84 research outputs found
On the Probabilistic Degree of OR over the Reals
We study the probabilistic degree over R of the OR function on n variables. For epsilon in (0,1/3), the epsilon-error probabilistic degree of any Boolean function f:{0,1}^n -> {0,1} over R is the smallest non-negative integer d such that the following holds: there exists a distribution of polynomials Pol in R[x_1,...,x_n] entirely supported on polynomials of degree at most d such that for all z in {0,1}^n, we have Pr_{P ~ Pol}[P(z) = f(z)] >= 1- epsilon. It is known from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that the epsilon-error probabilistic degree of the OR function is at most O(log n * log(1/epsilon)). Our first observation is that this can be improved to O{log (n atop <= log(1/epsilon))}, which is better for small values of epsilon.
In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials P in the support of the distribution Pol have the following special structure: P(x_1,...,x_n) = 1 - prod_{i in [t]} (1- L_i(x_1,...,x_n)), where each L_i(x_1,..., x_n) is a linear form in the variables x_1,...,x_n, i.e., the polynomial 1-P(bar{x}) is a product of affine forms. We show that the epsilon-error probabilistic degree of OR when restricted to polynomials of the above form is Omega(log (n over <= log(1/epsilon))/log^2 (log (n over <= log(1/epsilon))})), thus matching the above upper bound (up to polylogarithmic factors)
On the Probabilistic Degree of OR over the Reals
We study the probabilistic degree over reals of the OR function on
variables. For an error parameter in (0,1/3), the -error
probabilistic degree of any Boolean function over reals is the smallest
non-negative integer such that the following holds: there exists a
distribution of polynomials entirely supported on polynomials of degree at
most such that for all , we have . It is known from the works of Tarui ({Theoret.
Comput. Sci.} 1993) and Beigel, Reingold, and Spielman ({ Proc. 6th CCC} 1991),
that the -error probabilistic degree of the OR function is at most
. Our first observation is that this can be improved
to , which is better for small
values of .
In all known constructions of probabilistic polynomials for the OR function
(including the above improvement), the polynomials in the support of the
distribution have the following special structure:, where each is a linear form in
the variables , i.e., the polynomial is a
product of affine forms. We show that the -error probabilistic degree
of OR when restricted to polynomials of the above form is where . Thus
matching the above upper bound (up to poly-logarithmic factors)
OAM beam generation using all-fiber fused couplers
We demonstrate the orbital angular momentum (OAM) beam generation using an all-fiber fused coupler based on single mode fiber (SMF) and air-core fiber. The fabricated device is directly SMF compatible with ~80% power coupling efficiency
OAM Generation in optical fibre and free space devices
Orbital angular momentum (OAM) beam generation has been investigated using all-fibre and free space configurations. In the first approach, the composite fused coupler is based on a single mode fibre (SMF) and an air-core fibre. The second approach exploits geometrical phase introduced by nanostructuring of silica glass. Both approaches are demonstrated to achieve power coupling efficiencies in excess of 80% at telecom wavelengths
Blip-Up Blip-Down Circular EPI (BUDA-cEPI) for Distortion-Free dMRI with Rapid Unrolled Deep Learning Reconstruction
Purpose: We implemented the blip-up, blip-down circular echo planar imaging
(BUDA-cEPI) sequence with readout and phase partial Fourier to reduced
off-resonance effect and T2* blurring. BUDA-cEPI reconstruction with S-based
low-rank modeling of local k-space neighborhoods (S-LORAKS) is shown to be
effective at reconstructing the highly under-sampled BUDA-cEPI data, but it is
computationally intensive. Thus, we developed an ML-based reconstruction
technique termed "BUDA-cEPI RUN-UP" to enable fast reconstruction.
Methods: BUDA-cEPI RUN-UP - a model-based framework that incorporates
off-resonance and eddy current effects was unrolled through an artificial
neural network with only six gradient updates. The unrolled network alternates
between data consistency (i.e., forward BUDA-cEPI and its adjoint) and
regularization steps where U-Net plays a role as the regularizer. To handle the
partial Fourier effect, the virtual coil concept was also incorporated into the
reconstruction to effectively take advantage of the smooth phase prior, and
trained to predict the ground-truth images obtained by BUDA-cEPI with S-LORAKS.
Results: BUDA-cEPI with S-LORAKS reconstruction enabled the management of
off-resonance, partial Fourier, and residual aliasing artifacts. However, the
reconstruction time is approximately 225 seconds per slice, which may not be
practical in a clinical setting. In contrast, the proposed BUDA-cEPI RUN-UP
yielded similar results to BUDA-cEPI with S-LORAKS, with less than a 5%
normalized root mean square error detected, while the reconstruction time is
approximately 3 seconds.
Conclusion: BUDA-cEPI RUN-UP was shown to reduce the reconstruction time by
~88x when compared to the state-of-the-art technique, while preserving imaging
details as demonstrated through DTI application.Comment: Number: Figures: 8 Tables: 3 References: 7
Daksha: On Alert for High Energy Transients
We present Daksha, a proposed high energy transients mission for the study of
electromagnetic counterparts of gravitational wave sources, and gamma ray
bursts. Daksha will comprise of two satellites in low earth equatorial orbits,
on opposite sides of earth. Each satellite will carry three types of detectors
to cover the entire sky in an energy range from 1 keV to >1 MeV. Any transients
detected on-board will be announced publicly within minutes of discovery. All
photon data will be downloaded in ground station passes to obtain source
positions, spectra, and light curves. In addition, Daksha will address a wide
range of science cases including monitoring X-ray pulsars, studies of
magnetars, solar flares, searches for fast radio burst counterparts, routine
monitoring of bright persistent high energy sources, terrestrial gamma-ray
flashes, and probing primordial black hole abundances through lensing. In this
paper, we discuss the technical capabilities of Daksha, while the detailed
science case is discussed in a separate paper.Comment: 9 pages, 3 figures, 1 table. Additional information about the mission
is available at https://www.dakshasat.in
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