141 research outputs found
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 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
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