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 ${\cal S}_+^n \subset {\cal S}^n$ be the cone of positive semi-definite matrices as a subset of the vector space of real symmetric $n \times n$ matrices. The intersection of ${\cal S}_+^n$ with a linear subspace of ${\cal S}^n$ is called a spectrahedral cone. We consider spectrahedral cones $K$ such that every element of $K$ can be represented as a sum of rank 1 matrices in $K$. 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 $S$ 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 $S$ of rank $\ge 2$ there is an associated reductive linear group $G$ such that $S$ admits a holomorphic $G$-structure, corresponding to a reduction of the structure group of the tangent bundle. $S$ is characterized as the unique simply-connected compact complex manifold admitting such a $G$-structure which is at the same time integrable. To prove the deformation rigidity of $S$ it suffices that the corresponding integrable $G$-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 $X_0$ are linearly degenerate, thus defining a proper meromorphic distribution $W$ on $X_0$. We prove that such $W$ cannot possibly exist. On the one hand, integrability of $W$ would contradict the fact that $b_2(X)=1$. On the other hand, we prove that $W$ would be automatically integrable by producing families of integral complex surfaces of $W$ 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