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 $A$ be an element of the copositive cone ${\cal C}_n$. A zero $u$ of $A$ is a nonzero nonnegative vector such that $u^TAu = 0$. The support of $u$ is the index set \mbox{supp}u \subset \{1,\dots,n\} corresponding to the positive entries of $u$. A zero $u$ of $A$ is called minimal if there does not exist another zero $v$ of $A$ 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 $S_+(n)$ 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 $A$ with respect to $S_+(n)$ in terms of its minimal zeros. A similar condition is given for the irreducibility with respect to the cone ${\cal N}_n$ of entry-wise nonnegative matrices. For $n = 5$ matrices which are irreducible with respect to both $S_+(5)$ and ${\cal N}_5$ are extremal. For $n = 6$ 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

Centro-affine hypersurface immersions with parallel cubic form

We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, and K\"ochers conic omega-domains, which are the maximal connected sets consisting of invertible elements in a real semi-simple Jordan algebra. Every level surface of the omega function in an omega-domain is an affine complete, Euclidean complete proper affine hypersphere with parallel cubic form and with center in the origin. On the other hand, every proper affine hypersphere with parallel cubic form and with center in the origin can be represented as such a level surface. We provide a complete classification of proper affine hyperspheres with parallel cubic form based on the classification of semi-simple real Jordan algebras. Centro-affine hypersurface immersions with parallel cubic form are related to the wider class of real unital Jordan algebras. Every such immersion can be extended to an affine complete one, whose conic hull is the connected component of the unit element in the set of invertible elements in a real unital Jordan algebra. Our approach can be used to study also other classes of hypersurfaces with parallel cubic form.Comment: Fourth version, 35 pages. A missing case has been added to the classificatio