801 research outputs found
On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body
The interface problem describing the scattering of time-harmonic
electromagnetic waves by a dielectric body is often formulated as a pair of
coupled boundary integral equations for the electric and magnetic current
densities on the interface . In this paper, following an idea developed
by Kleinman and Martin \cite{KlMa} for acoustic scattering problems, we
consider methods for solving the dielectric scattering problem using a single
integral equation over for a single unknown density. One knows that
such boundary integral formulations of the Maxwell equations are not uniquely
solvable when the exterior wave number is an eigenvalue of an associated
interior Maxwell boundary value problem. We obtain four different families of
integral equations for which we can show that by choosing some parameters in an
appropriate way, they become uniquely solvable for all real frequencies. We
analyze the well-posedness of the integral equations in the space of finite
energy on smooth and non-smooth boundaries
Shape derivatives of boundary integral operators in electromagnetic scattering. Part I: Shape differentiability of pseudo-homogeneous boundary integral operators
In this paper we study the shape differentiability properties of a class of
boundary integral operators and of potentials with weakly singular
pseudo-homogeneous kernels acting between classical Sobolev spaces, with
respect to smooth deformations of the boundary. We prove that the boundary
integral operators are infinitely differentiable without loss of regularity.
The potential operators are infinitely shape differentiable away from the
boundary, whereas their derivatives lose regularity near the boundary. We study
the shape differentiability of surface differential operators. The shape
differentiability properties of the usual strongly singular or hypersingular
boundary integral operators of interest in acoustic, elastodynamic or
electromagnetic potential theory can then be established by expressing them in
terms of integral operators with weakly singular kernels and of surface
differential operators
Shape derivatives of boundary integral operators in electromagnetic scattering. Part II : Application to scattering by a homogeneous dielectric obstacle
We develop the shape derivative analysis of solutions to the problem of
scattering of time-harmonic electromagnetic waves by a bounded penetrable
obstacle. Since boundary integral equations are a classical tool to solve
electromagnetic scattering problems, we study the shape differentiability
properties of the standard electromagnetic boundary integral operators. The
latter are typically bounded on the space of tangential vector fields of mixed
regularity TH\sp{-1/2}(\Div_{\Gamma},\Gamma). Using Helmholtz decomposition,
we can base their analysis on the study of pseudo-differential integral
operators in standard Sobolev spaces, but we then have to study the G\^ateaux
differentiability of surface differential operators. We prove that the
electromagnetic boundary integral operators are infinitely differentiable
without loss of regularity. We also give a characterization of the first shape
derivative of the solution of the dielectric scattering problem as a solution
of a new electromagnetic scattering problem.Comment: arXiv admin note: substantial text overlap with arXiv:1002.154
Shape derivatives of boundary integral operators in electromagnetic scattering
We develop the shape derivative analysis of solutions to the problem of
scattering of time-harmonic electromagnetic waves by a bounded penetrable
obstacle. Since boundary integral equations are a classical tool to solve
electromagnetic scattering problems, we study the shape differentiability
properties of the standard electromagnetic boundary integral operators. Using
Helmholtz decomposition, we can base their analysis on the study of scalar
integral operators in standard Sobolev spaces, but we then have to study the
G\^ateaux differentiability of surface differential operators. We prove that
the electromagnetic boundary integral operators are infinitely differentiable
without loss of regularity and that the solutions of the scattering problem are
infinitely shape differentiable away from the boundary of the obstacle, whereas
their derivatives lose regularity on the boundary. We also give a
characterization of the first shape derivative as a solution of a new
electromagnetic scattering problem
Border collision bifurcations of stroboscopic maps in periodically driven spiking models
In this work we consider a general non-autonomous hybrid system based on the
integrate-and-fire model, widely used as simplified version of neuronal models
and other types of excitable systems. Our unique assumption is that the system
is monotonic, possesses an attracting subthreshold equilibrium point and is
forced by means of periodic pulsatile (square wave) function.\\ In contrast to
classical methods, in our approach we use the stroboscopic map (time- return
map) instead of the so-called firing-map. It becomes a discontinuous map
potentially defined in an infinite number of partitions. By applying theory for
