197,095 research outputs found
Heat conduction from irregular surfaces
The effect of irregularities on the rate of heat conduction from a two-dimensional isothermal surface into a semi infinite medium is considered. The effect of protrusions, depressions, and surface roughness is quantified in terms of the displacement of the linear temperature profile prevailing far from the surface. This shift, coined the displacement length, is designated as an appropriate global measure of the effect of the surface indentations incorporating the particular details of the possibly intricate geometry. To compute the displacement length, Laplace's equation describing the temperature distribution in the semi-infinite space above the surface is solved numerically by a modified Schwarz-Christoffel transformation whose computation requires solving a system of highly non-linear algebraic equations by iterative methods, and an integral equation method originating from the single-layer integral representation of a harmonic function involving the periodic Green's function. The conformal mapping method is superior in that it is capable of handling with high accuracy a large number of vertices and intricate wall geometries. On the other hand, the boundary integral method yields the displacement length as part of the solution. Families of polygonal wall shapes composed of segments in regular, irregular, and random arrangement are considered, and pre-fractal geometries consisting of large numbers of vertices are analyzed. The results illustrate the effect of wall geometry on the flux distribution and on the overall enhancement in the rate of transport for regular and complex wall shapes
A stochastic approach to reconstruction of faults in elastic half space
We introduce in this study an algorithm for the imaging of faults and of slip
fields on those faults. The physics of this problem are modeled using the
equations of linear elasticity. We define a regularized functional to be
minimized for building the image. We first prove that the minimum of that
functional converges to the unique solution of the related fault inverse
problem. Due to inherent uncertainties in measurements, rather than seeking a
deterministic solution to the fault inverse problem, we then consider a
Bayesian approach. In this approach the geometry of the fault is assumed to be
planar, it can thus be modeled by a three dimensional random variable whose
probability density has to be determined knowing surface measurements. The
randomness involved in the unknown slip is teased out by assuming independence
of the priors, and we show how the regularized error functional introduced
earlier can be used to recover the probability density of the geometry
parameter. The advantage of the Bayesian approach is that we obtain a way of
quantifying uncertainties as part of our final answer. On the downside, this
approach leads to a very large computation since the slip is unknown. To
contend with the size of this computation we developed an algorithm for the
numerical solution to the stochastic minimization problem which can be easily
implemented on a parallel multi-core platform and we discuss techniques aimed
at saving on computational time. After showing how this algorithm performs on
simulated data, we apply it to measured data. The data was recorded during a
slow slip event in Guerrero, Mexico.Comment: In this new version the second error functional is directly minimized
over a finite dimensional space leading to a more natural connection to the
stochastic formulatio
The Anomalous Scaling Exponents of Turbulence in General Dimension from Random Geometry
We propose an exact analytical formula for the anomalous scaling exponents of
inertial range structure functions in incompressible fluid turbulence. The
formula is a gravitational Knizhnik-Polyakov-Zamolodchikov (KPZ)-type relation,
and is valid in any number of space dimensions. It incorporates intermittency
by gravitationally dressing the Kolmogorov linear scaling via a coupling to a
random geometry. The formula has one real parameter that depends on
the number of space dimensions. The scaling exponents satisfy the convexity
inequality, and the supersonic bound constraint. They agree with the
experimental and numerical data in two and three space dimensions, and with
numerical data in four space dimensions. Intermittency increases with ,
and in the infinite limit the scaling exponents approach the value
one, as in Burgers turbulence. At large the th order exponent scales as
. We discuss the relation between fluid flows and black hole geometry
that inspired our proposal.Comment: 18 pages, 3 figures; v3: additional clarifications, added references;
v2: improved discussion, added one figur
Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories
The interplay rich between algebraic geometry and string and gauge theories
has recently been immensely aided by advances in computational algebra.
However, these symbolic (Gr\"{o}bner) methods are severely limited by
algorithmic issues such as exponential space complexity and being highly
sequential. In this paper, we introduce a novel paradigm of numerical algebraic
geometry which in a plethora of situations overcomes these short-comings. Its
so-called 'embarrassing parallelizability' allows us to solve many problems and
extract physical information which elude the symbolic methods. We describe the
method and then use it to solve various problems arising from physics which
could not be otherwise solved.Comment: 36 page
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