Computing Teichm\"{u}ller Maps between Polygons

By the Riemann-mapping theorem, one can bijectively map the interior of an $n$-gon $P$ to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of $P$ to those $Q$. In this case, one wants to find the ``best" mapping between these polygons, i.e., one that minimizes the maximum angle distortion (the dilatation) over \textit{all} points in $P$. From complex analysis such maps are known to exist and are unique. They are called extremal quasiconformal maps, or Teichm\"{u}ller maps. Although there are many efficient ways to compute or approximate conformal maps, there is currently no such algorithm for extremal quasiconformal maps. This paper studies the problem of computing extremal quasiconformal maps both in the continuous and discrete settings. We provide the first constructive method to obtain the extremal quasiconformal map in the continuous setting. Our construction is via an iterative procedure that is proven to converge quickly to the unique extremal map. To get to within $\epsilon$ of the dilatation of the extremal map, our method uses $O(1/\epsilon^{4})$ iterations. Every step of the iteration involves convex optimization and solving differential equations, and guarantees a decrease in the dilatation. Our method uses a reduction of the polygon mapping problem to that of the punctured sphere problem, thus solving a more general problem. We also discretize our procedure. We provide evidence for the fact that the discrete procedure closely follows the continuous construction and is therefore expected to converge quickly to a good approximation of the extremal quasiconformal map.Comment: 28 pages, 6 figure

Solar Flare Prediction and Feature Selection using Light Gradient Boosting Machine Algorithm

Solar flares are among the most severe space weather phenomena, and they have the capacity to generate radiation storms and radio disruptions on Earth. The accurate prediction of solar flare events remains a significant challenge, requiring continuous monitoring and identification of specific features that can aid in forecasting this phenomenon, particularly for different classes of solar flares. In this study, we aim to forecast C and M class solar flares utilising a machine-learning algorithm, namely the Light Gradient Boosting Machine. We have utilised a dataset spanning 9 years, obtained from the Space-weather Helioseismic and Magnetic Imager Active Region Patches (SHARP), with a temporal resolution of 1 hour. A total of 37 flare features were considered in our analysis, comprising of 25 active region parameters and 12 flare history features. To address the issue of class imbalance in solar flare data, we employed the Synthetic Minority Oversampling Technique (SMOTE). We used two labeling approaches in our study: a fixed 24-hour window label and a varying window that considers the changing nature of solar activity. Then, the developed machine learning algorithm was trained and tested using forecast verification metrics, with an emphasis on evaluating the true skill statistic (TSS). Furthermore, we implemented a feature selection algorithm to determine the most significant features from the pool of 37 features that could distinguish between flaring and non-flaring active regions. We found that utilising a limited set of useful features resulted in improved prediction performance. For the 24-hour prediction window, we achieved a TSS of 0.63 (0.69) and accuracy of 0.90 (0.97) for $\geq$C ($\geq$M) class solar flares.Comment: Accepted for publication in Solar Physics journa

Studies on the age and growth of Labeo calbasu (Hamilton) with an exploitation pattern from the Ganga River system, Uttar Pradesh (India)

Samples were collected to study the age and growth of Labeo calbasu (Hamilton) from the river Ghaghra (Guptarghat centre, Faizabad). The scales of L. calbasu have been used for age and growth studies in present paper. Study of the marginal rings on the scales of L. calbasu indicates their annual nature. The fish attained growth in 1st 18.7 cm, 2nd 27.8 cm, 3rd 35.7 cm, 4th 41.8 cm, 5th 46.9 cm, 6th 54.9 cm and 7th 57.4 cm years of the life. The growth rate was observed 18.7, 9.1, 7.9, 6.7, 5.1, 8.0 and 2.5 cm for 1st to 7th age classes respectively. The age groups 1+ to 4+ constituted 91.17% of the total exploited population and 8.83% of remaining age groups (5+ to 7+). The maximum exploited population was observed in 2+ age group with 33.68%. Overall exploitation pattern was systematic and a good indicator for heavy recruitment