What are the parities of photon-ring images near a black hole?

Light that grazes a black-hole event horizon can loop around one or more times before escaping again, resulting for distance observers in an infinite sequence of ever fainter and more delayed images near the black hole shadow. In the case of the M87 and Sgr A$^*$ back holes, the first of these so-called photon-ring images have now been observed. A question then arises: are such images minima, maxima, or saddle-points in the sense of Fermat's principle in gravitational lensing? or more briefly, the title question above. In the theory of lensing by weak gravitational fields, image parities are readily found by considering the time-delay surface (also called the Fermat potential or the arrival-time surface). In this work, we extend the notion of the time delay surface to strong gravitational fields and compute the surface for a Schwarzschild black hole. The time-delay surface is the difference of two wavefronts, one travelling forward from the source and one travelling backwards from the observer. Image parities are read off from the topography of the surface, exactly as in the weak-field regime, but the surface itself is more complicated. Of the images, furthest from the black hole and similar to the weak-field limit, are a minimum and a saddle point. The strong field repeats the pattern, corresponding to light taking one or more loops around the back hole. In between, there are steeply-rising walls in the time-delay surface, which can be interpreted as maxima and saddle points that are infinitely delayed and not observable -- these correspond to light rays taking a U-turn around the black hole.Comment: 11 pages. 7 figures. 3 appendix. Accepted in OJ

Exotic Image Formation in Strong Gravitational Lensing by Clusters of Galaxies -- IV. Elliptical NFW Lenses and Hyperbolic Umbilics

A source lying near hyperbolic umbilic (HU) leads to a ring-like image formation, constituting four images with high magnification factors and lying in a small region of the lens plane. Since (based on our earlier work) the observed number of HU image formations in cluster lenses is expected to increase in future, it is timely to investigate them in more detail. Like fold and cusp, HU also satisfies the magnification relation, i.e., the signed magnification sum of the four images equals zero. This work presents a detailed study of HU magnification relation ($R_{\rm hu}$) considering the elliptical Navarro-Frenk-White (eNFW) lens profile suitable for cluster scale dark matter halos. Our results show that for an isolated eNFW lens, $R_{\rm hu}$ is more sensitive to ellipticity than its mass or concentration parameter. An ellipticity greater than 0.3 results in $R_{\rm hu}$ lying close to zero with a small scatter around it. A substructure near the HU image formation causes the average $R_{\rm hu}$ value to deviate from zero and increases the scatter, with the amount of deviation depending on the image type near which the substructure lies. However, a population of substructures in the lens plane (equivalent to the galaxy lenses inside the cluster) does not significantly shift the average $R_{\rm hu}$ value from zero but increases the scatter around it. We find that $R_{\rm hu} \simeq 0$ for HU image formation in the Abell 1703 cluster. Repeating this test in other clusters where HU formations are discovered can be a useful indicator of substructure in cluster halos.Comment: 18 pages, 17 figures. Accepted in MNRA

Evaluation of functional outcome by modified mayo wrist score in intra- articular distal end radius fracture managed by plate osteosynthesis

Background: Fractures of distal end of radius continue to pose a therapeutic challenge. Intra articular and extra articular malalignment can lead to various complications like post traumatic osteoarthrosis, decreased grip strength and endurance, as well as limited motion and carpal instability.Methods: A prospective study of forty patients with intra-articular distal end radial fractures treated under the department of orthopaedics, PIMS, Udaipur, were included in this study. All patients managed by plate osteosynthesis were followed up to 9 months.Results: When outcome according to modified Mayo wrist scoring system was assessed, only one patient (2.50%) scored with poor grade. 11 (27.50%) were graded as excellent, 19 (22.50%) as grade good, 9 (22.50%) as grade fair. 32 (80%) patients had union within 2-3 months and 06 (15%) patients had union in 3-4 months. There were 2 (5%) case of delayed union.Conclusions: Assessment of functional outcome by modified mayo wrist score in intra-articular distal end radius fracture managed by plate osteosynthesis gives an excellent scoring system to assess outcome of communited intrarticular distal radius fractures with reference to function of the hand and wrist of the individual. Also, it will help to predict the likely functional outcome in relation to fracture pattern of distal radius.

Wave Effects in Double-Plane Lensing

We discuss the wave optical effects in gravitational lens systems with two point mass lenses in two different lens planes. We identify and vary parameters (i.e., lens masses, related distances, and their alignments) related to the lens system to investigate their effects on the amplification factor. We find that due to a large number of parameters, it is not possible to make generalized statements regarding the amplification factor. We conclude by noting that the best approach to study two-plane and multi-plane lensing is to study various possible lens systems case by case in order to explore the possibilities in the parameter space instead of hoping to generalize the results of a few test cases. We present a preliminary analysis of the parameter space for a two-lens system here.Comment: 13 pages, 8 Figures. Comments Welcom

Online Covariate Shift Detection based Adaptive Brain-Computer Interface to Trigger Hand Exoskeleton Feedback for Neuro-Rehabilitation

A major issue in electroencephalogram (EEG) based brain-computer interfaces (BCIs) is the intrinsic non-stationarities in the brain waves, which may degrade the performance of the classifier, while transitioning from calibration to feedback generation phase. The non-stationary nature of the EEG data may cause its input probability distribution to vary over time, which often appear as a covariate shift. To adapt to the covariate shift, we had proposed an adaptive learning method in our previous work and tested it on offline standard datasets. This paper presents an online BCI system using previously developed covariate shift detection (CSD)-based adaptive classifier to discriminate between mental tasks and generate neurofeedback in the form of visual and exoskeleton motion. The CSD test helps prevent unnecessary retraining of the classifier. The feasibility of the developed online-BCI system was first tested on 10 healthy individuals, and then on 10 stroke patients having hand disability. A comparison of the proposed online CSD-based adaptive classifier with conventional non-adaptive classifier has shown a significantly (p<0.01) higher classification accuracy in both the cases of healthy and patient groups. The results demonstrate that the online CSD-based adaptive BCI system is superior to the non-adaptive BCI system and it is feasible to be used for actuating hand exoskeleton for the stroke-rehabilitation applications

An EEG-EMG Correlation-based Brain-Computer Interface for Hand Orthosis Supported Neuro-Rehabilitation

Background Corticomuscular coupling has been investigated for long, to find out the underlying mechanisms behind cortical drives to produce different motor tasks. Although important in rehabilitation perspective, the use of corticomuscular coupling for driving brain-computer interface (BCI)-based neurorehabilitation is much ignored. This is primarily due to the fact that the EEG-EMG coherence popularly used to compute corticomuscular coupling, fails to produce sufficient accuracy in single-trial based prediction of motor tasks in a BCI system. New Method In this study, we have introduced a new corticomuscular feature extraction method based on the correlation between band-limited power time-courses (CBPT) associated with EEG and EMG. 16 healthy individuals and 8 hemiplegic patients participated in a BCI-based hand orthosis triggering task, to test the performance of the CBPT method. The healthy population was equally divided into two groups; one experimental group for CBPT-based BCI experiment and another control group for EEG-EMG coherence based BCI experiment. Results The classification accuracy of the CBPT-based BCI system was found to be 92.81±2.09% for the healthy experimental group and 84.53±4.58% for the patients’ group. Comparison with existing method The CBPT method significantly (p−value < 0.05) outperformed the conventional EEG-EMG coherence method in terms of classification accuracy. Conclusions The experimental results clearly indicate that the EEG-EMG CBPT is a better alternative as a corticomuscular feature to drive a BCI system. Additionally, it is also feasible to use the proposed method to design BCI-based robotic neurorehabilitation paradigms