piecewise-smooth systems, we avoid relying on particular computations and we
develop a novel approach that can be easily extended to systems with other
topologies (expansive dynamics) and higher dimensions.\\ More precisely, we
rigorously study the bifurcation structure in the two-dimensional parameter
space formed by the amplitude and the duty cycle of the pulse. We show that it
is covered by regions of existence of periodic orbits given by period adding
structures. They do not only completely describe all the possible spiking
asymptotic dynamics but also the behavior of the firing rate, which is a
devil's staircase as a function of the parameters
Convergence of a linearly transformed particle method for aggregation equations
We study a linearly transformed particle method for the aggregation equation
with smooth or singular interaction forces. For the smooth interaction forces,
we provide convergence estimates in and norms depending on the
regularity of the initial data. Moreover, we give convergence estimates in
bounded Lipschitz distance for measure valued solutions. For singular
interaction forces, we establish the convergence of the error between the
approximated and exact flows up to the existence time of the solutions in norm
Bourdieu, networks, and movements: Using the concepts of habitus, field and capital to understand a network analysis of gender differences in undergraduate physics
Current trends suggest that significant gender disparities exist within
Science, Technology, Engineering, and Mathematics (STEM) education at
university, with female students being underrepresented in physics, but more
equally represented in life sciences (e.g., biology, medicine). To understand
these trends, it is important to consider the context in which students make
decisions about which university courses to enrol in. The current study seeks
to investigate gender differences in STEM through a unique approach that
combines network analysis of student enrolment data with an interpretive lens
based on the sociological theory of Pierre Bourdieu. We generate a network of
courses taken by around 9000 undergraduate physics students (from 2009 to 2014)
to quantify Bourdieu's concept of field. We explore the properties of this
network to investigate gender differences in transverse movements (between
different academic fields) and vertical movements (changes in students'
achievement rankings within a field). Our findings indicate that female
students are more likely to make transverse movements into life science fields.
We also find that university physics does a poor job in attracting high
achieving students, and especially high achieving female students. Of the
students who do choose to study physics, low achieving female students are less
likely to continue than their male counterparts. The results and implications
are discussed in the context of Bourdieu's theory, and previous research. We
argue that in order to remove constraints on female student's study choices,
the field of physics needs to provide a culture in which all students feel like
they belong.Comment: 23 pages, 6 figures, 1 tabl
Uniform convergence of a linearly transformed particle method for the Vlasov-Poisson system
International audienceA particle method with linear transformation of the particle shape functions is studied for the 1d-1v Vlasov-Poisson equation, and a priori error estimates are proven which show that the approximated densities converge in the uniform norm. When compared to standard fixed-shape particle methods, the present approach can be seen as a way to gain one order in the convergence rate of the particle trajectories at the cost of linearly transforming each particle shape. It also allows to compute strongly convergent densities with particles that overlap in a bounded way
Mixed-mode oscillations in a multiple time scale phantom bursting system
In this work we study mixed mode oscillations in a model of secretion of GnRH
(Gonadotropin Releasing Hormone). The model is a phantom burster consisting of
two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The
forcing system (Regulator) evolves on the slowest scale and acts by moving the
slow nullcline of the forced system (Secretor). There are three modes of
dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady
state) and small oscillations related to the passage of the slow nullcline
through a fold point of the fast nullcline. We derive a variety of reductions,
taking advantage of the mentioned features of the system. We obtain two
results; one on the local dynamics near the fold in the parameter regime
corresponding to the presence of small oscillations and the other on the global
dynamics, more specifically on the existence of an attracting limit cycle. Our
local result is a rigorous characterization of small canards and sectors of
rotation in the case of folded node with an additional time scale, a feature
allowing for a clear geometric argument. The global result gives the existence
of an attracting unique limit cycle, which, in some parameter regimes, remains
attracting and unique even during passages through a canard explosion.Comment: 38 pages, 16 figure
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