Helical Magnetic Fields in Molecular Clouds? A New Method to Determine the Line-of-Sight Magnetic Field Structure in Molecular Clouds

Magnetic fields pervade in the interstellar medium (ISM) and are believed to be important in the process of star formation, yet probing magnetic fields in star formation regions is challenging. We propose a new method to use Faraday rotation measurements in small scale star forming regions to find the direction and magnitude of the component of magnetic field along the line-of-sight. We test the proposed method in four relatively nearby regions of Orion A, Orion B, Perseus, and California. We use rotation measure data from the literature. We adopt a simple approach based on relative measurements to estimate the rotation measure due to the molecular clouds over the Galactic contribution. We then use a chemical evolution code along with extinction maps of each cloud to find the electron column density of the molecular cloud at the position of each rotation measure data point. Combining the rotation measures produced by the molecular clouds and the electron column density, we calculate the line-of-sight magnetic field strength and direction. In California and Orion A, we find clear evidence that the magnetic fields at one side of these filamentary structures are pointing towards us and are pointing away from us at the other side. Even though the magnetic fields in Perseus might seem to suggest the same behavior, not enough data points are available to draw such conclusions. In Orion B, as well, there are not enough data points available to detect such behavior. This behavior is consistent with a helical magnetic field morphology. In the vicinity of available Zeeman measurements in OMC-1, OMC-B, and the dark cloud Barnard 1, we find magnetic field values of $-23\pm38~\mu$G, $-129\pm28~\mu$G, and $32\pm101~\mu$G, respectively, which are in agreement with the Zeeman Measurements

On the computation of zeros of Bessel functions

The zeros of some chosen Bessel functions of different orders is revised using the well-known bisection method , McMahon formula is also reviewed and the calculation of some zeros are carried out implementing a recent version of MATLAB software. The obtained results are analyzed and discussed on the lights of previous calculations

Smoothed Bootstrap Methods for Hypothesis Testing

This paper demonstrates the application of smoothed bootstrap methods and Efron’s methods for hypothesis testing on real-valued data, right-censored data and bivariate data. The tests include quartile hypothesis tests, two sample medians and Pearson and Kendall correlation tests. Simulation studies indicate that the smoothed bootstrap methods outperform Efron’s methods in most scenarios, particularly for small datasets. The smoothed bootstrap methods provide smaller discrepancies between the actual and nominal error rates, which makes them more reliable for testing hypotheses

Wavelets operational methods for fractional differential equations and systems of fractional differential equations

In this thesis, new and effective operational methods based on polynomials and wavelets for the solutions of FDEs and systems of FDEs are developed. In particular we study one of the important polynomial that belongs to the Appell family of polynomials, namely, Genocchi polynomial. This polynomial has certain great advantages based on which an effective and simple operational matrix of derivative was first derived and applied together with collocation method to solve some singular second order differential equations of Emden-Fowler type, a class of generalized Pantograph equations and Delay differential systems. A new operational matrix of fractional order derivative and integration based on this polynomial was also developed and used together with collocation method to solve FDEs, systems of FDEs and fractional order delay differential equations. Error bound for some of the considered problems is also shown and proved. Further, a wavelet bases based on Genocchi polynomials is also constructed, its operational matrix of fractional order derivative is derived and used for the solutions of FDEs and systems of FDEs. A novel approach for obtaining operational matrices of fractional derivative based on Legendre and Chebyshev wavelets is developed, where, the wavelets are first transformed into corresponding shifted polynomials and the transformation matrices are formed and used together with the polynomials operational matrices of fractional derivatives to obtain the wavelets operational matrix. These new operational matrices are used together with spectral Tau and collocation methods to solve FDEs and systems of FDEs