In this paper, we define soft exterior and study its basic properties. We establish several important results relating soft interior, soft exterior, soft closure, and soft boundary in soft topological spaces. Moreover, we characterize soft open sets, soft closed sets, and soft clopen sets via soft boundary. All these findings will provide a base to researchers who want to work in the field of soft topology and will help to establish a general framework for applications in practical fields.

In this paper, the concept of soft neighborhood of an element in a soft topological space is introduced, and its basic properties are studied.

This research offers a high step-up DC-DC converter using a two- winding coupled inductor and voltage multiplier circuit (VMC) including diodes and capacitors for renewable energy (RE) usages such as photovoltaic (PV) and fuel cell (FC). The advantages of this converter are: 1) High voltage gain with small duty cycle of the switch, 2) low voltage stress across switch/diodes, 3) Low number of components, 4) Lower volume and cost, 5) simple structure with only one power switch, 6) small current ripple of the input, 7) zero voltage and current (ZVS and ZCS) of the diodes, 8) improved efficiency, and 9) common grounding of the input and output. Due to the coupled inductor usage, the voltage gain is more flexible, and it can be enhanced by adjusting two different parameters: the turns ratio (N) of the coupled inductor and the duty cycle (D) of the switch. Furthermore, the voltage stresses of the semiconductors are decreased by increasing N. VMC is the other element for the power switch’s voltage stress reduction. The suggested topology could be an appropriate option for RE usage because of the small current ripple of the input and modified efficiency.

In this paper, a new bidirectional buck-boost dc-dc converter with capability of soft switching and zero input current ripple is proposed. The coupled inductor is used in the proposed converter to eliminate the input current ripple. In the proposed converter, zero voltage switching (ZVS) and zero current switching (ZCS) can be obtained for the main and auxiliary switches, respectively. In addition, the proposed topology is analyzed in all operating modes and all equations of voltage and current for components are obtained.Moreover, the required conditions for soft switching operation and also achieving zero input current ripple are calculated. Finally, the acuracy performance of the proposed converter is reconfirmed through simulation results in EMTDC/PSCAD software program.

We define a new metric on the set of all closed linear operators between Hilbert spaces and investigate its properties. In particular, we show that the set of all closed operators with a closed range is an open subset of the set of all closed operators and the map T®T+ is an isometry in this metric. We also investigate the relationships between the topology induced by this metric and the gap metric.

In this paper, we construct a topology using the concept of Hoop algebras and investigate its topological properties. These include examining specific types of topological spaces, such as Hausdorff, $T_0$-spaces, and $T_1$-spaces, as well as exploring the concept of connectedness. Additionally, we analyse the relationship between closed and compact sets within this topology. Finally, by incorporating the binary operation $\ri$ and the defined topology on Hoop algebras, we introduce the notion of semi-topological algebra and demonstrate that every Hoop algebra is a right semi-topological algebra.