11,376 research outputs found
Diffusion-Limited Aggregation on Curved Surfaces
We develop a general theory of transport-limited aggregation phenomena
occurring on curved surfaces, based on stochastic iterated conformal maps and
conformal projections to the complex plane. To illustrate the theory, we use
stereographic projections to simulate diffusion-limited-aggregation (DLA) on
surfaces of constant Gaussian curvature, including the sphere () and
pseudo-sphere (), which approximate "bumps" and "saddles" in smooth
surfaces, respectively. Although curvature affects the global morphology of the
aggregates, the fractal dimension (in the curved metric) is remarkably
insensitive to curvature, as long as the particle size is much smaller than the
radius of curvature. We conjecture that all aggregates grown by conformally
invariant transport on curved surfaces have the same fractal dimension as DLA
in the plane. Our simulations suggest, however, that the multifractal
dimensions increase from hyperbolic () geometry, which
we attribute to curvature-dependent screening of tip branching.Comment: 4 pages, 3 fig
Explicit methods for Hilbert modular forms
We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of
Hilbert modular forms over a totally real field. We provide many explicit
examples as well as applications to modularity and Galois representations.Comment: 52 pages, 10 figures, many table
Gradient-Based Estimation of Uncertain Parameters for Elliptic Partial Differential Equations
This paper addresses the estimation of uncertain distributed diffusion
coefficients in elliptic systems based on noisy measurements of the model
output. We formulate the parameter identification problem as an infinite
dimensional constrained optimization problem for which we establish existence
of minimizers as well as first order necessary conditions. A spectral
approximation of the uncertain observations allows us to estimate the infinite
dimensional problem by a smooth, albeit high dimensional, deterministic
optimization problem, the so-called finite noise problem in the space of
functions with bounded mixed derivatives. We prove convergence of finite noise
minimizers to the appropriate infinite dimensional ones, and devise a
stochastic augmented Lagrangian method for locating these numerically. Lastly,
we illustrate our method with three numerical examples
Branes, Calabi-Yau Spaces, and Toroidal Compactification of the N=1 Six-Dimensional E_8 Theory
We consider compactifications of the N=1, d=6, E_8 theory on tori to five,
four, and three dimensions and learn about some properties of this theory. As a
by-product we derive the SL(2,\IZ) duality of the N=2, d=4, SU(2) theory with
N_f=4. Using this theory on a D-brane probe we shed new light on the
singularities of F-theory compactifications to eight dimensions. As another
application we consider compactifications of F-theory, M-theory and the IIA
string on (singular) Calabi-Yau spaces where our theory appears in spacetime.
Our viewpoint leads to a new perspective on the nature of the singularities in
the moduli space and their spacetime interpretations. In particular, we have a
universal understanding of how the singularities in the classical moduli space
of Calabi--Yau spaces are modified by worldsheet instantons to singularities in
the moduli space of the corresponding conformal field theories.Comment: 40 pages, 2 figures, harvmac with epsf. Minor corrections, references
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A Dimension-Adaptive Multi-Index Monte Carlo Method Applied to a Model of a Heat Exchanger
We present an adaptive version of the Multi-Index Monte Carlo method,
introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with
coefficients that are random fields. A classical technique for sampling from
these random fields is the Karhunen-Lo\`eve expansion. Our adaptive algorithm
is based on the adaptive algorithm used in sparse grid cubature as introduced
by Gerstner and Griebel (2003), and automatically chooses the number of terms
needed in this expansion, as well as the required spatial discretizations of
the PDE model. We apply the method to a simplified model of a heat exchanger
with random insulator material, where the stochastic characteristics are
modeled as a lognormal random field, and we show consistent computational
savings
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