Exponential partial Bell polynomials are a main tool for computing explicit formula for a large family of numbers and polynomials. Using these polynomials, we review most of the polynomials already studied in the book of Djordjevic-Milovanovic (2014). This study allows to unify the writing of these polynomials and obtain other extensions. Consequently the obtained results find their application in Number Theory,notably for the arithmetical properties of power products, Fermat numbers, Mersenne numbers and Fermat Last Theorem.

In this paper, we mainly investigate the finite order transcendental meromorphic solutions of Fermat-type equations and also we consider here the linear difference operator of meromorphic function. In addition, we extend some recent results obtained in [1]. The example is exhibited to validate certain claims of the main result.

Public key cryptography is one of the common cryptosystems mainly because it does not have key agreement issue. One important operation in these cryptosystems is inversion. Therefore, improving its performance gains significant attention. In this paper, inversion operation over binary and prime fields are surveyed considering time and area complexity. Moreover, the implementation results on FPGA and ASIC platforms are investigated and analyzed.

The local limit Theorem describes how the density of a sum of random variables follows the normal curve. However the local limit Theorem is often seen as a curiosity of no particular importance when compared with the central limit Theorem. Nevertheless the local limit Theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fermat and was originally formalized by Jakob Bernoulli, Abraham DeMoivre and Pierre-Simon Laplace.Here we describe the historical roots of the local limit Theorem. We describe how it was supplanted by the central limit Theorem in applications. Then we review the revival started by B. V. Gnedenko and we describe modern developments.

The main purpose of this work is to introduce and investigate fuzzy quantum calculus. Our idea begins with a general definition of fuzzy $q$-derivative on arbitrary time scales using the generalized Hukuhara difference. It compiled some basic facts in the fields of the fuzzy $q$-derivative and the fuzzy $q$-integral and proved them in detail. Proceed with this work, specifying the particular concept of fuzzy $q$-Taylor's expansion, especially for continuous and fuzzy valued functions which are non-differentiable in the classical (usual) concept, as the best tool for approximating functions and solving the fuzzy initial value $q$-problems. Eventually, some numerical examples of fuzzy $q$-Taylor's expansion of special functions and functions with switching points, are solved for illustration.

Graham Higman published two important papers in 1960. In the first of these papers he proved that for any positive integer n the number of groups of order pn is bounded by a polynomial in p, and he formulated his famous PORC conjecture about the form of the function f(pn) giving the number of groups of order pn. In the second of these two papers he proved that the function giving the number of p-class two groups of order pn is PORC. He established this result as a corollary to a very general result about vector spaces acted on by the general linear group. This Theorem takes over a page to state, and is so general that it is hard to see what is going on. Higman's proof of this general Theorem contains several new ideas and is quite hard to follow. However in the Last few years several authors have developed and implemented algorithms for computing Higman's PORC formulae in special cases of his general Theorem. These algorithms give perspective on what are the key points in Higman's proof, and also simplify parts of the proof.

Many standard problems in calculus can be easily solved by an innovativevisual approach that makes no use of formulas. The method is based on Mamikon’ssweeping-tangent Theorem, a geometrically intuitive result that is easily understood byvery young students. In this paper, the method of sweeping tangents introduced andshown how it can be used to find (without the machinery of calculus) areas of manyplane regions, including an oval ring, a parabolic segment, a hyperbolic segment, theregion under a general power function, an exponential curve, a logarithmic curve, atractrix, the region between two curves traced by the rear and front wheels of a bicycle,the region enclosed by a cardioid, and by each member of a family of lima¸cons.The treatment of the parabolic segment and the exponential use geometric propertiesof subtangents to these curves, which can also be used to draw tangent lines. An unexpectedapplication of the method of sweeping tangents is to physics. In this application,conservation of angular momentum in a central force field is deduced as an elementaryconsequence of Mamikon’s sweeping-tangent Theorem.

The purpose of the current research is to investigate the what, dimensions and requirements as the cause for an advanced political system to be achieved. The problem of this research is the investigation of the what and types of goals for an advanced political system. The method used here is a descriptive-analytical one, and the results of the research show that, unlike developed political systems that only focus on and follow material and medium goals, the goals of an advanced political system are of two types: final goals and intermediate goals. The ultimate goal of this system is nearness to God and the intermediate goals are the five normal human needs, which are the five institutions of government, economy, family, education and religion or rights. This research, using the documentary method at the analytical level, identifies the ultimate cause of an advanced Islamic political system and differentiates it from the ultimate goal of the developed western political systems and mentions some of the levels and requirements for achieving the goal in an advanced Islamic political system