The problem of finding square roots of p-adic integers in Z p , p 2, has been a classic application of Hensel's lemma. A recent development on this problem is the application and analysis of convergence of numerical methods in approximating p-adic numbers. For a p-adic number a, Zerzaihi, Kecies, and Knapp (2010) introduced a fixedpoint method to find the square root of a in Q p . Zerzaihi and Kecies (2011) later extended this problem to finding the cube root of a using the secant method. In this paper, we compute for the square roots and cube roots of p-adic numbers in Q p , using the Newton-Raphson method. We present findings that confirm recent results on the square roots of p-adic numbers, and highlight the advantages of this method over the fixed point and secant methods. We also establish sufficient conditions for the convergence of this method, and determine the speed of its convergence. Finally, we detemine how many iterations are needed to obtain a specified number of correct digits in the approximate.

In this work, we present straightforward and concrete computations of the unitary irreducible representations of the Euclidean motion group M (2) employing the methods of deformation quantization. Deformation quantization is a quantization method of classical mechanics and is an autonomous approach to quantum mechanics, arising from the Wigner quasiprobability distributions and Weyl correspondence. We advertise the utility and power of deformation theory in Lie group representations. In implementing this idea, many aspects of the method of orbits is also learned, thus further adding to the mathematical toolkit of the beginning graduate student of physics. Furthermore, the essential unity of many topics in mathematics and physics (such as Lie groups and Lie algebras, quantization, functional analysis and symplectic geometry) is witnessed, an aspect seldom encountered in textbooks, in an elementary way.

This work presents quantization of time of arrival functions using generalized Stratonovich-Weyl quantization. We take into account the ordering problems involved, mainly the Born-Jordan and the symmetric ordering schemes. We call attention to the combination of the group theoretic methods usually employed in Weyl quantization with the implementation of different ordering schemes via integral kernel factors. It is possible to, and we do, apply the Pegg-Barnett method to the quantization of time to address physical issues such as boundedness and self-adjointness.

In this work, we perform exact and concrete computations of star-product of functions on the Euclidean motion group in the plane, and list its $C$-star-algebra properties. The star-product of phase space functions is one of the main ingredients in phase space quantum mechanics, which includes Weyl quantization and the Wigner transform, and their generalizations. These methods have also found extensive use in signal and image analysis. Thus, the computations we provide here should prove very useful for phase space models where the Euclidean motion groups play the crucial role, for instance, in quantum optics.

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