This study contributes to the theory of Riemann-Stieltjes integral. We prove that if all continuous piecewise linear functions are Riemann-Stieltjes integrable with respect to a bounded integrator α : [a,b] → R, then α must be of bounded variation on [a,b]. We also provide some other consequences.

Let R be a commutative Noetherian ring, M be a finitely generated R-module and a be an ideal of R. For an arbitrary integer k ,􀀀, 1, we introduce the concept of k-projective dimension of M denoted by k-pdRM. We show that the finite k-projective dimension of M is at least k-depth(a, R)􀀀, k-depth(a, M). As a generalization of the intersection theorem, we show that for any finitely generated R-module N, in certain conditions, k-pdRM is nearer upper bound for dimN than pdRM. Finally, if M is k-perfect, dimN ,k-gradeM that generalizes the Strong intersection theorem.

The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining probabilistic metric spaces. Then the Dirac function is presented as an important example of distributions functions. We also, introduced Sibley’ s metric or Levy metric on the set of distance distribution functions and so this space becomes a metric space. In the following, probabilistic metric spaces are defined from the Serstnev’ s view, and some examples such as Menger probabilistic metric spaces, are introduced. In this paper, the strong topology induced by distance distribution functions is introduced, and then probabilistic diameter, probabilistic bounded, probabilistic semi-bounded, probabilistic unbounded and probabilistic totally bounded sets are introduced. Also, we prove that in every probabilistic metric, every probabilistic totally bounded set is probabilistic bounded set. We also present the cantor intersection theorem and a formulation of Bair’ s theorem in complete probabilistic metric spaces. In addition, we prove that the Heine-Borel property and the Bolzano-Wierestrass property are equivalent.

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.