For each simple Lie algebra g (excluding, for trivial reasons, type C) we find the lowest possible degree of an invariant 2 nd order PDE over the adjoint variety in Pg, a homogeneous contact manifold. Here a PDE F (x i , u, u i , u ij ) = 0 has degree ≤ d if F is a polynomial of degree ≤ d in the minors of (u ij ), with coefficients functions of the contact coordinates x i , u, u i (e.g., Monge-Ampère equations have degree 1). For g of type A or G 2 we show that this gives all invariant 2 nd order PDEs. For g of type B and D we provide an explicit formula for the lowest-degree invariant 2 nd order PDEs. For g of type E and F 4 we prove uniqueness of the lowest-degree invariant 2 nd order PDE; we also conjecture that uniqueness holds in type D.
Abstract. By studying the development of shock waves out of discontinuity waves, in 1954 P. Lax discovered a class of PDEs, which he called "completely exceptional", where such a transition does not occur after a finite time. A straightforward integration of the completely exceptionality conditions allowed Boillat to show that such PDEs are actually of Monge-Ampère type. In this paper, we first recast these conditions in terms of characteristics, and then we show that the completely exceptional PDEs, with 2 or 3 independent variables, can be described in terms of the conformal geometry of the Lagrangian Grassmannian, where they are naturally embedded. Moreover, for an arbitrary number of independent variables, we show that the space of r th degree sections of the Lagrangian Grassmannian can be resolved via a BGG operator. In the particular case of 1 st degree sections, i.e., hyperplane sections or, equivalently, Monge-Ampère equations, such operator is a close analog of the trace-free second fundamental form.
Fermionic linear optics is a model of quantum computation which is efficiently simulable on a classical probabilistic computer. We study the problem of a classical simulation of fermionic linear optics augmented with noisy auxiliary states. If the auxiliary state can be expressed as a convex combination of pure Fermionic Gaussian states, the corresponding computation scheme is classically simulable. We present an analytic characterisation of the set of convex-Gaussian states in the first non-trivial case, in which the Hilbert space of the ancilla is a four-mode Fock space. We use our result to solve an open problem recently posed by De Melo et al. [1] and to study in detail the geometrical properties of the set of convex-Gaussian states.
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