrA in itself. We prove that all
one-dimensional self-interpretations are definably isomorphic to the identity
self-interpretation. In order to prove the results we show that all linear
orders that are interpretable in (N,+) are scattered orders with the finite
Hausdorff rank and that the ranks are bounded in terms of the dimension of the
respective interpretations. From our result about self-interpretations of PrA
it follows that PrA isn't one-dimensionally interpretable in any of its finite
subtheories. We note that the latter was conjectured by A. Visser.Comment: Published in proceedings of LFCS 201
Tropical surface singularities
In this paper, we study tropicalisations of singular surfaces in toric
threefolds. We completely classify singular tropical surfaces of
maximal-dimensional type, show that they can generically have only finitely
many singular points, and describe all possible locations of singular points.
More precisely, we show that singular points must be either vertices, or
generalized midpoints and baricenters of certain faces of singular tropical
surfaces, and, in some cases, there may be additional metric restrictions to
faces of singular tropical surfaces.Comment: A gap in the classification was closed. 37 pages, 29 figure
The determination of integral closures and geometric applications
We express explicitly the integral closures of some ring extensions; this is
done for all Bring-Jerrard extensions of any degree as well as for all general
extensions of degree < 6; so far such an explicit expression is known only for
degree < 4 extensions. As a geometric application we present explicitly the
structure sheaf of every Bring-Jerrard covering space in terms of coefficients
of the equation defining the covering; in particular, we show that a degree-3
morphism f : Y --> X is quasi-etale if and only if the first Chern class of the
sheaf f_*(O_Y) is trivial (details in Theorem 5.3). We also try to get a
geometric Galoisness criterion for an arbitrary degree-n finite morphism; this
is successfully done when n = 3 and less satifactorily done when n = 5.Comment: Advances in Mathematics, to appear (no changes, just add this info
New and Old Results in Resultant Theory
Resultants are getting increasingly important in modern theoretical physics:
they appear whenever one deals with non-linear (polynomial) equations, with
non-quadratic forms or with non-Gaussian integrals. Being a subject of more
than three-hundred-year research, resultants are of course rather well studied:
a lot of explicit formulas, beautiful properties and intriguing relationships
are known in this field. We present a brief overview of these results,
including both recent and already classical. Emphasis is made on explicit
formulas for resultants, which could be practically useful in a future physics
research.Comment: 50 pages, 15 figure
Design and analysis of fractional factorial experiments from the viewpoint of computational algebraic statistics
We give an expository review of applications of computational algebraic
statistics to design and analysis of fractional factorial experiments based on
our recent works. For the purpose of design, the techniques of Gr\"obner bases
and indicator functions allow us to treat fractional factorial designs without
distinction between regular designs and non-regular designs. For the purpose of
analysis of data from fractional factorial designs, the techniques of Markov
bases allow us to handle discrete observations. Thus the approach of
computational algebraic statistics greatly enlarges the scope of fractional
factorial designs.Comment: 16 page
Parameter estimation for Boolean models of biological networks
Boolean networks have long been used as models of molecular networks and play
an increasingly important role in systems biology. This paper describes a
software package, Polynome, offered as a web service, that helps users
construct Boolean network models based on experimental data and biological
input. The key feature is a discrete analog of parameter estimation for
continuous models. With only experimental data as input, the software can be
used as a tool for reverse-engineering of Boolean network models from
experimental time course data.Comment: Web interface of the software is available at
http://polymath.vbi.vt.edu/polynome
Marginally stable circular orbits in stationary axisymmetric spacetimes
16 pages, 1 figure. Published in Phys. Rev. D 94, 024015 (2016)E. G. gratefully acknowledges support from Consejo Nacional de Ciencia y Tecnonología (CONACyT Scholarship No. 494039/218141). S. B. also thanks the London Mathematical Society for their support (Grace Chisholm Young Fellowship No. GCY 13-14 02)
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