In this paper, we introduce and explore the class of $\omega _{\alpha }$-% open sets which are strictly weaker than open and prove that the collection of all $\omega _{\alpha }$-open subsets of a space forms a topology that is finer than the original one. We also introduce what we call $\omega _{\alpha }$- continuity and $\omega _{\alpha _{X}}$-continuity and we give several characterizations and two decompositions of $\omega _{\alpha }$-continuity. Finally, new decompositions of continuity are provided.

Let R be a commutative ring with unity. In this article, we introduce the concept of prime principal right ideal rings, A prime ideal P of R is said to be a prime principal right ideal given by $P =\{ ar: r\in R\}$ for some element a in R. The ring R is said to be the prime principal right ideal ring if every prime ideal of R is a prime principal right ideal. A prime principal right ideal ring R is called a prime principal right ideal domain if R is a domain. Several properties and characteristics of prime principal right ideal rings are studied.

Analyzing the brain signals generated by brain neurons can help identify the severe chronic neurological condition known as epilepsy. To interact with human organs and produce signals, neurons are intricately connected to one another. Electroencephalogram (EEG) media are generally used for the recording of these brain impulses. These signals are highly data-intensive, complicated, noisy, non-linear, and non-stationary. Therefore, it is difficult to detect seizures and learn about the brain. However, they may fail to classify correctly due to complex data. In this study, the `Fourier Bessel' series expansion-based empirical wavelet transform (FBSE-EWT) is used to first decompose the EEG data into sub-bands. The five best-performing ensemble learning classifiers such as Na\"ive Bayes, SVM, RF, XG Boost, and CNN are given here as a result of experiments using the obtained ranked features on various classifiers.

We present novel combinatorial proofs for some cases of a classic result of Kummer, that the highest power of $p$ dividing the binomial coefficient $n$ choose $m$ equals the number of carries when adding $m$ to $n-m$ in base $p$. We prove the case $p=2$ by considering permutations of binary vectors, and extend this approach to arbitrary primes for the case of no carries.

If $X$ is considered as a topological local group, then topological group $H$ will be found to be an enlargement of $X$. We prove that the cohomology of local topological group $X$ results in an isomorphic form along with the cohomology of quotient topological group $H$.

In this research article, we investigate the uniqueness problems concerning with L-functions of differential polynomials of meromorphic functions sharing one finite value. This article extends and generalizes the earlier works due to X. M. et al \cite{5}.

In this paper sufficient conditions are given for determining when the product of functions of different spaces belong to $H_{\omega_{1}}$ class. The estimation of the Fourier coefficients of the functions of this class is also provided.

Let $R$ be a commutative ring with unity and $K$ be a submodule of a non-zero left $R$-module $D$. This paper introduces a new concept as a generalization of the prime submodule, which is called $R$-prime. We prove some properties of the $R$-prime submodule and study the relationship between the $R$-prime submodules and other submodules as prime and $J$-semiprime. Also, we will see the $R$-prime submodule with the multiplication module and the injective envelope.

On a finite probability space, we consider a problem of {\it fair pricing} of contingent claims in the sense of \cite{FS89} and its sensitivity to a distortion of information, where we follow the {\it weak information} modeling approach from \cite{Fabrice03}. We show that, in complete models, or more generally, for replicable contingent claims, the weak information does not affect the fair price. For incomplete models, this is not the case for non-replicable claims, where we obtain explicit formulas for the {\it information premium} and {\it correction to an optimal trading strategy}. We illustrate our results by an example, where we demonstrate that under weak information, the fair price can increase, stay the same, or decrease. Finally, we perform the stability analysis for the information premium and the correction of the optimal trading strategy to perturbations of the contingent claim payoff, stock price dynamics, and the reference probability measure.

We explore with zero distribution of $k^{th}$ derivative of non-vanishing difference operators of finite order entire functions. The result of this study may be viewed as being analogous to the Hayman Conjecture. This work generalises and improves the results due to M. Andasmas - Z. Latreuch and C. Song, K. Liu, L. Ma.

The concepts of an operation $\gamma_{\tilde{\theta}_{\mu}}$ and $\tilde{\theta}_{\mu}$-open ($\tilde{\theta}_{\mu}$-closed) set on generalized topological space (GTS) were introduced by Won Keun Min \cite{a13}. In this paper, we introduce and study an operation $\gamma_{\tilde{\theta}_{\omega}}$ as a new operation in GTS. Some relationships between this new operation and each of $\gamma_{\tilde{\theta}_{\mu}}$ operation and $\mu$-closure operator are obtained. Moreover, we give some sufficient conditions for the equivalence between $\gamma_{\tilde{\theta}_{\omega}}$ operation and $\gamma_{\tilde{\theta}_{\mu}}$ operations, and between $\gamma_{\tilde{\theta}_{\omega}}$ and $\mu$-closure operator. We use the $\omega_{\mu}$-closure operator to introduce $\tilde{\theta}_{\omega}$-open sets as a new class of sets and we prove that this class of sets lies strictly between the class of $\mu$-open sets and the class of $\tilde{\theta}_{\mu}$-open sets. Several properties of $\tilde{\theta}_{\omega}$-open sets and connections to other well-known generalized open sets are discussed.

Through a minor adaptation of a result that is outlined in a work on infinite series by Konrad Knopp, the natural logarithm function can be represented by an infinite series of rational expressions that converges over the function's domain. In this article, we illustrate how this approach can be utilized to represent a radical function, allowing for its approximation by ratios of first degree polynomials. We describe how these ratios may yield more accurate numerical estimates than the partial sums of the corresponding series obtained from the Binomial Theorem while requiring fewer computations. This material would be suitable as an enrichment topic in second-semester calculus courses.

An $r$-hued coloring of a graph $G$ is a proper $k$-coloring of graph $G$ such that the neighbours of any vertex $u$ accept at least $\min \{r, d(u)\}$ different colors. The $r$-hued chromatic number is the minimum $k$ such that graph $G$ has an $r$-hued $k$-coloring, it is denoted by $\chi_r(G)$. The corona product $G\circ H$ of two graphs $G$ and $H$ is obtained by taking one copy of $G$ and $|V\left(G)\right|$ copies of $H$ and joining every vertex of $i^{th}$ copy of $H$ to the $i^{th}$ copy of $G$, where $1\le i\leq |V\left(G\right)|$. In this paper, we obtain the $r$-hued chromatic number of corona product of two graphs $G$ and $H$, denoted by $G\circ H$. First, we consider $G\circ H$, where $G$ is the path graph and $H$ is any simple graph like complete graph. Secondly, we consider $G$ as the complete graph and $H$ as the path graph. Finally we consider $G$ as the cycle graph and $H$ as the path graph.