In this paper, the Gradual real numbers are considered and the notion of the Gradual normed linear space is given. Also some topological properties of such spaces are studied, and it is shown that the Gradual normed linear space is a locally convex space, in classical sense. So the results in locally convex spaces can be translated in Gradual normed linear spaces. Finally, we give an example of a Gradual normed linear space which is not normable in classical analysis.

The term norm inflation appears to be a generalization, towards all sources of law, of legislative inflation, defined as a disproportionate increase in the volume and length of legislation. As with monetary inflation, the overabundance of sings risk causing their devaluation (Carbonnier). Is this a benign illness or the symptom of systemic pathology of the legal system? Is it specific to our time or has it been known before? Are the effects visible throughout the law or confined to particular areas of it? If norm inflation produces undesirable effects that deserve our attention, we should look at this scope, now and in the past, its foreseeable effects, its possible causes and remedies that might counter it. This article looks at the phenomenon of norm inflation, with reference mostly to Canada, France, the UK and the USA, and with a glance at the Iranian situation. The article demonstrates how economic analysis can be used to broach the matter.

Criminal responsibility age has always been a controversial issue in criminal law. One of such controversies is Graduality or suddenness of criminal responsibility. According to most Shiite jurisprudents, criminal responsibility age in Islamic penal law is based on religious maturity age, i.e. 9 for girls and 15 for boys. Now, a number of questions arise here: Is founding the basis of criminal responsibility age on religious maturity correct? Is the age of religious maturity a religious matter or a natural one? Is it Gradual or does it happens suddenly? This paper studies the criminal responsibility age ofthe girls from different aspects, evaluates the existing related laws, suggests that this age is Gradual and varies according to girls' rational growth.

This contribution deals with the mathematical modeling of R-norm entropy and R-norm divergence in quantum logics. We extend some results concerning the R-norm entropy and conditional R-norm entropy given in (Inf. Control 45, 1980), to the quantum logics. Firstly, the concepts of R-norm entropy and conditional R-norm entropy in quantum logics are introduced. We prove the concavity property for the notion of R-norm entropy in quantum logics and we show that this entropy measure does not have the property of sub-additivity in a true sense. It is proven that the monotonicity property for the suggested type of conditional version of R-norm entropy, holds. Furthermore, we introduce the concept of R-norm divergence of states in quantum logics and we derive basic properties of this quantity. In particular, a relationship between the R-norm divergence and the R-norm entropy of partitions is provided.

This paper is devoted to study $\mathcal{I}$-convergent,$\mathcal{I-}$null, $\mathcal{I-}$bounded and bounded sequence spaces in Gradual normed linear spaces, denoted by $c_{\| \cdot \|_G} ^\mathcal{I} ,c_{0 \| \cdot \|_G} ^\mathcal{I} ,\ell_{\infty \| \cdot \|_G} ^\mathcal{I}, \ell_{\infty \| \cdot\|_G}, m_{\| \cdot \|_G} ^\mathcal{I}$ and $m_{0 \| \cdot \|_G} ^\mathcal{I}$ respectively. We discussed some algebraic and topological properties of these classes. Also, we studied some inclusions involving these spaces.

This paper focuses on the norm retrieval problem for wavelet transform. The wavelet transform associated with an admissible wavelet is norm retrieval. However, this problem is difficult in special cases, i.e., for wavelet systems. We give some results of the norm retrieval for wavelet systems.

The main objective of this article is to introduce Gradual normed Riesz spaces and their properties. In fact, by adding a property to Gradual normed spaces as discussed in reference [20], we have defined the Gradual Banach lattices. Next, we focus on the stability of the supremum-preserving functional equation in these spaces, utilizing the fixed-point theorem to examine the conditions ensuring stability. Also, we can verify the stability of various functional equations in these spaces.

Islam is a comprehensive religion belonging to all times and places. One of its characteristics is that it was revealed and introduced little by little, its fundamental principles to be believed in, being given priority over its commandments, and, in turn, some of its commandments over the others. Even, for several reasons, some of the obligatory commandments were not introduced at once unless adequate preparations were made for them.This article is trying to show that when an Islamic government IS established in a society, it is obliged (by the shariah) to implement the divine commandments immediately unless doing so will bring dishonor on Islam, due to lack of appropriate ground, even among the Muslims. Therefore, for some reasons, the most important of which is a contradiction between commandments, the implementation of commandments may be postponed to future date, but the Islamic government is obliged to make preparations for them to be fulfilled.It seems that non-usury banking and Islamic system of taxes, in the Islamic economic literature, follow this regulation.

In this paper we present some properties of set-norm exhaustive set multi functions and also of atoms and pseudo-atoms of set multi functions taking values in the family of non-empty subsets of a commutative semi group with unity.