We unify and generalize several known results about systems of random polynomials. We first classify all orthogonally invariant normal measures for spaces of polynomial mappings. For each such measure we calculate the expected number of real zeros. The results for invariant measures extend to underdetermined systems, giving the expected volume for orthogonally invariant random real projective v...

The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for σ-finite invariant measures (Corollary 1). For infinite measures the ergodic decomposition is not unique, but the measure class of the decomposing measure on the space of projective measures is uniquely defined by ...

Given an epimorphism φ: Π → Π' between projective planes Π and Π', it is an open question how the groups of projectivities of Π and Π' (regarded as permutation groups on projective lines) are related. Within this note we will not answer this sophisticated and hard problem in full, but we will address the question to which extend the projectivities of Π induce permutations on the lines of Π ' wh...

We study the action of the group of automorphisms of the projective plane on the Maruyama scheme of sheaves MP 2(r, c1,c2) of rank r and Chern classes c1 = 0 and c2 = n and obtain sufficient conditions for unstability in the sense of Mumford’s geometric invariant theory. The conditions are in terms of the splitting behaviour of sheaves when restricted to lines in the projective plane. A very st...

An l-group G is an abelian group equipped with a translation invariant lattice order. Baker and Beynon proved that G is finitely generated projective iff it is finitely presented. A unital l-group is an l-group G with a distinguished order unit, i.e., an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. While every finitely generated projective unital l-...

The spinor representation is developed and used to investigate minimal surfaces in R with embedded planar ends. The moduli spaces of planar-ended minimal spheres and real projective planes are determined, and new families of minimal tori and Klein bottles are given. These surfaces compactify in S to yield surfaces critical for the Möbius invariant squared mean curvature functional W . On the ot...

A novel method is presented for distributed matching of curves across widely varying viewpoints. The fundamental projective joint-invariants for curves in the real projective space are the volume cross-ratios. A curve in m-dimensional projective space is represented by a signature manifold comprising n-point projective joint invariants, where n is at least m + 2. The signature manifold can be u...

The following structure theorems are proved: An orbit of a real solvable Lie group in projective space that is a complex submanifold is isomorphic to C x (C*)m x Í2 , where Í2 is an open orbit of a real solvable Lie group in a projective rational variety. Also, any homogeneous space of a complex Lie group that is isomorphic to C" can be realized as an orbit in some projective space. As a conseq...

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikeš, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sec...

In this paper, the statistical theory of shape for ordered nite point conngurations is studied. A general, invariant based, approach for deening shape and nding its density is developed. Some examples, that have been possible to compute analytically, are given, including both aane and positive similarity shape. Projective shape and the related projective invariants are important topics in compu...