This paper shows analytically and numerically that a vortex plate coupled to a neutral density filter can deliver a true optical spatial derivative when placed at the focal plane of a 2
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f lens pair. This technique turns any intensity or phase variations of coherent light into an intensity that is proportional to the square of the norm of the initial variation gradient. Since the optical derivative removes the uniform background, it is possible to measure the mode numbers of spatial phase gradients or fluctuations optically, without using any interferometer. 
Radial basis functions are typically used when discretization schemes require inhomogeneous node distributions. While spawning from a desire to interpolate functions on a random set of nodes, they have found successful applications in solving many types of differential equations. However, the weights of the interpolated solution, used in the linear superposition of basis functions to interpolate the solution, and the actual value of the solution are completely different. In fact, these weights mix the value of the solution with the geometrical location of the nodes used to discretize the equation. In this paper, we used nodal radial basis functions, which are interpolants of the impulse function at each node inside the domain. This transformation allows to solve a linear hyperbolic partial differential equation using series expansion rather than the explicit computation of a matrix inverse. This transformation effectively yields an implicit solver which only requires the multiplication of vectors with matrices. Because the solver requires neither matrix inverse nor matrixmatrix products, this approach is numerically more stable and reduces the error by at least two orders of magnitude, compared to solvers using radial basis functions directly. Further, boundary conditions are integrated directly inside the solver, at no extra cost. The method is locally conservative, keeping the error virtually constant throughout the computation.more » « less

Abstract Gravitational lensing by massive objects along the line of sight to the source causes distortions to gravitational wave (GW) signals; such distortions may reveal information about fundamental physics, cosmology, and astrophysics. In this work, we have extended the search for lensing signatures to all binary black hole events from the third observing run of the LIGOVirgo network. We search for repeated signals from strong lensing by (1) performing targeted searches for subthreshold signals, (2) calculating the degree of overlap among the intrinsic parameters and sky location of pairs of signals, (3) comparing the similarities of the spectrograms among pairs of signals, and (4) performing dualsignal Bayesian analysis that takes into account selection effects and astrophysical knowledge. We also search for distortions to the gravitational waveform caused by (1) frequencyindependent phase shifts in strongly lensed images, and (2) frequencydependent modulation of the amplitude and phase due to point masses. None of these searches yields significant evidence for lensing. Finally, we use the nondetection of GW lensing to constrain the lensing rate based on the latest mergerrate estimates and the fraction of dark matter composed of compact objects.
Free, publiclyaccessible full text available July 31, 2025 
Abstract We report the observation of a coalescing compact binary with component masses 2.5–4.5
M _{⊙}and 1.2–2.0M _{⊙}(all measurements quoted at the 90% credible level). The gravitationalwave signal GW230529_181500 was observed during the fourth observing run of the LIGO–Virgo–KAGRA detector network on 2023 May 29 by the LIGO Livingston observatory. The primary component of the source has a mass less than 5M _{⊙}at 99% credibility. We cannot definitively determine from gravitationalwave data alone whether either component of the source is a neutron star or a black hole. However, given existing estimates of the maximum neutron star mass, we find the most probable interpretation of the source to be the coalescence of a neutron star with a black hole that has a mass between the most massive neutron stars and the least massive black holes observed in the Galaxy. We provisionally estimate a merger rate density of for compact binary coalescences with properties similar to the source of GW230529_181500; assuming that the source is a neutron star–black hole merger, GW230529_181500like sources may make up the majority of neutron star–black hole coalescences. The discovery of this system implies an increase in the expected rate of neutron star–black hole mergers with electromagnetic counterparts and provides further evidence for compact objects existing within the purported lower mass gap. ${55}_{47}^{+127}\phantom{\rule{0.25em}{0ex}}{\mathrm{Gpc}}^{3}\phantom{\rule{0.25em}{0ex}}{\mathrm{yr}}^{1}$Free, publiclyaccessible full text available July 26, 2025