We prove as for the real case that a martingale with values in a separabale real Hilbert space is extremal if and only if it satisfies the predictable representation property.

1 Lecture 1: Model Theory seca The basic idea behind model theory is to associate a Hilbert space construction with a function, and then use Hilbert space theory to illuminate the function theory. In one variable, one approach is to study the de Branges-Rovnyak space associated with a function φ in the ball of H(D). This is the Hilbert space of analytic functions on the disk D with reproducing ...

In the first part of the paper, we develop a general method for converting derivability problems, from a broad range of deductive systems, into the derivability problem in a quite specific system, namely the Datalog fragment of universal Horn logic. In this generality, the derivability problems may not be recursively solvable, let alone feasible; in particular, we may get Datalog “programs” wit...

A cut ideal of a graph records the relations among the cuts of the graph. These toric ideals have been introduced by Sturmfels and Sullivant who also posed the problem of relating their properties to the combinatorial structure of the graph. We study the cut ideals of the family of ring graphs, which includes trees and cycles. We show that they have quadratic Gröbner bases and that their coordi...

We show that the Wigner-Yanase-Dyson-Lieb concavity is a general property of an interpolation theory which works between pairs of (hilbertian) seminorms. As an application, the theory extends the relevant work of Lieb and Araki to positive linear forms of arbitrary *-algebras. In this context a "relative entropy" is defined for every pair of positive linear forms of a *-algebra with identity. F...

Propositional primal logic, as defined by Gurevich and Neeman, has two kinds of quotations: p said φ, and p implied φ. Note 1. The derivation problem for propositional primal logic with one kind of quotations is solvable linear time. Note 2. In the Hilbertian calculus for propositional primal logic, the shortest derivation of a formula φ from hypotheses H may be exponential in the length of (H,φ).

Quantum Kripke frames and other related kinds of Kripke frames are introduced. The inner structures of these Kripke frames are studied in detail, and many of them turn out to form nice geometries. To be precise, geometric frames, which are more general than quantum Kripke frames, correspond to projective geometries with a pure polarity; and quantum Kripke frames correspond to irreducible Hilber...

The purpose of the project was to attempt an interpretation of the nature of digital quantum information from a geometrical perspective. This lead to designing an algorithm for building the equivalents of Platonic solids in a 2n dimensional Hilbert space, structures called uniform Hilbertian polytopes. The long-term goal was to better understand quantum entanglement, and possibly find a measure...