Introduction: ALTERNATIVE punishments of imprisonment, which are called as a new approach of the penal system to fight against the negative consequences of prison, have many effects and consequences that need to be investigated. Therefore, in this study, while examining the types of ALTERNATIVE punishments, the moral and social effects of the implementation of these punishments have been explored. Materials and Methods: The present study was conducted using an analytical-descriptive method. In this study, in order to provide an appropriate answer to the question presented, by referring to the theoretical bases (books) and articles published in the ISC, Magiran, Civilica, SID and Ensani in the years 2011 to 2023 and with the method of extracting the existing legal provisions, the effectiveness of this type of punishment was also criticized in reaching the intended goals. Conclusion: The results of this study show that with the implementation of such punishments, the level of public trust, the growth of morals and social capital in the society will also improve. Therefore, the implementation of these punishments, while reducing the criminal population of prisoners and reforming them, provides better implementation of social justice and improvement of public trust, emotional and financial support for the families of criminals

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.

Let p be a prime number and let n be a positive integer prime to p. By an Ihara-result, one means the existence of an injection with torsion-free cokernel, from a full lattice, in the space of p-old modular forms, into a full lattice, in the space of all modular forms of level np. In this paper, Ihara-results are proven for genus two Siegel modular forms, Siegel-Jacobi forms and Hilbert modular forms. Ihara did the genus one case of elliptic modular forms[1]. A geometric formulation is proposed for the notion of p-old Siegel modular forms of genus two, using clarifying comments by R. Schmidt[2] and, then, following suggestions in an earlier paper[3] on how to prove Ihara results. The main THEOREM in[3] is used, where an argument by G. Pappas has been extended to prove the torsion-freeness of certain cokernel, using the density of Hecke-orbits in the moduli space of principally polarized abelian varieties and in the Hilbert-Blumenthal moduli space, which was proved by C. Chai[4].