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)

Piezoelectric vibration damping using resonant shunt circuits: an exact solution

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