134,387 research outputs found
On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields
Recently, Gupta et.al. [GKKS2013] proved that over Q any -variate
and -degree polynomial in VP can also be computed by a depth three
circuit of size . Over fixed-size
finite fields, Grigoriev and Karpinski proved that any
circuit that computes (or ) must be of size
[GK1998]. In this paper, we prove that over fixed-size finite fields, any
circuit for computing the iterated matrix multiplication
polynomial of generic matrices of size , must be of size
. The importance of this result is that over fixed-size
fields there is no depth reduction technique that can be used to compute all
the -variate and -degree polynomials in VP by depth 3 circuits of
size . The result [GK1998] can only rule out such a possibility
for depth 3 circuits of size .
We also give an example of an explicit polynomial () in
VNP (not known to be in VP), for which any circuit computing
it (over fixed-size fields) must be of size . The
polynomial we consider is constructed from the combinatorial design. An
interesting feature of this result is that we get the first examples of two
polynomials (one in VP and one in VNP) such that they have provably stronger
circuit size lower bounds than Permanent in a reasonably strong model of
computation.
Next, we prove that any depth 4
circuit computing
(over any field) must be of size . To the best of our knowledge, the polynomial is the
first example of an explicit polynomial in VNP such that it requires
size depth four circuits, but no known matching
upper bound
Upper limit on spontaneous supercurrents in SrRuO
It is widely believed that the perovskite SrRuO is an unconventional
superconductor with broken time reversal symmetry. It has been predicted that
superconductors with broken time reversal symmetry should have spontaneously
generated supercurrents at edges and domain walls. We have done careful imaging
of the magnetic fields above SrRuO single crystals using scanning Hall
bar and SQUID microscopies, and see no evidence for such spontaneously
generated supercurrents. We use the results from our magnetic imaging to place
upper limits on the spontaneously generated supercurrents at edges and domain
walls as a function of domain size. For a single domain, this upper limit is
below the predicted signal by two orders of magnitude. We speculate on the
causes and implications of the lack of large spontaneous supercurrents in this
very interesting superconducting system.Comment: 9 page
Microlensing toward crowded fields: Theory and applications to M31
We present a comprehensive treatment of the pixel-lensing theory and apply it
to lensing experiments and their results toward M31. Using distribution
functions for the distances, velocities, masses, and luminosities of stars, we
derive lensing event rates as a function of the event observables. In contrast
to the microlensing regime, in the pixel-lensing regime (crowded or unresolved
sources) the observables are the maximum excess flux of the source above a
background and the full width at half-maximum (FWHM) time of the event. To
calculate lensing event distribution functions depending on these observables
for the specific case of M31, we use data from the literature to construct a
model of M31, reproducing consistently photometry, kinematics and stellar
population. We predict the halo- and self-lensing event rates for bulge and
disk stars in M31 and treat events with and without finite source signatures
separately. We use the M31 photon noise profile and obtain the event rates as a
function of position, field of view, and S/N threshold at maximum
magnification. We calculate the expected rates for WeCAPP and for a potential
Advanced Camera for Surveys (ACS) lensing campaign. The detection of two events
with a peak signal-to-noise ratio larger than 10 and a timescale larger than 1
day in the WeCAPP 2000/2001 data is in good agreement with our theoretical
calculations. We investigate the luminosity function of lensed stars for noise
characteristics of WeCAPP and ACS. For the pixel-lensing regime, we derive the
probability distribution for the lens masses in M31 as a function of the FWHM
timescale, flux excess and color, including the errors of these observables.Comment: 45 pages, 27 figures LaTeX; corrected typos; published in the
Astrophysical Journal Supplemen
The Nonlinear Meissner Effect in Unconventional Superconductors
We examine the long-wavelength current response in anisotropic
superconductors and show how the field-dependence of the Meissner penetration
length can be used to detect the structure of the order parameter. Nodes in the
excitation gap lead to a nonlinear current-velocity constitutive equation at
low temperatures which is distinct for each symmetry class of the order
parameter. The effective Meissner penetration length is linear in and
exhibits a characteristic anisotropy for fields in the -plane that is
determined by the positions of the nodes in momentum space. The nonlinear
current-velocity relation also leads to an intrinsic magnetic torque for
in-plane fields that are not parallel to a nodal or antinodal direction. The
torque scales as for and has a characteristic angular
dependence. We analyze the effects of thermal excitations, impurity scattering
and geometry on the current response of a superconductor, and
discuss our results in light of recent measurements of the low-temperature
penetration length and in-plane magnetization of single-crystals of
and .Comment: 30 pages, RevTeX file with 16 postscript figures. Submitted to Phys.
