276 research outputs found
Detecting multipartite entanglement
We discuss the problem of determining whether the state of several quantum
mechanical subsystems is entangled. As in previous work on two subsystems we
introduce a procedure for checking separability that is based on finding state
extensions with appropriate properties and may be implemented as a semidefinite
program. The main result of this work is to show that there is a series of
tests of this kind such that if a multiparty state is entangled this will
eventually be detected by one of the tests. The procedure also provides a means
of constructing entanglement witnesses that could in principle be measured in
order to demonstrate that the state is entangled.Comment: 9 pages, REVTE
The Łojasiewicz exponent over a field of arbitrary characteristic
Let K be an algebraically closed field and let K((XQ)) denote the field
of generalized series with coefficients in K. We propose definitions of the local
Łojasiewicz exponent of F = ( f1, . . . , fm) ∈ K[[X, Y ]]m as well as of the
Łojasiewicz exponent at infinity of F = ( f1, . . . , fm) ∈ K[X, Y ]m, which generalize
the familiar case of K = C and F ∈ C{X, Y }m (resp. F ∈ C[X, Y ]m), see
Cha˛dzy´nski and Krasi´nski (In: Singularities, 1988; In: Singularities, 1988; Ann Polon
Math 67(3):297–301, 1997; Ann Polon Math 67(2):191–197, 1997), and prove some
basic properties of such numbers. Namely, we show that in both cases the exponent
is attained on a parametrization of a component of F (Theorems 6 and 7), thus being
a rational number. To this end, we define the notion of the Łojasiewicz pseudoexponent
of F ∈ (K((XQ))[Y ])m for which we give a description of all the generalized
series that extract the pseudoexponent, in terms of their jets. In particular, we show
that there exist only finitely many jets of generalized series giving the pseudoexponent
of F (Theorem 5). The main tool in the proofs is the algebraic version of Newton’s
Polygon Method. The results are illustrated with some explicit examples
Tameness of holomorphic closure dimension in a semialgebraic set
Given a semianalytic set S in a complex space and a point p in S, there is a
unique smallest complex-analytic germ at p which contains the germ of S, called
the holomorphic closure of S at p. We show that if S is semialgebraic then its
holomorphic closure is a Nash germ, for every p, and S admits a semialgebraic
filtration by the holomorphic closure dimension. As a consequence, every
semialgebraic subset of a complex vector space admits a semialgebraic
stratification into CR manifolds satisfying a strong version of the condition
of the frontier.Comment: Published versio
Monotone functions and maps
In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2
(2011)] we defined semi-monotone sets, as open bounded sets, definable in an
o-minimal structure over the reals, and having connected intersections with all
translated coordinate cones in R^n. In this paper we develop this theory
further by defining monotone functions and maps, and studying their fundamental
geometric properties. We prove several equivalent conditions for a bounded
continuous definable function or map to be monotone. We show that the class of
graphs of monotone maps is closed under intersections with affine coordinate
subspaces and projections to coordinate subspaces. We prove that the graph of a
monotone map is a topologically regular cell. These results generalize and
expand the corresponding results obtained in Basu et al. for semi-monotone
sets.Comment: 30 pages. Version 2 appeared in RACSAM. In version 3 Corollaries 1
and 2 were corrected. In version 4 Theorem 3 is correcte
Generic Global Rigidity in Complex and Pseudo-Euclidean Spaces
In this paper we study the property of generic global rigidity for frameworks of graphs embedded in d-dimensional complex space and in a d-dimensional pseudo-Euclidean space R with a metric of indefinite signature). We show that a graph is generically globally rigid in Euclidean space iff it is generically globally rigid in a complex or pseudo-Euclidean space. We also establish that global rigidity is always a generic property of a graph in complex space, and give a sufficient condition for it to be a generic property in a pseudo-Euclidean space. Extensions to hyperbolic space are also discussed.Engineering and Applied Science
Uniqueness of Bessel models: the archimedean case
In the archimedean case, we prove uniqueness of Bessel models for general
linear groups, unitary groups and orthogonal groups.Comment: 22 page
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