23,356 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 natureâ€”one of the most important desiderata for a theory of laws and propertiesâ€”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

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