In this paper, a vector version of the intermediate value theorem is established. The main theorem of this article can be considered as an improvement of the main results have been appeared in [On  xed point theorems for monotone increasing vector valued mappings via scalarizing, Positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the  xed point, relaxation of the relatively compactness and the continuity on the map with replacing topological interior of the cone by the algebraic interior. Moreover, by applying Ascoli-Arzela's theorem an example in order to show that the main theorem of the paper [An intermediate value theorem for monotone operators in ordered Banach spaces, Fixed point theory and applications, 2012 (1) (2012) 1-4] may fail, is established.

Pregnancies and their related problems are involved in a remarkable percent of abnormal uterine bleeding (AUB). Thus, for finding the etiology of AUB, it is very important to be sure about the presence or absence of pregnancy. In aborted cases, the absence of placental villi, as a definite criterion of pregnancy, mistaken the diagnosis of pregnancy. In such cases, the presence of intermediate trophoblastic cells (ITC) could be used as another valuable criterion for diagnosis of pregnancy. Accordingly, in the present study both morphological characteristics of ITCs and correlation of their presence with other diagnostic values of pregnancy are examined. The specimens include 100 uterine curetting tissues which were referred to pathology lab for diagnosing the etiology of AUB. The specimens are sectioned, stained with H&E or PAS and studied with light microscopy. The results showed that ITCs are large cells having irregular, hyperchromatic and indented nuclei. These cells were present in all cases of pregnancies (61 cases), but were not found in nonpregnants. It is concluded that the presence of ITCs is a definite criterion for diagnosis of pregnancy. We also showed that the morphological characteristics of nuclei of these cells is a valuable tool for their diagnosis.    

In this paper, we study fuzzy calculus in two main branches differential and integral. Some rules for finding limit and gH -derivative of gH -difference, constant multiple of two fuzzy-valued functions are obtained and we also present fuzzy chain rule for calculating gH -derivative of a composite function. Two techniques namely, Leibniz’s rule and integration by parts are introduced for fuzzy integrals. Furthermore, we prove three essential theorems such as a fuzzy intermediate value theorem, fuzzy mean value theorem for integral and mean value theorem for gH -derivative. We derive a Bolzano’s theorem, Rolle’s theorem and some properties for gH -differentiable functions. To illustrate and explain these rules and theorems, we have provided several examples in details.

The current definition of the derivative makes the set of differentiable functions much smaller than the set of continuous functions, such that most of the real single variable functions are not differentiable and the surveying the rate of their growth is not possible with the available definition. In the present paper, using Caratheodory definition, we extend the set of differentiable functions by introducing a definition for generalized differentiation of a single variable function and its generalized derivative, in such a way that the validity of the basic theorems of this theory such as Rolle's theorem, Cauchy's mean value theorem, mean value theorem and Taylor's theorem would be hold. Finally we give some examples.

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.