23,551 research outputs found
PPT from spectra
In this contribution we solve the following problem. Let H_{nm} be a Hilbert
space of dimension nm, and let A be a positive semidefinite self-adjoint linear
operator on H_{nm}. Under which conditions on the spectrum has A a positive
partial transpose (is PPT) with respect to any partition H_n \otimes H_m of the
space H_{nm} as a tensor product of an n-dimensional and an m-dimensional
Hilbert space? We show that the necessary and sufficient conditions can be
expressed as a set of linear matrix inequalities on the eigenvalues of A.Comment: 6 pages, no figure
Platonic Laws of Nature
David Armstrong accepted the following three theses: universals are immanent, laws are relations between universals, and laws govern. Taken together, they form an attractive position, for they promise to explain regularities in nature—one of the most important desiderata for a theory of laws and properties—while remaining compatible with naturalism. However, I argue that the three theses are incompatible. The basic idea is that each thesis makes an explanatory claim, but the three claims can be shown to run in a problematic circle. I then consider which thesis we ought to reject and suggest some general lessons for the metaphysics of laws
Minimal zeros of copositive matrices
Let be an element of the copositive cone . A zero of
is a nonzero nonnegative vector such that . The support of is
the index set \mbox{supp}u \subset \{1,\dots,n\} corresponding to the
positive entries of . A zero of is called minimal if there does not
exist another zero of such that its support \mbox{supp}v is a strict
subset of \mbox{supp}u. We investigate the properties of minimal zeros of
copositive matrices and their supports. Special attention is devoted to
copositive matrices which are irreducible with respect to the cone of
positive semi-definite matrices, i.e., matrices which cannot be written as a
sum of a copositive and a nonzero positive semi-definite matrix. We give a
necessary and sufficient condition for irreducibility of a matrix with
respect to in terms of its minimal zeros. A similar condition is given
for the irreducibility with respect to the cone of entry-wise
nonnegative matrices. For matrices which are irreducible with respect
to both and are extremal. For a list of candidate
combinations of supports of minimal zeros which an exceptional extremal matrix
can have is provided.Comment: Some conditions and proofs simplifie
Analytic formulas for complete hyperbolic affine spheres
We classify all regular three-dimensional convex cones which possess an
automorphism group of dimension at least two, and provide analytic expressions
for the complete hyperbolic affine spheres which are asymptotic to the
boundaries of these cones. The affine spheres are represented by explicit
hypersurface immersions into three-dimensional real space. The generic member
of the family of immersions is given by elliptic integrals.Comment: 16 page
Graph immersions with parallel cubic form
We consider non-degenerate graph immersions into affine space whose cubic form is parallel with respect to the Levi-Civita
connection of the affine metric. There exists a correspondence between such
graph immersions and pairs , where is an -dimensional real
Jordan algebra and is a non-degenerate trace form on . Every graph
immersion with parallel cubic form can be extended to an affine complete
symmetric space covering the maximal connected component of zero in the set of
quasi-regular elements in the algebra . It is an improper affine hypersphere
if and only if the corresponding Jordan algebra is nilpotent. In this case it
is an affine complete, Euclidean complete graph immersion, with a polynomial as
globally defining function. We classify all such hyperspheres up to dimension
5. As a special case we describe a connection between Cayley hypersurfaces and
polynomial quotient algebras. Our algebraic approach can be used to study also
other classes of hypersurfaces with parallel cubic form.Comment: some proofs have been simplified with respect to the first versio
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