74,958 research outputs found
Collapsed heteroclinic snaking near a heteroclinic chain in dragged meniscus problems
We study a liquid film that is deposited onto a flat plate that is inclined
at a constant angle to the horizontal and is extracted from a liquid bath at a
constant speed. We additionally assume that there is a constant temperature
gradient along the plate that induces a Marangoni shear stress. We analyse
steady-state solutions of a long-wave evolution equation for the film
thickness. Using centre manifold theory, we first obtain an asymptotic
expansion of solutions in the bath region. The presence of the temperature
gradient significantly changes these expansions and leads to the presence of
logarithmic terms that are absent otherwise. Next, we obtain numerical
solutions of the steady-state equation and analyse the behaviour of the
solutions as the plate velocity is changed. We observe that the bifurcation
curve exhibits snaking behaviour when the plate inclination angle is beyond a
certain critical value. Otherwise, the bifurcation curve is monotonic. The
solutions along these curves are characterised by a foot-like structure that is
formed close to the meniscus and is preceded by a thin precursor film further
up the plate. The length of the foot increases along the bifurcation curve.
Finally, we explain that the snaking behaviour of the bifurcation curves is
caused by the existence of an infinite number of heteroclinic orbits close to a
heteroclinic chain that connects in an appropriate three-dimensional phase
space the fixed point corresponding to the precursor film with the fixed point
corresponding to the foot and then with the fixed point corresponding to the
bath.Comment: Final revised version. 18 pages. To be published in Eur. Phys. J.
Biological control via "ecological" damping: An approach that attenuates non-target effects
In this work we develop and analyze a mathematical model of biological
control to prevent or attenuate the explosive increase of an invasive species
population in a three-species food chain. We allow for finite time blow-up in
the model as a mathematical construct to mimic the explosive increase in
population, enabling the species to reach "disastrous" levels, in a finite
time. We next propose various controls to drive down the invasive population
growth and, in certain cases, eliminate blow-up. The controls avoid chemical
treatments and/or natural enemy introduction, thus eliminating various
non-target effects associated with such classical methods. We refer to these
new controls as "ecological damping", as their inclusion dampens the invasive
species population growth. Further, we improve prior results on the regularity
and Turing instability of the three-species model that were derived in earlier
work. Lastly, we confirm the existence of spatio-temporal chaos
On selection criteria for problems with moving inhomogeneities
We study mechanical problems with multiple solutions and introduce a
thermodynamic framework to formulate two different selection criteria in terms
of macroscopic energy productions and fluxes. Studying simple examples for
lattice motion we then compare the implications for both resting and moving
inhomogeneities.Comment: revised version contains new introduction, numerical simulations of
Riemann problems, and a more detailed discussion of the causality principle;
18 pages, several figure
Maximum entropy methods as the bridge between macroscopic and microscopic theory
This paper investigates a function of macroscopic variables known as the
singular potential, building on previous work by Ball and Majumdar. The
singular potential is a function of the admissible statistical averages of
probability distributions on a state space, defined so that it corresponds to
the maximum possible entropy given known observed statistical averages,
although non-classical entropy-like objective functions will also be
considered. First the set of admissible moments must be established, and under
the conditions presented in this work the set is open, bounded and convex
allowing a description in terms of supporting hyperplanes, which provides
estimates on the development of singularities for related probability
distributions. Under appropriate conditions it is shown that the singular
potential is strictly convex, as differentiable as the microscopic entropy and
blows up uniformly as the macroscopic variable tends to the boundary of the set
of admissible moments. Applications of the singular potential are then
discussed, and particular consideration will be given to certain free-energy
functionals typical in mean-field theory, demonstrating an equivalence between
certain microscopic and macroscopic free-energy functionals. This allows
statements about L^1-local minimisers of Onsager's free energy to be obtained
which cannot be given by two-sided variations, and overcomes the need to ensure
local minimisers are bounded away from zero and infinity before taking bounded
variations. The analysis also permits the definition of a dual order parameter
for which Onsager's free energy allows an explicit representation. Also the
difficulties in approximating the singular potential by everywhere defined
functions, in particular by polynomials, are addressed with examples
demonstrating the failure of the Taylor approximation to preserve shape
properties of the singular potential
The Fermi-Pasta-Ulam problem: 50 years of progress
A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with
its suggested resolutions and its relation to other physical problems. We focus
on the ideas and concepts that have become the core of modern nonlinear
mechanics, in their historical perspective. Starting from the first numerical
results of FPU, both theoretical and numerical findings are discussed in close
connection with the problems of ergodicity, integrability, chaos and stability
of motion. New directions related to the Bose-Einstein condensation and quantum
systems of interacting Bose-particles are also considered.Comment: 48 pages, no figures, corrected and accepted for publicatio
The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools
On September 14, 2015, the newly upgraded Laser Interferometer
Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW)
signal, emitted a billion light-years away by a coalescing binary of two
stellar-mass black holes. The detection was announced in February 2016, in time
for the hundredth anniversary of Einstein's prediction of GWs within the theory
of general relativity (GR). The signal represents the first direct detection of
GWs, the first observation of a black-hole binary, and the first test of GR in
its strong-field, high-velocity, nonlinear regime. In the remainder of its
first observing run, LIGO observed two more signals from black-hole binaries,
one moderately loud, another at the boundary of statistical significance. The
detections mark the end of a decades-long quest, and the beginning of GW
astronomy: finally, we are able to probe the unseen, electromagnetically dark
Universe by listening to it. In this article, we present a short historical
overview of GW science: this young discipline combines GR, arguably the
crowning achievement of classical physics, with record-setting, ultra-low-noise
laser interferometry, and with some of the most powerful developments in the
theory of differential geometry, partial differential equations,
high-performance computation, numerical analysis, signal processing,
statistical inference, and data science. Our emphasis is on the synergy between
these disciplines, and how mathematics, broadly understood, has historically
played, and continues to play, a crucial role in the development of GW science.
We focus on black holes, which are very pure mathematical solutions of
Einstein's gravitational-field equations that are nevertheless realized in
Nature, and that provided the first observed signals.Comment: 41 pages, 5 figures. To appear in Bulletin of the American
Mathematical Societ
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