American Institute of Mathematics

Context. Hyperspectral images are data cubes with two spatial dimensions and a third spectral dimension, providing a spectrum for each pixel, and thus allowing the mapping of extended sources’ physical properties. Aims. In this article, we present the Semi-blind Unmixing with Sparsity for Hyperspectral Images (SUSHI), an algorithm for non-stationary unmixing of hyperspectral images with spatial regularization of spectral parameters. The method allows for the disentangling of physical components without the assumption of a unique spectrum for each component. Thus, unlike most source separation methods used in astrophysics, all physical components obtained by SUSHI vary in spectral shape and in amplitude across the data cube. Methods. Non-stationary source separation is an ill-posed inverse problem that needs to be constrained. We achieve this by training a spectral model and applying a spatial regularization constraint on its parameters. For the spectral model, we used an Interpolatory Auto-Encoder, a generative model that can be trained with limited samples. For spatial regularization, we applied a sparsity constraint on the wavelet transform of the model parameter maps. Results. We applied SUSHI to a toy model meant to resemble supernova remnants in X-ray astrophysics, though the method may be used on any extended source with any hyperspectral instrument. We compared this result to the one obtained by a classic 1D fit on each individual pixel. We find that SUSHI obtains more accurate results, particularly when it comes to reconstructing physical parameters. We then applied SUSHI to real X-ray data from the supernova remnant Cassiopeia A and to the Crab Nebula. The results obtained are realistic and in accordance with past findings but have a much better spatial resolution. Thanks to spatial regularization, SUSHI can obtain reliable physical parameters at fine scales that are out of reach for pixel-by-pixel methods.

We prove that an innocent looking inequality implies the Riemann Hypothesis and show a way to approach this inequality through sums of Legendre symbols. The "1/4" in Theorem 1 can be replaced by any positive constant. So the real issue is trying to prove that f (x) > 0 for small positive x. Research supported by an FRG grant from NSF. 1 λ is completely multiplicative and takes the value −1 on primes so that λ(p e1 1 . . . p er r ) = (−1) e1+ • ••+er .

Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least 67.9% of the nontrivial zeros are simple. Here we obtain an unconditional form of Montgomery's theorem and show how to apply it to prove the following result on simple zeros: Assuming all the zeros ρ = β + iγ of the Riemann zeta-function such that T 3/8 < γ ≤ T satisfy |β − 1/2| < 1/(2 log T ), then, as T tends to infinity, at least 61.7% of these zeros are simple. The method of proof neither requires nor provides any information on whether any of these zeros are on or not on the critical line where β = 1/2. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis.

ABSTRACTBackgroundThe coronavirus disease 2019 (COVID-19) pandemic has resulted in severe shortages of personal protective equipment (PPE) necessary to protect front-line healthcare personnel. These shortages underscore the urgent need for simple, efficient, and inexpensive methods to decontaminate SARS-CoV-2-exposed PPE enabling safe reuse of masks and respirators. Efficient decontamination must be available not only in low-resourced settings, but also in well-resourced settings affected by PPE shortages. Methylene blue (MB) photochemical treatment, hitherto with many clinical applications including those used to inactivate virus in plasma, presents a novel approach for widely applicable PPE decontamination. Dry heat (DH) treatment is another potential low-cost decontamination method.MethodsMB and light (MBL) and DH treatments were used to inactivate coronavirus on respirator and mask material. We tested three N95 filtering facepiece respirators (FFRs), two medical masks (MMs), and one cloth community mask (CM). FFR/MM/CM materials were inoculated with SARS-CoV-2 (a Betacoronavirus), murine hepatitis virus (MHV) (a Betacoronavirus), or porcine respiratory coronavirus (PRCV) (an Alphacoronavirus), and treated with 10 µM MB followed by 50,000 lux of broad-spectrum light or 12,500 lux of red light for 30 minutes, or with 75°C DH for 60 minutes. In parallel, we tested respirator and mask integrity using several standard methods and compared to the FDA-authorized vaporized hydrogen peroxide plus ozone (VHP+O3) decontamination method. Intact FFRs/MMs/CM were subjected to five cycles of decontamination (5CD) to assess integrity using International Standardization Organization (ISO), American Society for Testing and Materials (ASTM) International, National Institute for Occupational Safety and Health (NIOSH), and Occupational Safety and Health Administration (OSHA) test methods.FindingsOverall, MBL robustly and consistently inactivated all three coronaviruses with at least a 4-log reduction. DH yielded similar results, with the exception of MHV, which was only reduced by 2-log after treatment. FFR/MM integrity was maintained for 5 cycles of MBL or DH treatment, whereas one FFR failed after 5 cycles of VHP+O3. Baseline performance for the CM was variable, but reduction of integrity was minimal.InterpretationMethylene blue with light and DH treatment decontaminated masks and respirators by inactivating three tested coronaviruses without compromising integrity through 5CD. MBL decontamination of masks is effective, low-cost and does not require specialized equipment, making it applicable in all-resource settings. These attractive features support the utilization and continued development of this novel PPE decontamination method.

We obtain an asymptotic formula, as $n\to\infty$, for the monomer-monomer correlation function $K_2(x,y)$ in the classical dimer model on a triangular lattice, with the horizontal and vertical weights $w_h=w_v=1$ and the diagonal weight $w_d=t>0$, where $x$ and $y$ are sites $n$ spaces apart in adjacent rows. We find that $t_c=\frac{1}{2}$ is a critical value of $t$. We prove that in the subcritical case, $0<t<\frac{1}{2}$, as $n\to\infty$, $K_2(x,y)=K_2(\infty)\left[1-\frac{e^{-n/\xi}}{n}\,\Big(C_1+C_2(-1)^n+\mathcal O(n^{-1})\Big)\right]$, with explicit formulae for $K_2(\infty)$, $\xi$, $C_1$, and $C_2$. In the supercritical case, $\frac{1}{2} < t < 1$, we prove that as $n\to\infty$, $K_2(x,y)=K_2(\infty)\Bigg[1- \frac{e^{-n/\xi}}{n}\, \Big(C_1\cos(\omega n+\varphi_1)+C_2(-1)^n\cos(\omega n+\varphi_2)+ C_3+C_4(-1)^n$ $+\mathcal O(n^{-1})\Big)\Bigg]$, with explicit formulae for $K_2(\infty)$, $\xi$, $\omega$, and $C_1$, $C_2$, $C_3$, $C_4$, $\varphi_1$, $\varphi_2$. The proof is based on an extension of the Borodin-Okounkov-Case-Geronimo formula to block Toeplitz determinants and on an asymptotic analysis of the Fredholm determinants in hand.

Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire graph using the fewest number of initial vertices. Chang et al. introduced $k$-power domination in [Generalized power domination in graphs, {\it Discrete Applied Math.} 160 (2012) 1691-1698] as a generalization of power domination and standard graph domination. Independently, Amos et al. defined $k$-forcing in [Upper bounds on the $k$-forcing number of a graph, {\it Discrete Applied Math.} 181 (2015) 1-10] to generalize zero forcing. In this paper, we combine the study of $k$-forcing and $k$-power domination, providing a new approach to analyze both processes. We give a relationship between the $k$-forcing and the $k$-power domination numbers of a graph that bounds one in terms of the other. We also obtain results using the contraction of subgraphs that allow the parallel computation of $k$-forcing and $k$-power dominating sets.