Fundamentals of optical design in rotating mirror cameras

The elementary optics of rotating-mirror high-speed cameras are analysed geometrically. The images in an off-axis scanning mirror are shown to lie on Pascal's limacon. The Miller principle by which shuttering is accomplished is described. A simple estimate of motion blurring is compared with the Rayleigh criterion.     

Free

It has been conjectured that every free algebraic action of the additive group of complex numbers on complex affine three space is conjugate to a global translation. The main result lends support to this conjecture by showing that the morphism to the variety defined by the ring of invariants of the associated action on the coordinate ring is smooth. As a consequence, the graph morphism is an open immersion, and simple proofs of certain cases of the conjecture are obtained.

     

Algebras and differential equations with additive symmetry

The set of all m-ary algebra structures on a given vector space affords, by the change of basis action, a representation of the general linear group. The invariants of a given subgroup are identified with those algebras admitting that subgroup as algebra automorphisms. Any finite dimensional representation of the additive group as automorphisms is obtained as the exponential of a nilpotent derivation. The latter can be embedded in the Lie algebra sl(2) so that the maximal vectors in an irreducible decomposition of the set of algebras as an sl(2) module are the invariants of the given action of the additive group. Dimension formulas and explicit bases are computed for the space of algebras with certain additive group actions. Employing the equivalence of the categories of m-ary algebras and systems of autonomous mth order homogeneous differential equations, the algebraic results are connected to the construction of first integrals and semi-invariants.     

G a Invariants and Slices

Criteria for local triviality of algebraic actions of the additive group of complex numbers on complex affine space are extended to more general varieties. Finite generation of rings of invariants of locally trivial actions on factorial affine varieties is dicussed, giving some sufficient conditions for finite generation and examples where finite generation fails. A missing hypothesis in a theorem of Miyanishi is identified, and an example is given to demonstrate the necessity of the hypothesis. The corrected theorem is shown to hold for a class of triangular G _a actions on C⁴, with the consequence that all these actions are conjugate to translations. A new criterion for an action to be conjugate to a global translation is given.     

Triangular actions on

Every locally trivial action of the additive group of complex numbers on four-dimensional complex affine space that is given by a triangular derivation is conjugate to a translation. A criterion for a proper action on complex affine -space to be locally trivial is given, along with an example showing that the hypotheses of the criterion are sharp.

     

On locally trivial G a -actions

If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1). We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety. If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (Theorem 3). Equivalently, ifC ⁿ , is the total space for a principalG _a -bundle over some open subset ofC ^n–1 then the bundle is trivial. On the other hand, there is a locally trivialG _a -action on a normal affine variety with nonfinitely generated ring of invariants (Theorem 2).Supported in part by NSA Grant No. MDA904-96-1-0069     

Locally trivial 𝔾a−actions on ℂ5 with singular algebraic quotients

Simple examples are given of proper algebraic actions of the additive group of complex numbers on ?⁵ whose geometric quotients are, respectively, a?ne, strictly quasia?ne, and algebraic spaces which are not schemes. Moreover, a Zariski locally trivial action is given whose ring of invariant regular functions defines a singular factorial a?ne fourfold embedded in ?¹². The geometric quotient for the action embeds as a strictly quasia?ne variety in the smooth locus of the algebraic quotient with complement isomorphic to the normal a?ne surface with the A₂?singularity at the origin.     

Rigidity and compact real forms of semisimple complex jordan algebras

Finite dimensional semisimple complex Jordan algebras are shown to be rigid. This result is applied to obtain a new classification free proof of the existence of a compact real (i.e. formally real) form for any such algebra.