68 research outputs found
Stability analysis of a class of two-dimensional multipolar vortex equilibria
Published versio
Analytical solutions for distributed multipolar vortex equilibria on a sphere
Published versio
Conformal mapping methods for interfacial dynamics
The article provides a pedagogical review aimed at graduate students in
materials science, physics, and applied mathematics, focusing on recent
developments in the subject. Following a brief summary of concepts from complex
analysis, the article begins with an overview of continuous conformal-map
dynamics. This includes problems of interfacial motion driven by harmonic
fields (such as viscous fingering and void electromigration), bi-harmonic
fields (such as viscous sintering and elastic pore evolution), and
non-harmonic, conformally invariant fields (such as growth by
advection-diffusion and electro-deposition). The second part of the article is
devoted to iterated conformal maps for analogous problems in stochastic
interfacial dynamics (such as diffusion-limited aggregation, dielectric
breakdown, brittle fracture, and advection-diffusion-limited aggregation). The
third part notes that all of these models can be extended to curved surfaces by
an auxilliary conformal mapping from the complex plane, such as stereographic
projection to a sphere. The article concludes with an outlook for further
research.Comment: 37 pages, 12 (mostly color) figure
Conformal Mappings and Dispersionless Toda hierarchy
Let be the space consists of pairs , where is a
univalent function on the unit disc with , is a univalent function
on the exterior of the unit disc with and
. In this article, we define the time variables , on which are holomorphic with respect to the natural
complex structure on and can serve as local complex coordinates
for . We show that the evolutions of the pair with
respect to these time coordinates are governed by the dispersionless Toda
hierarchy flows. An explicit tau function is constructed for the dispersionless
Toda hierarchy. By restricting to the subspace consists
of pairs where , we obtain the integrable hierarchy
of conformal mappings considered by Wiegmann and Zabrodin \cite{WZ}. Since
every homeomorphism of the unit circle corresponds uniquely to
an element of under the conformal welding
, the space can be naturally
identified as a subspace of characterized by . We
show that we can naturally define complexified vector fields \pa_n, n\in \Z
on so that the evolutions of on
with respect to \pa_n satisfy the dispersionless Toda
hierarchy. Finally, we show that there is a similar integrable structure for
the Riemann mappings . Moreover, in the latter case, the time
variables are Fourier coefficients of and .Comment: 23 pages. This is to replace the previous preprint arXiv:0808.072
Scaling Limits for Internal Aggregation Models with Multiple Sources
We study the scaling limits of three different aggregation models on Z^d:
internal DLA, in which particles perform random walks until reaching an
unoccupied site; the rotor-router model, in which particles perform
deterministic analogues of random walks; and the divisible sandpile, in which
each site distributes its excess mass equally among its neighbors. As the
lattice spacing tends to zero, all three models are found to have the same
scaling limit, which we describe as the solution to a certain PDE free boundary
problem in R^d. In particular, internal DLA has a deterministic scaling limit.
We find that the scaling limits are quadrature domains, which have arisen
independently in many fields such as potential theory and fluid dynamics. Our
results apply both to the case of multiple point sources and to the
Diaconis-Fulton smash sum of domains.Comment: 74 pages, 4 figures, to appear in J. d'Analyse Math. Main changes in
v2: added "least action principle" (Lemma 3.2); small corrections in section
4, and corrected the proof of Lemma 5.3 (Lemma 5.4 in the new version);
expanded section 6.
Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations
In this article, we classify the solutions of the dispersionless Toda
hierarchy into degenerate and non-degenerate cases. We show that every
non-degenerate solution is determined by a function of
two variables. We interpret these non-degenerate solutions as defining
evolutions on the space of pairs of conformal mappings ,
where is a univalent function on the exterior of the unit disc, is a
univalent function on the unit disc, normalized such that ,
and . For each solution, we show how to define the
natural time variables , as complex coordinates on the space
. We also find explicit formulas for the tau function of the
dispersionless Toda hierarchy in terms of . Imposing
some conditions on the function , we show that the
dispersionless Toda flows can be naturally restricted to the subspace
of defined by . This recovers
the result of Zabrodin.Comment: 25 page
Disparate SAR Data of Griseofulvin Analogues for the Dermatophytes Trichophyton mentagrophytes, T. rubrum, and MDA-MB-231 Cancer Cells
The influence of early aging on eye movements during motor simulation
Movement based interventions such as imagery and action observation are used increasingly to support physical rehabilitation of adults during early aging. The efficacy of these more covert approaches is based on an intuitively appealing assumption that movement execution, imagery and observation share neural substrate; alteration of one influences directly the function of the other two. Using eye movement metrics this paper reports findings that question the congruency of the three conditions. The data reveal that simulating movement through imagery and action observation may offer older adults movement practice conditions that are not constrained by the age-related decline observed in physical conditions. In addition, the findings provide support for action observation as a more effective technique for movement reproduction in comparison to imagery. This concern for imagery was also seen in the less congruent temporal relationship in movement time between imagery and movement execution suggesting imagery inaccuracy in early aging
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