Application of the inhomogeneous Lippmann-Schwinger equation to inverse scattering problems

In this paper we present a hybrid approach to numerically solve two-dimensional electromagnetic inverse scattering problems, whereby the unknown scatterer is hosted by a possibly inhomogeneous background. The approach is `hybrid' in that it merges a qualitative and a quantitative method to optimize the way of exploiting the a priori information on the background within the inversion procedure, thus improving the quality of the reconstruction and reducing the data amount necessary for a satisfactory result. In the qualitative step, this a priori knowledge is utilized to implement the linear sampling method in its near-field formulation for an inhomogeneous background, in order to identify the region where the scatterer is located. On the other hand, the same a priori information is also encoded in the quantitative step by extending and applying the contrast source inversion method to what we call the `inhomogeneous Lippmann-Schwinger equation': the latter is a generalization of the classical Lippmann-Schwinger equation to the case of an inhomogeneous background, and in our paper is deduced from the differential formulation of the direct scattering problem to provide the reconstruction algorithm with an appropriate theoretical basis. Then, the point values of the refractive index are computed only in the region identified by the linear sampling method at the previous step. The effectiveness of this hybrid approach is supported by numerical simulations presented at the end of the paper.Comment: accepted in SIAM Journal on Applied Mathematic

Electron-Electron Bremsstrahlung Emission and the Inference of Electron Flux Spectra in Solar Flares

Although both electron-ion and electron-electron bremsstrahlung contribute to the hard X-ray emission from solar flares, the latter is normally ignored. Such an omission is not justified at electron (and photon) energies above $\sim 300$ keV, and inclusion of the additional electron-electron bremsstrahlung in general makes the electron spectrum required to produce a given hard X-ray spectrum steeper at high energies. Unlike electron-ion bremsstrahlung, electron-electron bremsstrahlung cannot produce photons of all energies up to the maximum electron energy involved. The maximum possible photon energy depends on the angle between the direction of the emitting electron and the emitted photon, and this suggests a diagnostic for an upper cutoff energy and/or for the degree of beaming of the accelerated electrons. We analyze the large event of January 17, 2005 observed by RHESSI and show that the upward break around 400 keV in the observed hard X-ray spectrum is naturally accounted for by the inclusion of electron-electron bremsstrahlung. Indeed, the mean source electron spectrum recovered through a regularized inversion of the hard X-ray spectrum, using a cross-section that includes both electron-ion and electron-electron terms, has a relatively constant spectral index $\delta$ over the range from electron kinetic energy $E = 200$ keV to $E = 1$ MeV. However, the level of detail discernible in the recovered electron spectrum is not sufficient to determine whether or not any upper cutoff energy exists.Comment: 7 pages, 5 figures, submitted to Astrophysical Journa

The sub-arcsecond hard X-ray structure of loop footpoints in a solar flare

The newly developed X-ray visibility forward fitting technique is applied to Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) data of a limb flare to investigate the energy and height dependence on sizes, shapes, and position of hard X-ray chromospheric footpoint sources. This provides information about the electron transport and chromospheric density structure. The spatial distribution of two footpoint X-ray sources is analyzed using PIXON, Maximum Entropy Method, CLEAN and visibility forward fit algorithms at nonthermal energies from $\sim 20$ to $\sim 200$ keV. We report, for the first time, the vertical extents and widths of hard X-ray chromospheric sources measured as a function of energy for a limb event. Our observations suggest that both the vertical and horizontal sizes of footpoints are decreasing with energy. Higher energy emission originates progressively deeper in the chromosphere consistent with downward flare accelerated streaming electrons. The ellipticity of the footpoints grows with energy from $\sim 0.5$ at $\sim 20$ keV to $\sim 0.9$ at $\sim 150$ keV. The positions of X-ray emission are in agreement with an exponential density profile of scale height $\sim 150$~km. The characteristic size of the hard X-ray footpoint source along the limb is decreasing with energy suggesting a converging magnetic field in the footpoint. The vertical sizes of X-ray sources are inconsistent with simple collisional transport in a single density scale height but can be explained using a multi-threaded density structure in the chromosphere.Comment: 7 pages, 7 figures, submitted to Ap

