204 research outputs found
Well-posedness in Maxwell systems with distributions of polarization relaxation parameters
polarization relaxation parameter
Dynamics of Current, Charge and Mass
Electricity plays a special role in our lives and life. Equations of electron
dynamics are nearly exact and apply from nuclear particles to stars. These
Maxwell equations include a special term the displacement current (of vacuum).
Displacement current allows electrical signals to propagate through space.
Displacement current guarantees that current is exactly conserved from inside
atoms to between stars, as long as current is defined as Maxwell did, as the
entire source of the curl of the magnetic field. We show how the Bohm
formulation of quantum mechanics allows easy definition of current. We show how
conservation of current can be derived without mention of the polarization or
dielectric properties of matter. Matter does not behave the way physicists of
the 1800's thought it does with a single dielectric constant, a real positive
number independent of everything. Charge moves in enormously complicated ways
that cannot be described in that way, when studied on time scales important
today for electronic technology and molecular biology. Life occurs in ionic
solutions in which charge moves in response to forces not mentioned or
described in the Maxwell equations, like convection and diffusion. Classical
derivations of conservation of current involve classical treatments of
dielectrics and polarization in nearly every textbook. Because real dielectrics
do not behave in a classical way, classical derivations of conservation of
current are often distrusted or even ignored. We show that current is conserved
exactly in any material no matter how complex the dielectric, polarization or
conduction currents are. We believe models, simulations, and computations
should conserve current on all scales, as accurately as possible, because
physics conserves current that way. We believe models will be much more
successful if they conserve current at every level of resolution, the way
physics does.Comment: Version 4 slight reformattin
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
On a poroviscoelastic model for cell crawling
In this paper a minimal, one–dimensional, two–phase, viscoelastic, reactive, flow model for a crawling cell is presented. Two–phase models are used with a variety of constitutive assumptions in the literature to model cell motility. We use an upper–convected Maxwell model and demonstrate that even the simplest of two–phase, viscoelastic models displays features relevant to cell motility. We also show care must be exercised in choosing parameters for such models as a poor choice can lead to an ill–posed problem. A stability analysis reveals that the initially stationary, spatially uniform strip of cytoplasm starts to crawl in response to a perturbation which breaks the symmetry of the network volume fraction or network stress. We also demonstrate numerically that there is a steady travelling–wave solution in which the crawling velocity has a bell–shaped dependence on adhesion strength, in agreement with biological observation
Dielectric relaxation time spectroscopy for tissue characterisation
This thesis is concerned with Electrical Impedance Spectroscopy (EIS), a noninvasive
technique for characterising biological tissue and distinguishing pathology.
The thesis is focused on the development of an improved method for extracting
physiologically related parameters from the measured impedance data in vivo and
instrumentation for spectroscopic measurements.
In EIS, the electrical properties of physiological tissues, defined by their composition
and structure, are measured as functions of frequency. Experimental
observations of the existence of dielectric dispersions caused by distributions of
dielectric relaxation time (DRT) constants were made on different types of biological
material. It is postulated that widely used approaches for modelling these
electrical properties are fundamentally flawed. The research work concentrates on
the reconstruction of DRT spectra directly from the measured frequency response.
The reconstruction problem involves inversion of a linear operator and like many
inverse problems, is complicated by the ill-posed nature of the problem. In this
thesis an inversion algorithm - Galerkin Regularised Inverse Method (GRIM) -
based on standard mathematical methods is developed. The DRT spectrum establishes
a link between the raw impedance data and the physiological structure
and function of biological tissues. The GRIM yields a large number of independent
parameters each related to process on a different scale. Special care was
taken in testing the method on simulated data and improving its resolution.
The thesis is also concerned with the design and practical implementation of
EIS systems. Two approaches are considered: systems based on commercially
available Impedance Analysers and systems designed specially for studies in vivo.
To evaluate the GRIM, an Impedance Analyser, benefitting from a higher accuracy
and a wider frequency range, is used. To meet the more rigorous specification
demanded for studies on living human tissues, an electrical impedance spectrometer
is developed. The suitability of different current sources is investigated.
