Search Results (24)

The phononic crystals composed of soft materials have received extensive attention owing to the extraordinary behavior when undergoing large deformations, making it possible to provide tunable band gaps actively. However, the inverse designs of them mainly rely on the gradient-driven or gradient-free optimization schemes, which require sensitivity analysis or cause time-consuming, lacking intelligence and flexibility. To this end, a deep learning-based framework composed of a conditional variational autoencoder and multilayer perceptron is proposed to discover the mapping relation from the band gaps to the topology layout applied with prestress. The nonlinear superelastic neo-Hookean model is employed to describe the constitutive characteristics, based on which the band structures are obtained via the transfer matrix method accompanied with Bloch theory. The results show that the proposed data-driven approach can efficiently and rapidly generate multiple candidates applied with predicted prestress. The band gaps are in accord with each other and also consistent with the prescribed targets, verifying the accuracy and flexibility simultaneously. Furthermore, based on the generalization performance, the design space is deeply exploited to obtain desired soft structures whose stop bands are characterized by wider bandwidth, lower location, and enhanced wave attenuation performance. Full article

In this paper, we present a novel family of soft sets named “soft ${\omega }_{\delta }$-open sets”. We find that this class constitutes a soft topology that lies strictly between the soft topologies of soft $\delta$-open sets and soft ${\omega }^0$-open sets. Also, we introduce certain sufficient conditions for the equivalence between this new soft topology and several existing soft topologies. Moreover, we verify several relationships that contain soft covering properties, such as soft compactness and soft Lindelofness, which are related to this new soft topology. Furthermore, in terms of the soft interior operator in certain soft topologies, we define four classes of soft sets. Via them, we obtain new decomposition theorems for soft $\delta$-openness and soft $\theta$-openness, and we characterize the soft topological spaces that have the soft “semi-regularization property”. In addition, via soft ${\omega }_{\delta }$-open sets, we introduce and investigate a new class of soft functions named “soft ${\omega }_{\delta }$-continuous functions”. Finally, we look into the connections between the newly proposed soft concepts and their counterparts in classical topological spaces. Full article

Soft continuity can contribute to the development of digital images and computational topological applications other than the field of soft topology. In this work, we study a new class of generalized soft continuous functions defined on the class of soft open sets modulo soft sets of the first category, which is called soft functions with the Baire property. This class includes all soft continuous functions. More precisely, it contains various classes of weak soft continuous functions. The essential properties and operations of the soft functions with the Baire property are established. It is shown that a soft continuous with values in a soft second countable space is identical to a soft function with the Baire property, apart from a topologically negligible soft set. Then we introduce two more subclasses of soft functions with the Baire property and examine their basic properties. Furthermore, we characterize these subclasses in terms of soft continuous functions. At last, we present a diagram that shows the relationships between the classes of soft functions defined in this work and those that exist in the literature. Full article

In this paper, we study soft sets of the first and second Baire categories. The soft sets of the first Baire category are examined to be small soft sets from the point of view of soft topology, while the soft sets of the second Baire category are examined to be large. The family of soft sets of the first Baire category in a soft topological space forms a soft $\sigma$-ideal. This contributes to the development of the theory of soft ideal topology. The main properties of these classes of soft sets are discussed. The concepts of soft points where soft sets are of the first or second Baire category are introduced. These types of soft points are subclasses of non-cluster and cluster soft sets. Then, various results on the first and second Baire category soft points are obtained. Among others, the set of all soft points at which a soft set is of the second Baire category is soft regular closed. Moreover, we show that there is symmetry between a soft set that is of the first Baire category and a soft set in which each of its soft points is of the first Baire category. This is equivalent to saying that the union of any collection of soft open sets of the first Baire category is again a soft set of the first Baire category. The last assertion can be regarded as a generalized version of one of the fundamental theorems in topology known as the Banach Category Theorem. Furthermore, it is shown that any soft set can be represented as a disjoint soft union of two soft sets, one of the first Baire category and the other not of the first Baire category at each of its soft points. Full article

The concept of continuity in topological spaces has a very important place. For this reason, a great deal of work has been done on continuity, and many generalizations of continuity have been obtained. In this work, we seek to find a new approach to the study of soft continuity in soft topological spaces in connection with an induced mapping based on soft sets. By defining the *-image of a soft set, we define an induced soft mapping and present its related properties. To elaborate on the obtained results and relationships, we furnish a number of illustrative examples. Full article

