$pi$N two-body scattering data. I. A user's guide to the Lovelace-- Almehed data tape

Direct electron density determination from SAXS data opens up new opportunities. The ability to model density at high resolution and the implicit direct estimation of solvent terms such as the hydration shell may enable high-resolution wide angle scattering data to be used to calculate density when combined with additional structural information. Other diffraction methods that do not measure three-dimensional intensities, such as fiber diffraction, may also be able to take advantage of iterative structure factor retrieval. While the ability to reconstruct electron density ab initio is a major breakthrough in the field of solution scattering, the potential of the technique has yet to be fully uncovered. Additional structural information from techniques such as crystallography, NMR, and electron microscopy and density modification procedures can now be integrated to perform advanced modeling of the electron density function at high resolution, pushing the boundaries of solution scattering further than ever before.

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Quantities Directly Measurable by Scattering

10.1093/oso/9780199670871.003.0003 ◽

2018 ◽

Author(s):

Eaton E. Lattman ◽

Thomas D. Grant ◽

Edward H. Snell

Keyword(s):

Molecular Weight ◽

Particle Shape ◽

Particle Surface ◽

Scattering Data ◽

Radius Of Gyration ◽

Particle Volume ◽

Particle Surface Area ◽

Scattering Angle ◽

Particle Dimension ◽

Maximum Particle

In this chapter we note that solution scattering data can be divided into four regions. At zero scattering angle, the scattering provides information on molecular weight of the particle in solution. Beyond that, the scattering is influenced by the radius of gyration. As the scattering angle increases, the scattering is influenced by the particle shape, and finally by the interface with the particle and the solution. There are a number of important invariants that can be calculated directly from the data including molecular mass, radius of gyration, Porod invariant, particle volume, maximum particle dimension, particle surface area, correlation length, and volume of correlation. The meaning of these is described in turn along with their mathematical derivations.

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Spin effects in the effective field theory approach to Post-Minkowskian conservative dynamics

Journal of High Energy Physics ◽

10.1007/jhep06(2021)012 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Zhengwen Liu ◽

Rafael A. Porto ◽

Zixin Yang

Keyword(s):

Field Theory ◽

Effective Field Theory ◽

Scattering Problem ◽

Finite Size ◽

Effective Field ◽

Compact Objects ◽

Scattering Data ◽

Theory Approach ◽

Spin Effects ◽

Radial Action

Abstract Building upon the worldline effective field theory (EFT) formalism for spinning bodies developed for the Post-Newtonian regime, we generalize the EFT approach to Post-Minkowskian (PM) dynamics to include rotational degrees of freedom in a manifestly covariant framework. We introduce a systematic procedure to compute the total change in momentum and spin in the gravitational scattering of compact objects. For the special case of spins aligned with the orbital angular momentum, we show how to construct the radial action for elliptic-like orbits using the Boundary-to-Bound correspondence. As a paradigmatic example, we solve the scattering problem to next-to-leading PM order with linear and bilinear spin effects and arbitrary initial conditions, incorporating for the first time finite-size corrections. We obtain the aligned-spin radial action from the resulting scattering data, and derive the periastron advance and binding energy for circular orbits. We also provide the (square of the) center-of-mass momentum to $$ \mathcal{O}\left({G}^2\right) $$ O G 2 , which may be used to reconstruct a Hamiltonian. Our results are in perfect agreement with the existent literature, while at the same time extend the knowledge of the PM dynamics of compact binaries at quadratic order in spins.

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Structure of the amorphous titania precursor phase of N-doped photocatalysts

RSC Advances ◽

10.1039/d0ra08886b ◽

2021 ◽

Vol 11 (15) ◽

pp. 8619-8627

Author(s):

I. E. Grey ◽

P. Bordet ◽

N. C. Wilson

Keyword(s):

Thermal Analyses ◽

Real Space ◽

Ammonia Solution ◽

Scattering Data ◽

X Ray ◽

X Ray Scattering ◽

Amorphous Titania ◽

Precursor Phase ◽

Titania Precursor ◽

Titanyl Sulphate

Amorphous titania samples prepared by ammonia solution neutralization of titanyl sulphate have been characterized by chemical and thermal analyses, and with reciprocal-space and real-space fitting of wide-angle synchrotron X-ray scattering data.

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Stability of the Non-abelian X-ray Transform in Dimension $$\ge 3$$

Journal of Geometric Analysis ◽

10.1007/s12220-021-00679-0 ◽

2021 ◽

Author(s):

Jan Bohr

Keyword(s):

Stability Estimate ◽

Scattering Data ◽

Regularity Conditions ◽

Uniform Bounds ◽

Ray Transform ◽

Shallow Layer ◽

Matrix Potential ◽

Novel Method ◽

Statistical Consistency ◽

Appl Math

AbstractNon-abelian X-ray tomography seeks to recover a matrix potential $$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$ Φ : M → C m × m in a domain M from measurements of its so-called scattering data $$C_\Phi $$ C Φ at $$\partial M$$ ∂ M . For $$\dim M\ge 3$$ dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map $$\Phi \mapsto C_\Phi $$ Φ ↦ C Φ was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for $$\dim M =2$$ dim M = 2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below $$\partial M$$ ∂ M and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular, proving uniform bounds on layer depth and stability constants.

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Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions

Advances in Difference Equations ◽

10.1186/s13662-021-03374-0 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Bo Xu ◽

Yufeng Zhang ◽

Sheng Zhang

Keyword(s):

Spectral Problem ◽

Fractional Order ◽

Point Of View ◽

Adjoint Equations ◽

Scattering Data ◽

Bilinear Method ◽

Soliton Equation ◽

Scattering Transform ◽

First Time ◽

Hirota Bilinear

AbstractAblowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies.

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