Determination of the Acceleration Region Size in a Loop-structured Solar Flare

In order to study the acceleration and propagation of bremsstrahlung-producing electrons in solar flares, we analyze the evolution of the flare loop size with respect to energy at a variety of times. A GOES M3.7 loop-structured flare starting around 23:55 on 2002 April 14 is studied in detail using \textit{Ramaty High Energy Solar Spectroscopic Imager} (\textit{RHESSI}) observations. We construct photon and mean-electron-flux maps in 2-keV energy bins by processing observationally-deduced photon and electron visibilities, respectively, through several image-processing methods: a visibility-based forward-fit (FWD) algorithm, a maximum entropy (MEM) procedure and the uv-smooth (UVS) approach. We estimate the sizes of elongated flares (i.e., the length and width of flaring loops) by calculating the second normalized moments of the intensity in any given map. Employing a collisional model with an extended acceleration region, we fit the loop lengths as a function of energy in both the photon and electron domains. The resulting fitting parameters allow us to estimate the extent of the acceleration region which is between $\sim 13 \rm{arcsec}$ and $\sim 19 \rm{arcsec}$. Both forward-fit and uv-smooth algorithms provide substantially similar results with a systematically better fit in the electron domain.The consistency of the estimates from these methods provides strong support that the model can reliably determine geometric parameters of the acceleration region. The acceleration region is estimated to be a substantial fraction ($\sim 1/2$) of the loop extent, indicating that this dense flaring loop incorporates both acceleration and transport of electrons, with concurrent thick-target bremsstrahlung emission.Comment: 8 pages, 5 figures, accepted to Astronomy and Astrophysics journa

The effect of wave-particle interactions on low energy cutoffs in solar flare electron spectra

Solar flare hard X-ray spectra from RHESSI are normally interpreted in terms of purely collisional electron beam propagation, ignoring spatial evolution and collective effects. In this paper we present self-consistent numerical simulations of the spatial and temporal evolution of an electron beam subject to collisional transport and beam-driven Langmuir wave turbulence. These wave-particle interactions represent the background plasma's response to the electron beam propagating from the corona to chromosphere and occur on a far faster timescale than coulomb collisions. From these simulations we derive the mean electron flux spectrum, comparable to such spectra recovered from high resolution hard X-rays observations of solar flares with RHESSI. We find that a negative spectral index (i.e. a spectrum that increases with energy), or local minima when including the expected thermal spectral component at low energies, occurs in the standard thick-target model, when coulomb collisions are only considered. The inclusion of wave-particle interactions does not produce a local minimum, maintaining a positive spectral index. These simulations are a step towards a more complete treatment of electron transport in solar flares and suggest that a flat spectrum (spectral index of 0 to 1) down to thermal energies maybe a better approximation instead of a sharp cut-off in the injected electron spectrum.Comment: 6 pages, 5 figures, accepted by ApJ

Go-with-the-flow: Tracking, Analysis and Sonification of Movement and Breathing to Build Confidence in Activity Despite Chronic Pain

Chronic (persistent) pain (CP) affects one in ten adults; clinical resources are insufficient, and anxiety about activity restricts lives. Technological aids monitor activity but lack necessary psychological support. This paper proposes a new sonification framework, Go-with-the-Flow, informed by physiotherapists and people with CP. The framework proposes articulation of user-defined sonified exercise spaces (SESs) tailored to psychological needs and physical capabilities that enhance body and movement awareness to rebuild confidence in physical activity. A smartphone-based wearable device and a Kinect-based device were designed based on the framework to track movement and breathing and sonify them during physical activity. In control studies conducted to evaluate the sonification strategies, people with CP reported increased performance, motivation, awareness of movement and relaxation with sound feedback. Home studies, a focus group and a survey of CP patients conducted at the end of a hospital pain management session provided an in-depth understanding of how different aspects of the SESs and their calibration can facilitate self-directed rehabilitation and how the wearable version of the device can facilitate transfer of gains from exercise to feared or demanding activities in real life. We conclude by discussing the implications of our findings on the design of technology for physical rehabilitation

Occlusal traits in children with neurofibromatosis type 1

Literature is poor of data about the occlusion in children affected by neurofibromatosis type 1 (NF1). This case-control study investigated the occlusal traits in a group of children with NF1

Limb reconstruction with knee mega-prosthesis in patients with distal femur primary tumours: gait analysis and alignment evaluation

Introduction: The aim of this study was the functional evaluation and lower limb alignment assessment of patients with a modular knee prosthesis after distal femur resection for primary bone tumour. Materials and methods: 15 patients affected by distal femur tumor and treated with a megaprosthesis implant (6 females and 9 males, mean age: 41 years, range: 15-74 years) and 15 controls were recruited for the study. For each subject the function evaluation included an instrumented gait analysis, while only patients underwent a teleradiography and a latero-lateral X-ray projection of the knee. Results: The duration of the stance phase of gait was shortened in the prosthetic limb with respect to the contralateral limb (57.5±3.6 % gait cycle vs. 60.9±4.8 % gait cycle, P = 0.01), with a correspondent increase of the swing phase. The prosthetic limb also showed an altered knee joint kinematic curve during gait, with a flexion deficit at load response with respect to the contralateral limb (4.5±3.6° vs. 13.4±5.0°, P = 0.000003). Abnormal timing in the muscle activation intervals were observed for tibialis anterior, gastrocnemius lateralis and rectus femoris of the prosthetic limb. The prosthetic limb was found to be misaligned with respect to the contralateral limb, both for the femorotibial (P<0.05) and the ankle joints (P<0.05). Conclusions: Gait analysis performed widely in reference centres could lead to a change in the design of megaprostheses to improve the function and prevent degenerative changes in not involved joint. A multicentric expertise is mandator