This paper studies the Ratliff-Rush closure and the integral closure of an ideal in a valuation and Prüfer domains. By definition, the RatliffRush closure (respectively the integral closure) of an ideal I of a domain R is the ideal Given by Ĩ := S (I :R I ) (respectively I := {x ∈ R|x satisfies an equation of the form x + a1x + · · ·+ ak = 0, where ai ∈ I i for each i ∈ {1, . . . , k}}). An ide...

This paper studies the Ratliff-Rush closure and the integral closure of an ideal in a valuation and Prüfer domains. By definition, the RatliffRush closure (respectively the integral closure) of an ideal I of a domain R is the ideal given by Ĩ := S (I :R I ) (respectively I := {x ∈ R|x satisfies an equation of the form x + a1x + · · ·+ ak = 0, where ai ∈ I i for each i ∈ {1, . . . , k}}). An ide...

An external barrier of an option contract is a stochastic variable which determines whether the option is knocked in or out when the value of the variable is above or below some predetermined level, but itself is not the price of an asset which underlies the option. In this paper, we present analytic formulation for the valuation of European options on one or multiple assets with single externa...

This chapter introduces a rule-based perspective on the framework of fuzzy lattices, and the Fuzzy Lattice Reasoning (FLR) classifier. The notion of fuzzy lattice rules is introduced, and a training algorithm for inducing a fuzzy lattice rule engine from data is specified. The role of positive valuation functions for specifying fuzzy lattices is underlined and non-linear (sigmoid) positive valu...

The aim of the present work is to introduce the concept of $lambda _{r}$-almost convergence of sequences. We define the spaces $fleft( lambda _{r}right) $ and $f_{0}left( lambda _{r}right) $ of $ lambda _{r}$-almost convergent and $lambda _{r}$-almost null sequences. We investigate some inclusion relations concerning those spaces with examples and we determine the $beta $- and $gamma $-duals of...

In this paper, a neutral multi-species logarithmic population model is investigated. By applying the matrixs spectral theory which is different from the methods employed in the literature, a set of sufficient conditions are obtained for the existence and uniqueness of almost periodic solution of the neutral multi-species logarithmic population model. The obtained sufficient conditions are given...