Serre Duality, Abel's Theorem, and Jacobi Inversion for Supercurves Over a Thick Superpoint

The principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1|q over a finite-dimensional local supercommutative C-algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincare duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic Groups and Class Fields, combined with an invariance result for the topology on the dual of the space of repartitions. Our Abel map, taking Cartier divisors of degree zero to the dual of the space of sections of the Berezinian sheaf, modulo periods, is defined via Penkov's characterization of the Berezinian sheaf as the cohomology of the de Rham complex of the sheaf D of differential operators, as a right module over itself. We discuss the Jacobi inversion problem for the Abel map and give an example demonstrating that if n is an integer sufficiently large that the generic divisor of degree n is linearly equivalent to an effective divisor, this need not be the case for all divisors of degree n.Comment: 14 page

Periodicity and Growth in a Lattice Gas with Dynamical Geometry

We study a one-dimensional lattice gas "dynamical geometry model" in which local reversible interactions of counter-rotating groups of particles on a ring can create or destroy lattice sites. We exhibit many periodic orbits and and show that all other solutions have asymptotically growing lattice length in both directions of time. We explain why the length grows as $\sqrt{t}$ in all cases examined. We completely solve the dynamics for small numbers of particles with arbitrary initial conditions.Comment: 18 pages, LaTe

D-modules on 1|1 Supercurves

It is known that to every 1|1 dimensional supercurve X there is associated a dual supercurve \hat{X}, and a superdiagonal \Delta in their product. We establish that the categories of D-modules on X, \hat{X}, and \Delta are equivalent. This follows from a more general result about D-modules and purely odd submersions. The equivalences preserve tensor products, and take vector bundles to vector bundles. Line bundles with connection are studied, and examples are given where X is a superelliptic curve.Comment: 18 page

Intellectual need and problem-free activity in the mathematics classroom

Intellectual need, a key part of the DNR theoretical framework, is posited to be necessary for significant learning to occur. This paper provides a theoretical examination of intellectual need and its absence in mathematics classrooms. Although this is not an empirical study, we use data from observed high school algebra classrooms to illustrate four categories of activity students engage in while feeling little or no intellectual need. We present multiple examples for each category in order to draw out different nuances of the activity, and we contrast the observed situations with ones that would provide various types of intellectual need. Finally, we offer general suggestions for teaching with intellectual need