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1. Monotonicity Under Local Operations: Linear Entropic Formulas

   https://doi.org/10.1109/TIT.2020.2986728  

   Alhejji, Mohammad A.; Smith, Graeme (April 2020, IEEE Transactions on Information Theory)

   All correlation measures, classical and quantum, must be monotonic under local operations. In this paper, we characterize monotonic formulas that are linear combinations of the von Neumann entropies associated with the quantum state of a physical system that has n parts. We show that these formulas form a polyhedral convex cone, which we call the monotonicity cone, and enumerate its facets. We illustrate its structure and prove that it is equivalent to the cone of monotonic formulas implied by strong subadditivity. We explicitly compute its extremal rays for n ≤ 5. We also consider the symmetric monotonicity cone, in which the formulas are required to be invariant under subsystem permutations. We describe this cone fully for all n. We also show that these results hold when states and operations are constrained to be classical. 

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2. Microlocal properties of seven-dimensional lemon and apple Radon transforms with applications in Compton scattering tomography

   https://doi.org/10.1088/1361-6420/ac65ad  

   Webber, James W; Quinto, Eric Todd (May 2022, Inverse Problems)

   Abstract We present a microlocal analysis of two novel Radon transforms of interest in Compton scattering tomography, which map compactly supported L 2 functions to their integrals over seven-dimensional sets of apple and lemon surfaces. Specifically, we show that the apple and lemon transforms are elliptic Fourier integral operators, which satisfy the Bolker condition. After an analysis of the full seven-dimensional case, we focus our attention on n D subsets of apple and lemon surfaces with fixed central axis, where n < 7. Such subsets of surface integrals have applications in airport baggage and security screening. When the data dimensionality is restricted, the apple transform is shown to violate the Bolker condition, and there are artifacts which occur on apple–cylinder intersections. The lemon transform is shown to satisfy the Bolker condition, when the support of the function is restricted to the strip 0 < z < 1 . 

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3. Second moment of the light-cone Siegel transform and applications

   https://doi.org/10.1016/j.aim.2023.109270  

   Kelmer, Dubi; Yu, Shucheng (November 2023, Advances in Mathematics)

   We study the light-cone Siegel transform, transforming functions on the light cone of a rational indefinite quadratic form Q to a function on the homogenous space $$\SO_Q(\Z)\backslash \SO_Q(\R)$$. In particular, we prove a second moment formula for this transform for forms of signature (n+1,1), and show how it can be used for various applications involving counting integer points on the light cone. In particular, we prove some new results on intrinsic Diophantine approximations on ellipsoids as well as on the distribution of values of random linear and quadratic forms on the light cone. 

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4. Development of Abel's inversion method to extract radially resolved optical emission maps from spectral data cubes collected via push-broom hyperspectral imaging with sub-pixel shifting sampling

   https://doi.org/10.1039/C9JA00239A  

   Shi, Songyue; Finch, Kevin; She, Yue; Gamez, Gerardo (January 2020, Journal of Analytical Atomic Spectrometry)

   null (Ed.)

   Optical emission spectroscopy (OES) imaging is often used for diagnostics for better understanding of the underlying mechanisms of plasmas. Typical spectral images, however, contain intensity maps that are integrated along the line-of-sight. A widespread method to extract the radial information is Abel's inversion, but most approaches result in accumulation of error toward the plasma axial position, which is often the region of most interest. Here, a Fourier-transform based Abel's inversion algorithm, which spreads the error evenly across the radial profile, is optimized for OES images collected on a push-broom hyperspectral imaging system (PbHSI). Furthermore, a sub-pixel shifting (SPS) sampling protocol is employed on the PbHSI in the direction of the radial reconstruction to allow improved fidelity from the increased number of data points. The accuracy and fidelity of the protocol are characterized and optimized with a software-based 3-dimensional hyperspectral model datacube. A systematic study of the effects of varying levels of representative added noise, different noise filters, number of data points and cosine expansions used in the inversion, as well as the spatial intensity distribution shapes of the radial profile are presented. A 3D median noise filter with 3-pixel radius, a minimum of 50 points and 8 cosine expansions is needed to keep the relative root mean squared error (rRMSE) <8%. The optimized protocol is implemented for the first time on OES images of a micro-capillary dielectric barrier discharge (μDBD) source obtained via SPS PbHSI system and the extracted radial emission of different plasma species (He, N 2 , N 2 + ) are shown. 

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5. Set superpartitions and superspace duality modules

   https://doi.org/10.1017/fms.2022.90  

   Rhoades, Brendon; Wilson, Andrew Timothy (January 2022, Forum of Mathematics, Sigma)

   Abstract The superspace ring $$\Omega _n$$ is a rank n polynomial ring tensored with a rank n exterior algebra. Using an extension of the Vandermonde determinant to $$\Omega _n$$ , the authors previously defined a family of doubly graded quotients $${\mathbb {W}}_{n,k}$$ of $$\Omega _n$$ , which carry an action of the symmetric group $${\mathfrak {S}}_n$$ and satisfy a bigraded version of Poincaré Duality. In this paper, we examine the duality modules $${\mathbb {W}}_{n,k}$$ in greater detail. We describe a monomial basis of $${\mathbb {W}}_{n,k}$$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called ordered set superpartitions . These are ordered set partitions $$(B_1 \mid \cdots \mid B_k)$$ of $$\{1,\dots ,n\}$$ in which the nonminimal elements of any block $$B_i$$ may be barred or unbarred. 

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