Faculty Research Publications (Mathematics)
http://hdl.handle.net/1957/13818
2014-04-24T01:56:45ZDecay rates to equilibrium for nonlinear plate equations with degenerate, geometrically-constrained damping
http://hdl.handle.net/1957/47321
Decay rates to equilibrium for nonlinear plate equations with degenerate, geometrically-constrained damping
Geredeli, Pelin G.; Webster, Justin T.
We analyze the convergence to equilibrium of solutions to the nonlinear Berger plate evolution
equation in the presence of localized interior damping (also referred to as geometrically constrained
damping). Utilizing the results in [24], we have that any trajectory converges to the set of stationary
points N. Employing standard assumptions from the theory of nonlinear unstable dynamics on
the set N, we obtain the rate of convergence to an equilibrium. The critical issue in the proof
of convergence to equilibria is a unique continuation property (which we prove for the Berger
equation) that provides a gradient structure for the dynamics. We also consider the more involved
von Karman evolution, and show that the same results hold assuming a unique continuation
property for solutions, which is presently a challenging open problem.
This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Springer and can be found at: http://link.springer.com/journal/245.
2013-12-01T00:00:00ZAdvection–Dispersion Across Interfaces
http://hdl.handle.net/1957/46498
Advection–Dispersion Across Interfaces
Ramirez, Jorge M.; Thomann, Enrique A.; Waymire, Edward C.
This article concerns a systemic manifestation of small scale interfacial
heterogeneities in large scale quantities of interest to a variety of
diverse applications spanning the earth, biological and ecological sciences.
Beginning with formulations in terms of partial differential equations governing
the conservative, advective-dispersive transport of mass concentrations
in divergence form, the specific interfacial heterogeneities are introduced
in terms of (spatial) discontinuities in the diffusion coefficient across a
lower-dimensional hypersurface. A pathway to an equivalent stochastic formulation
is then developed with special attention to the interfacial effects in
various functionals such as first passage times, occupation times and local
times. That an appreciable theory is achievable within a framework of applications
involving one-dimensional models having piecewise constant coefficients
greatly facilitates our goal of a gentle introduction to some rather
dramatic mathematical consequences of interfacial effects that can be used to
predict structure and to inform modeling.
This is the publisher’s final pdf. The published article is copyrighted by the Institute of Mathematical Statistics and can be found at: http://www.imstat.org/sts/.
2013-11-01T00:00:00ZMetrics of positive scalar curvature and generalised Morse functions, Part II
http://hdl.handle.net/1957/46373
Metrics of positive scalar curvature and generalised Morse functions, Part II
Walsh, Mark
The surgery technique of Gromov and Lawson may be used to
construct families of positive scalar curvature metrics which are parameterised
by Morse functions. This has played an important role in the study of the
space of metrics of positive scalar curvature on a smooth manifold and its
corresponding moduli spaces. In this paper, we extend this technique to work
for families of generalised Morse functions, i.e. smooth functions with both
Morse and birth-death singularities.
First published in Transactions of the American Mathematical Society in Vol. 366 no. 1, published by the American Mathematical Society. This is the publisher’s final pdf. The published article is copyrighted by the American Mathematical Society and can be found at: http://www.ams.org/publications/journals/journalsframework/tran.
2014-01-01T00:00:00ZInequivalent Cantor sets in 𝑅³ whose complements have the same fundamental group
http://hdl.handle.net/1957/46328
Inequivalent Cantor sets in 𝑅³ whose complements have the same fundamental group
Garity, Dennis J.; Repovs, Dusan
For each Cantor set C in R³, all points of which have bounded
local genus, we show that there are infinitely many inequivalent Cantor sets in
R³ with the complement having the same fundamental group as the complement
of C. This answers a question from Open Problems in Topology and has
as an application a simple construction of nonhomeomorphic open 3-manifolds
with the same fundamental group. The main techniques used are analysis of
local genus of points of Cantor sets, a construction for producing rigid Cantor
sets with simply connected complement, and manifold decomposition theory.
The results presented give an argument that for certain groups G, there
are uncountably many nonhomeomorphic open 3-manifolds with fundamental
group G.
First published in Proceedings of the American Mathematical Society in Vol. 141 no. 8, published by the American Mathematical Society. This is the publisher’s final pdf. The published article is copyrighted by the American Mathematical Society and can be found at: http://www.ams.org/publications/journals/journalsframework/proc.
2013-08-01T00:00:00Z