Mathematics Theses and Dissertations
http://hdl.handle.net/2429/32622
2014-04-24T07:43:21ZThe behavior of the Hilbert scheme of points under the derived McKay correspondence
http://hdl.handle.net/2429/45044
In this thesis, we completely determine the image of structure sheaves of zero-dimensional, torus invariant, closed subschemes on the minimal, crepant resolution Y of the Kleinian quotient singularity C²/Z/n under the Fourier-Mukai equivalence of categories, between derived category of coherent sheaves on Y and Z/n-equivariant derived category of coherent sheaves on C². As a consequence, we obtain a combinatorial correspondence between partitions and Z/n-colored skew partitions.
2013-09-09T00:00:00ZHarnack inequality for nondivergent linear elliptic operators on Riemannian manifolds : a self-contained proof
http://hdl.handle.net/2429/45031
In this paper, a self-contained proof is given to a well-known Harnack inequality of second order nondivergent uniformly elliptic operators on Riemannian manifolds with the condition that M-[R(v )]>0, following the ideas of M. Safonov [5]. Basically, the proof consists of three parts: 1)
Critical Density Lemma, 2) Power-Decay of the Distribution Functions of Solutions, and 3)Harnack Inequality.
2013-09-05T00:00:00ZAn exactly divergence-free finite element method for non-isothermal flow problems
http://hdl.handle.net/2429/45013
In this thesis the exactly divergence-free finite element method developed by Cockburn, Scheotzau and Kanschat in [13] and [14] is studied. This method is first reviewed in the context of Stokes problem. An interior
penalty discontinuous Galerkin approach is formulated and analysed in the unified framework established in
[13], [14] and [37]. Then we extend the method to non-isothermal flow problems, in particular, to a generalised Boussinesq equation. Following the work by Ricardo, Scheotzau and Qin in [34], the method is formulated and
the numerical analysis is reviewed. Numerical examples are implemented and presented,
which verify the theoretical error estimates and the exactly divergence-free property.
2013-09-03T00:00:00ZTransport and dispersion of particles in visco-plastic fluids
http://hdl.handle.net/2429/45002
This thesis focuses on development of a model to predict “spreading” of the
solids (i.e. proppant) fraction during the fracturing operation. We develop
a 1D model that allows us to estimate dispersion of solid particles along a
vertical pipe in a fully turbulent flow of a shear thinning yield stress fluid (i.e.,
visco-plastic fluid), as well as slip relative to the mean flow. In dimensionless
form, this results in a quasilinear advection-diffusion equation. Advection by
the mean flow, particle settling relative to the mean, in the direction of gravity,
turbulent particle dispersivity and Taylor dispersion are the 4 main transport
phenomena modelled in the 1D model. We provide a simple analysis of the
1D model, suitable for spreadsheet-type field design purposes, in which we
estimate “mixing lengths” due to both settling and dispersion. Secondly, we
provide an accurate numerical algorithm for solution of the 1D model and
show how pulses of proppant (i.e. slugs) may or may not interact for typical
process parameters.
2013-09-03T00:00:00Z