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.

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.

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.