52 research outputs found
Minimizing optimal transport for functions with fixed-size nodal sets
Consider the class of zero-mean functions with fixed and
norms and exactly nodal points. Which functions minimize
, the Wasserstein distance between the measures whose densities
are the positive and negative parts? We provide a complete solution to this
minimization problem on the line and the circle, which provides sharp constants
for previously proven ``uncertainty principle''-type inequalities, i.e., lower
bounds on . We further show that, while such
inequalities hold in many metric measure spaces, they are no longer sharp when
the non-branching assumption is violated; indeed, for metric star-graphs, the
optimal lower bound on is not inversely proportional to the size
of the nodal set, . Based on similar reductions, we make connections between
the analogous problem of minimizing for defined on
with an equivalent optimal domain partition
problem
Probing the Magnetic Field Structure in Gamma-Ray Bursts through Dispersive Plasma Effects on the Afterglow Polarization
(Abr) The origin and structure of magnetic fields in Gamma-Ray Burst (GRB)
fireball plasmas are two of the most important open questions in all GRB
models. We show that the structure and strength of the magnetic field may be
constrained by radio and IR observations of the early afterglow, where plasma
effects on the polarization of propagating radiation are significant. We
calculate these propagation effects for cold and relativistic plasmas, and find
that in the presence of a uniform equipartition field the degree of linear
polarization is suppressed, and circular polarization prevails at low
frequencies, nu < 1-3 GHz, (2x10^11 Hz < nu < few x 10^14 Hz) in the forward
(reverse) shock. At higher frequencies linear polarization dominates. At the
frequency of the transition between circular and linear polarization, the net
level of polarization is minimal, ~10-20%. These features are nearly
independent of the circumburst density. The transition frequency is smaller by
a factor of ~10 when the uniform field is much weaker than equipartition. The
dependence of these results on viewing geometry, outflow collimation and
magnetic field orientation is discussed. When the configuration of the field is
entangled over length scales much smaller than the extent of the emitting
plasma, the aforementioned effects should not be observed and a linear
polarization at the few % level is expected. Polarimetric observations during
the early afterglow, and particularly of the reverse shock emission, may
therefore place strong constraints on the structure and strength of the
magnetic field within the fireball plasma.Comment: 12 pages, 6 figures. Accepted for publication in ApJ. Revised version
includes improved discussion of viewing and fireball geometry, with
implications to resulting polarizatio
Relativistic Radiation Mediated Shocks
The structure of relativistic radiation mediated shocks (RRMS) propagating
into a cold electron-proton plasma is calculated and analyzed. A qualitative
discussion of the physics of relativistic and non relativistic shocks,
including order of magnitude estimates for the relevant temperature and length
scales, is presented. Detailed numerical solutions are derived for shock
Lorentz factors in the range , using a novel
iteration technique solving the hydrodynamics and radiation transport equations
(the protons, electrons and positrons are argued to be coupled by collective
plasma processes and are treated as a fluid). The shock transition
(deceleration) region, where the Lorentz factor drops from to , is characterized by high plasma temperatures and highly anisotropic radiation, with characteristic
shock-frame energy of upstream and downstream going photons of a few~ and , respectively.Photon scattering is dominated
by e pairs, with pair to proton density ratio reaching
. The width of the deceleration region, in terms of
Thomson optical depths for upstream going photons, is large,
( neglecting the contribution of
pairs) due to Klein Nishina suppression of the scattering cross section. A high
energy photon component, narrowly beamed in the downstream direction, with a
nearly flat power-law like spectrum, , and an energy
cutoff at carries a fair fraction of the energy flux
at the end of the deceleration region. An approximate analytic model of RRMS,
reproducing the main features of the numerical results, is provided
Near invariance of quasi-energy spectrum of Floquet Hamiltonians
The spectral analysis of the unitary monodromy operator, associated with a
time-periodically (paramatrically) forced Schrodinger equation, is a question
of longstanding interest. Here, we consider this question for Hamiltonians of
the form
where is an unperturbed autonomous Hamiltonian, , and
has a period of . In particular, in the small
regime, we seek a comparison between the spectral properties of
the monodromy operator, the one-period flow map associated with the
dynamics, and that of the autonomous (unforced) flow,
. We consider which is
spatially periodic on with respect to a lattice. Using the
decomposition of and into their actions on spaces
(Floquet-Bloch fibers) of pseudo-periodic functions, we establish a near
spectral-invariance property for the monodromy operator, when acting data which
are -localized in energy and quasi-momentum. Our analysis requires
the following steps: (i) spectrally-localized data are approximated by {\it
band-limited (Floquet-Bloch) wavepackets}; (ii) the envelope dynamics of such
wavepackets is well approximated by an effective (homogenized) PDE, and (iii)
an exact invariance property for band-limited Floquet-Bloch wavepackets, which
follows from the effective dynamics. We apply our general results to a number
of periodic Hamiltonians, , of interest in the study of photonic and
quantum materials
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