Rev.
Unsteady undular bores in fully nonlinear shallow-water theory
We consider unsteady undular bores for a pair of coupled equations of
Boussinesq-type which contain the familiar fully nonlinear dissipationless
shallow-water dynamics and the leading-order fully nonlinear dispersive terms.
This system contains one horizontal space dimension and time and can be
systematically derived from the full Euler equations for irrotational flows
with a free surface using a standard long-wave asymptotic expansion.
In this context the system was first derived by Su and Gardner. It coincides
with the one-dimensional flat-bottom reduction of the Green-Naghdi system and,
additionally, has recently found a number of fluid dynamics applications other
than the present context of shallow-water gravity waves. We then use the
Whitham modulation theory for a one-phase periodic travelling wave to obtain an
asymptotic analytical description of an undular bore in the Su-Gardner system
for a full range of "depth" ratios across the bore. The positions of the
leading and trailing edges of the undular bore and the amplitude of the leading
solitary wave of the bore are found as functions of this "depth ratio". The
formation of a partial undular bore with a rapidly-varying finite-amplitude
trailing wave front is predicted for ``depth ratios'' across the bore exceeding
1.43. The analytical results from the modulation theory are shown to be in
excellent agreement with full numerical solutions for the development of an
undular bore in the Su-Gardner system.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9
figure
Flood propagation modelling with the Local Inertia Approximation: theoretical and numerical analysis of its physical limitations
Attention of the researchers has increased towards a simplification of the
complete Shallow water Equations called the Local Inertia Approximation (LInA),
which is obtained by neglecting the advection term in the momentum conservation
equation. In the present paper it is demonstrated that a shock is always
developed at moving wetting-drying frontiers, and this justifies the study of
the Riemann problem on even and uneven beds. In particular, the general exact
solution for the Riemann problem on horizontal frictionless bed is given,
together with the exact solution of the non-breaking wave propagating on
horizontal bed with friction, while some example solution is given for the
Riemann problem on discontinuous bed. From this analysis, it follows that
drying of the wet bed is forbidden in the LInA model, and that there are
initial conditions for which the Riemann problem has no solution on smoothly
varying bed. In addition, propagation of the flood on discontinuous sloping bed
is impossible if the bed drops height have the same order of magnitude of the
moving-frontier shock height. Finally, it is found that the conservation of the
mechanical energy is violated. It is evident that all these findings pose a
severe limit to the application of the model. The numerical analysis has proven
that LInA numerical models may produce numerical solutions, which are
unreliable because of mere algorithmic nature, also in the case that the LInA
mathematical solutions do not exist. The applicability limits of the LInA model
are discouragingly severe, even if the bed elevation varies continuously. More
important, the non-existence of the LInA solution in the case of discontinuous
topography and the non-existence of receding fronts radically question the
viability of the LInA model in realistic cases. It is evident that classic SWE
models should be preferred in the majority of the practical applications
Impact of the Casimir-Polder Potential and Johnson Noise on Bose-Einstein Condensate Stability near Surfaces
We investigate the stability of magnetically trapped atomic Bose-Einstein
condensates and thermal clouds near the transition temperature at small
distances 0.5 microns < d < 10 microns from a microfabricated silicon chip. For
a 2 microns thick copper film the trap lifetime is limited by Johnson-noise
induced currents and falls below 1 s at a distance of 4 microns. A dielectric
surface does not adversely affect the sample until the attractive
Casimir-Polder potential significantly reduces the trap depth.Comment: 4 pages, 5 figures, and submitted to Physical Review Letter
- …