In this paper, we define and investigate soft ${\omega }_{\theta }$-open sets as a novel type of soft set. We characterize them and demonstrate that they form a soft topology that lies strictly between the soft topologies of soft $\theta$-open sets and soft ${\omega }^0$-open sets. Moreover, we show that soft ${\omega }_{\theta }$-open sets and soft ${\omega }^0$-open sets are equivalent for soft regular spaces. Furthermore, we investigate the connections between particular types of soft sets in a given soft anti-locally countable space and the soft topological space of soft ${\omega }_{\theta }$-open sets generated by it. In addition to these, we define soft $\left({\omega }_{\theta },\omega \right)$-sets and soft $\left({\omega }_{\theta },\theta \right)$-sets as two classes of sets, and via these sets, we introduce two decompositions of soft $\theta$-open sets and soft ${\omega }_{\theta }$-open sets, respectively. Finally, the relationships between these three new classes of soft sets and their analogs in general topology are examined. Full article

Soft topological spaces (STSs) have received a lot of attention recently, and numerous soft topological ideas have been created from differing viewpoints. Herein, we put forth a new class of generalizations of soft open sets called “weakly soft semi-open subsets” following an approach inspired by the components of a soft set. This approach opens the door to reformulating the existing soft topological concepts and examining their behaviors. First, we deliberate the main structural properties of this class and detect its relationships with the previous generalizations with the assistance of suitable counterexamples. In addition, we probe some features that are obtained under some specific stipulations and elucidate the properties of the forgoing generalizations that are missing in this class. Next, we initiate the interior and closure operators with respect to the classes of weakly soft semi-open and weakly soft semi-closed subsets and look at some of their fundamental characteristics. Ultimately, we pursue the concept of weakly soft semi-continuity and furnish some of its descriptions. By a counterexample, we elaborate that some characterizations of soft continuous functions are invalid for weakly soft semi-continuous functions. Full article

We define the notions of weakly $\mu$-countably compactness and nearly $\mu$-countably compactness denoted by $\mathcal{W}\mu$-CC and $\mathcal{N}\mu$-CC as generalizations of $\mu$-compact spaces in the sense of Csaśzaŕ generalized topological spaces. To obtain a more general setting, we define $\mathcal{W}\mu$-CC and $\mathcal{N}\mu$-CC via hereditary classes. Using ${\mu }_{\theta }$-open sets, $\mu$-regular open sets, and $\mu$-regular spaces, many results and characterizations have been presented. Moreover, we use the properties of functions to investigate the effects of some types of continuities on $\mathcal{W}\mu$-CC and $\mathcal{N}\mu$-CC. Finally, we define soft $\mathcal{W}\mu$-CC and $\mathcal{N}\mu$-CC as generalizations of soft $\mu$-compactness in soft generalized topological spaces. Full article

This paper contributes to the field of supra-soft topology. We introduce and investigate supra $pp$-soft $T_j$ and supra $pt$-soft $T_j$-spaces $(j=0,1,2,3,4)$. These are defined in terms of different ordinary points; they rely on partial belong and partial non-belong relations in the first type, and partial belong and total non-belong relations in the second type. With the assistance of examples, we reveal the relationships among them as well as their relationships with classes of supra-soft topological spaces such as supra $tp$-soft $T_j$ and supra $tt$-soft $T_j$-spaces $(j=0,1,2,3,4)$. This work also investigates both the connections among these spaces and their relationships with the supra topological spaces that they induce. Some connections are shown with the aid of examples. In this regard, we prove that for $i=0,1$, possessing the $T_i$ property by a parametric supra-topological space implies possessing the $pp$-soft $T_i$ property by its supra-soft topological space. This relationship is invalid for the other types of soft spaces introduced in previous literature. We derive some results of $pp$-soft $T_i$-spaces from the cardinality numbers of the universal set and a set of parameters. We also demonstrate how these spaces behave as compared to their counterparts studied in soft topology and its generalizations (such as infra-soft topologies and weak soft topologies). Moreover, we investigated whether subspaces, finite product spaces, and soft