90,406 research outputs found
On the linear structure of cones
For encompassing the limitations of probabilistic coherence spaces which do not seem to provide natural interpretations of continuous data types such as the real line, Ehrhard and al. introduced a model of probabilistic higher order computation based on (positive) cones, and a class of totally monotone functions that they called "stable". Then Crubillé proved that this model is a conservative extension of the earlier probabilistic coherence space model. We continue these investigations by showing that the category of cones and linear and Scott-continuous functions is a model of intuitionistic linear logic. To define the tensor product, we use the special adjoint functor theorem, and we prove that this operation is and extension of the standard tensor product of probabilistic coherence spaces. We also show that these latter are dense in cones, thus allowing to lift the main properties of the tensor product of probabilistic coherence spaces to general cones. Last we define in the same way an exponential of cones and extend measurability to these new operations
Spectrahedral cones generated by rank 1 matrices
Let be the cone of positive semi-definite
matrices as a subset of the vector space of real symmetric
matrices. The intersection of with a linear subspace of is called a spectrahedral cone. We consider spectrahedral cones such
that every element of can be represented as a sum of rank 1 matrices in
. We shall call such spectrahedral cones rank one generated (ROG). We show
that ROG cones which are linearly isomorphic as convex cones are also
isomorphic as linear sections of the positive semi-definite matrix cone, which
is not the case for general spectrahedral cones. We give many examples of ROG
cones and show how to construct new ROG cones from given ones by different
procedures. We provide classifications of some subclasses of ROG cones, in
particular, we classify all ROG cones for matrix sizes not exceeding 4. Further
we prove some results on the structure of ROG cones. We also briefly consider
the case of complex or quaternionic matrices. ROG cones are in close relation
with the exactness of semi-definite relaxations of quadratically constrained
quadratic optimization problems or of relaxations approximating the cone of
nonnegative functions in squared functional systems.Comment: Version 2: section on complex and quaternionic case added, many
sections completely rewritte
A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles
We use convex polyhedral cones to study a large class of multivariate
meromorphic germs, namely those with linear poles, which naturally arise in
various contexts in mathematics and physics. We express such a germ as a sum of
a holomorphic germ and a linear combination of special non-holomorphic germs
called polar germs. In analyzing the supporting cones -- cones that reflect the
pole structure of the polar germs -- we obtain a geometric criterion for the
non-holomorphicity of linear combinations of polar germs. This yields the
uniqueness of the above sum when required to be supported on a suitable family
of cones and assigns a Laurent expansion to the germ. Laurent expansions
provide various decompositions of such germs and thereby a uniformized proof of
known results on decompositions of rational fractions. These Laurent expansions
also yield new concepts on the space of such germs, all of which are
independent of the choice of the specific Laurent expansion. These include a
generalization of Jeffrey-Kirwan's residue, a filtered residue and a coproduct
in the space of such germs. When applied to exponential sums on rational convex
polyhedral cones, the filtered residue yields back exponential integrals.Comment: 30 page
The geometry of convex cones associated with the Lyapunov inequality and the common Lyapunov function problem
In this paper, the structure of several convex cones that arise in the study of Lyapunov functions is investigated. In particular, the cones associated with quadratic Lyapunov functions for both linear and non-linear systems are considered, as well as cones that arise in connection with
diagonal and linear copositive Lyapunov functions for positive linear systems. In each of these cases, some technical results are presented on the structure of individual cones and it is shown how these insights can lead to new results on the problem of common Lyapunov function existence
The geometry of convex cones associated with the Lyapunov inequality and the common Lyapunov function problem
In this paper, the structure of several convex cones that arise in the study of Lyapunov functions is investigated. In particular, the cones associated with quadratic Lyapunov functions for both linear and non-linear systems are considered, as well as cones that arise in connection with
diagonal and linear copositive Lyapunov functions for positive linear systems. In each of these cases, some technical results are presented on the structure of individual cones and it is shown how these insights can lead to new results on the problem of common Lyapunov function existence
Rigidity of irreducible Hermitian symmetric spaces of the compact type under K"ahler deformation
We study deformations of irreducible Hermitian symmetric spaces of the
compact type, known to be locally rigid, as projective-algberaic manifolds and
prove that no jump of complex structures can occur. For each of rank there is an associated reductive linear group such that admits a
holomorphic -structure, corresponding to a reduction of the structure group
of the tangent bundle. is characterized as the unique simply-connected
compact complex manifold admitting such a -structure which is at the same
time integrable. To prove the deformation rigidity of it suffices that the
corresponding integrable -structures converge.
We argue by contradiction using the deformation theory of rational curves.
Assuming that a jump of complex structures occurs, cones of vectors tangent to
degree-1 rational curves on the special fiber are linearly degenerate,
thus defining a proper meromorphic distribution on . We prove that
such cannot possibly exist. On the one hand, integrability of would
contradict the fact that . On the other hand, we prove that would
be automatically integrable by producing families of integral complex surfaces
of as pencils of degree-1 rational curves. For the verification that there
are enough integral surfaces we need a description of generic cones on the
special fiber. We show that they are in fact images of standard cones under
linear projections. We achieve this by studying deformations of normalizations
of Chow spaces of minimal rational curves marked at a point, which are
themselves Hermitian symmetric, irreducible except in the case of
Grassmannians
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