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Discrete Exterior Calculus
Topological methods can deduce qualitative behaviors
of discrete mathematical models for phenomena including incompressible
fluid flows in disordered porous media, coupled elastodynamic/electrodynamic
excitations, coupled electron-phonon transport in nanoscale composites,
and so on. Most vector calculus theorems, including the Stokes, Green and
Ostrogradski integral theorems and all vector identities, have exact analogs
in a new discrete math formalism. The point of this formulation is to construct
simplified lattice models of physical phenomena that are better suited
to both computation and mathematical analysis. Curiously, the concept of
a limit is absent. The discrete calculus is based on algebraic topology
and the boundary and coboundary mappings do the work of gradient, divergence
and curl. In computation, much of the usual linear algebra algorithms are
replaced discrete operations such as list sort and table search.
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Hierarchical Simulation of Nanostructures
Challenges of nanotechnology include small size
(down to atomic scale), complex organization, strong lateral interaction,
and so on. New nanometer-scale devices, from nanostructured composite materials
to one-electron transistors, molecular computers, and even molecular motors,
open the prospect of using engineered quantum effects to achieve desirable
macroscopic properties. New simulation techniques are required that traverse
a series of length scales (atomic, nanoscale, mesoscale, perhaps macroscale)
and treat coupled phenomena (electronic, mechanical, electromagnetic, ?)
correctly.
We are developing software related to three novel
simulation concepts: (1) a hierarchical Green function renormalization
method for multiscale simulations (2) discrete vector calculus applied
to transport in nanoscale composites, and (3) use of context-sensitive
L-systems to define complex, hierarchical nanoscale systems and structures
and to write appropriate computer code for performing simulations. These
address nanoscale crossover phenomena, where significant transport or quantum
lengths can exceed system component size, in the context of integrated
systems with significant structural features spanning many different length
scales.
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Transport and wave dynamics in fractal structures
This work performed with Mizuho Schwalm and others
is on electron propagation on fractal lattices, in randomly disordered
structures, and in ballistic device geometries. Our contributions include
an algebraic extension theory for lattice Green functions [1] and a streamlined
renormalization procedure for lattice dynamical problems on fractals. With
the extension theory one can find Green functions on higher dimensional
lattices, including infinitely ramified fractal lattices which are not
yet well understood.
In 1993 I met M. Giona who has adapted the
formalism to treat practical problems in mass and heat transport in porous
media [2]. This resulted in cooperation between UND and the
Italian Center for Fractal and Disordered Structures in Rome and was supported
by a NATO Collaborative Research Grant.
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Lie groups and reduction of order of discrete
dynamical systems
The Lie theory of integration of differential
equations subsumes all other integration methods. All known solution methods
are equivalent to finding Lie Groups admitted by the equation and using
these groups to reduce the order.
There is quite a bit of current interest in extending
Lie theory to systems of finite difference equations, i.e. discrete dynamical
systems. Given a group that leaves a dynamical system invariant, it is
not difficult to show that this leads to a reduction of order, exactly
as in the differential equation case. The difficulty however is to find
such a group. Unlike the situation for differential equations, there is
no known algorithm.
Recently we have discovered an algorithm for finding
Lie groups for a given discrete dynamical system. The systems of interest
come from physics problems. Specifically we have dealt with the recursions
that come from real-space renormalization of lattice models. Often
we can find one or more Lie groups which such systems admit. Thus we can
reduce the order. In many instances this leads to explicit solutions. The
net result has been to produce several new classes of exactly solvable
lattice problems. Such exactly solvable problems are of great interest
in theoretical physics because the insight they provide is different from
the insight afforded by approximate solution of more realistic models.
But beyond that, the general group theoretic methods we discovering for
integrating difference equations will be of interest in a much wider area
of applied mathematics.