This research work includes studies of animal tissue in vitro and in vivo. Optimal
experiments are defined in terms of the measurement frequency range and
the entire experimental protocol for dielectric spectroscopy is established. These
biological data are used to evaluate the GRIM. A comparison between different
tissue classes in vivo is made. From studying ischemic tissues, it is postulated
and verified that physiological differences and changes can be measured using the
technique of DRT spectroscopy
Time-dependent electromagnetic scattering from dispersive materials
This paper studies time-dependent electromagnetic scattering from metamaterials that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave-material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation that is fully given on the surface of the scatterer via a time-dependent boundary integral equation. Discretizing this equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we derive error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method
Dynamics of Current, Charge and Mass
abstract: Electricity plays a special role in our lives and life. The dynamics of electrons allow light to flow through a vacuum. The equations of electron dynamics are nearly exact and apply from nuclear particles to stars. These Maxwell equations include a special term, the displacement current (of a vacuum). The displacement current allows electrical signals to propagate through space. Displacement current guarantees that current is exactly conserved from inside atoms to between stars, as long as current is defined as the entire source of the curl of the magnetic field, as Maxwell did.We show that the Bohm formulation of quantum mechanics allows the easy definition of the total current, and its conservation, without the dificulties implicit in the orthodox quantum theory. The orthodox theory neglects the reality of magnitudes, like the currents, during times that they are not being explicitly measured.We show how conservation of current can be derived without mention of the polarization or dielectric properties of matter. We point out that displacement current is handled correctly in electrical engineering by ‘stray capacitances’, although it is rarely discussed explicitly. Matter does not behave as physicists of the 1800’s thought it did. They could only measure on a time scale of seconds and tried to explain dielectric properties and polarization with a single dielectric constant, a real positive number independent of everything. Matter and thus charge moves in enormously complicated ways that cannot be described by a single dielectric constant,when studied on time scales important today for electronic technology and molecular biology. When classical theories could not explain complex charge movements, constants in equations were allowed to vary in solutions of those equations, in a way not justified by mathematics, with predictable consequences. Life occurs in ionic solutions where charge is moved by forces not mentioned or described in the Maxwell equations, like convection and diffusion. These movements and forces produce crucial currents that cannot be described as classical conduction or classical polarization. Derivations of conservation of current involve oversimplified treatments of dielectrics and polarization in nearly every textbook. Because real dielectrics do not behave in that simple way-not even approximately-classical derivations of conservation of current are often distrusted or even ignored. We show that current is conserved inside atoms. We show that current is conserved exactly in any material no matter how complex are the properties of dielectric, polarization, or conduction currents. Electricity has a special role because conservation of current is a universal law.Most models of chemical reactions do not conserve current and need to be changed to do so. On the macroscopic scale of life, conservation of current necessarily links far spread boundaries to each other, correlating inputs and outputs, and thereby creating devices.We suspect that correlations created by displacement current link all scales and allow atoms to control the machines and organisms of life. Conservation of current has a special role in our lives and life, as well as in physics. We believe models, simulations, and computations should conserve current on all scales, as accurately as possible, because physics conserves current that way. We believe models will be much more successful if they conserve current at every level of resolution, the way physics does.We surely need successful models as we try to control macroscopic functions by atomic interventions, in technology, life, and medicine. Maxwell’s displacement current lets us see stars. We hope it will help us see how atoms control life.View the article as published at https://www.degruyter.com/view/j/mlbmb.2017.5.issue-1/mlbmb-2017-0006/mlbmb-2017-0006.xm
A stability criterion for high-frequency oscillations
We show that a simple Levi compatibility condition determines stability of
WKB solutions to semilinear hyperbolic initial-value problems issued from
highly-oscillating initial data with large amplitudes. The compatibility
condition involves the hyperbolic operator, the fundamental phase associated
with the initial oscillation, and the semilinear source term; it states roughly
that hyperbolicity is preserved around resonances.
If the compatibility condition is satisfied, the solutions are defined over
time intervals independent of the wavelength, and the associated WKB solutions
are stable under a large class of initial perturbations. If the compatibility
condition is not satisfied, resonances are exponentially amplified, and
arbitrarily small initial perturbations can destabilize the WKB solutions in
small time.
The amplification mechanism is based on the observation that in frequency
space, resonances correspond to points of weak hyperbolicity. At such points,
the behavior of the system depends on the lower order terms through the
compatibility condition.
The analysis relies, in the unstable case, on a short-time Duhamel
representation formula for solutions of zeroth-order pseudo-differential
equations.
Our examples include coupled Klein-Gordon systems, and systems describing
Raman and Brillouin instabilities.Comment: Final version, to appear in M\'em. Soc. Math. F
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