Vol. 14, No. 4 (2025): Regular Issue

In this work, we present the necessary notations and background information that will be used throughout our paper. We begin by introducing some foundational concepts regarding Fredholm operators, providing the essential context for our subsequent analysis. Next, we delve into fundamental principles from the theory of a specific class of Fourier integral operators, focusing on symbols and phase functions. These elements form the cornerstone of our primary objective. Furthermore, we prove significant results concerning the composition of Fourier integral operators with their $L^{2}$-adjoints. These findings are crucial as they enable us to derive important conclusions about the Fredholm properties of these operators.

This article introduces a new fractional operator named Fractional Ratio Derivative ($FRD$). The motivation for developing new fractional operators is usually driven by the constraints of traditional integer-order calculus, which inappropriately expresses complex systems and phenomena. $FRD$ can explain geometry, minimizing techniques, and features such as the chain rule, Leibniz rule, linearity, and semigroup structure. This allows us to see the geometry of the fractional derivative, which solves the problem with fractional derivatives caused by irregular geometry. $FRD$ is defined solely in the order $(0<\alpha<1)$, and it is assumed to be differentiable in the same way that the Caputo derivative is. This article strengthened the theory of fractional ratio derivatives by explaining the fundamental features and providing an overview of the mean-value theorems and fractional partial derivatives. Also, The neural network model proposed here shows better accuracy while using $FRD$.

In this study, we employ rational expression techniques via $\mathcal{F}$-contractions to establish fixed point theorems within the context of an orthogonal complete metric space. We extend and generalize several established results from prior literature. Additionally, we provide illustrative examples to validate the robustness of our findings. Moreover, our results facilitate the determination of unique solutions for both differential and integral equations.

123 Topological indices are powerful tools in various fields of science and engineering. It also assist in assessing the environmental impact, biodegradability, and toxicity of chemical substances, supporting the creation of safer and more sustainable chemicals. It can be defined as a mapping from the molecular structure of a substance to a real number. Its purpose is to characterize the physicochemical properties of specific substances while maintaining its invariance under graph transformations. Molecular descriptors play a crucial role in mathematical chemistry, particularly in investigations involving quantitative structure-property relationships and quantitative structure-activity relationships. In the context of this study, we investigated the chemical composition of pentacene, focusing on various indices, including the general randić connectivity index, the first and second multiple zagreb indices, the first general zagreb index, the atomic bond connectivity index, the hyper-zagreb index, and the geometric-arithmetic index for propane paraline graphs of linear-pentacene, multi-pentacene and linear [n]-pentacene.

Reducing the planar systems into their normal form, diffeomorphism of the plane with a smooth discriminant curve for neighborhoods of each critical point is used. In this study, using co-ordinate transformations, a generalization of the normal form is developed to understand the dynamics at infinity of dynamical systems. With the obtained normal form of dynamical planar systems, the equilibrium points and their stability can be studied quite easily, as normal form simplifies the involved equations of dynamical systems. We study behavior at infinity of a predator-prey model (Rosenzweig McArthur Model) through Poincaré compactification using the normal form of the model. The extensive analysis of dynamics at infinity of the model reveals that due to the presence of a saddle at the poles, the flow of the system solutions is trapped in the positive plane, and hence the system has an attracting region in $R^2_+$. The qualitative behavior of equilibrium points at infinity can be understood by using the Poincaré compactification through its generalized normal form. These methods are frequently used to analyze the behavior of the escapes to infinity in a family of Hamiltonian systems so-called Manev-type problems. Further, the classical results of the Kepler problem can be obtained using Poincare Compactification in Manev problems. A variation of the Newtonian gravitational n-body problem with a potential that depends on the distance and velocity was studied due to Poincare compactification.

In this study, we will formulate semianalytic solutions to nonlinear integro-fractional differential equations of the Volterra Fredhlom-Hammerstein type using a different kernel and a combination of the Laplace transform and the Modified Adomian Decomposition Method. The kernels in this case are the difference kernel and the first-order simple degenerate kernel. We describe the higher-multifractal derivative in the Caputo sense. These approaches conceptualise the solution to a functional equation as an infinite series of components that converge to the solution upon applying the inverse of the Laplace transformation. Numerical calculations typically use a reduced number of terms when obtaining a closed-form solution is not possible. Finally, examples demonstrate these points.

In the present paper, we announced a new notion of ``generalized $N$-fuzzy metric spaces" which explores a more broad view of fuzzy metric spaces mentioned in the reviewed literatures. A fixed point theorem in the setting of Generalized $N$-fuzzy metric space via fuzzy $n$-Banach Contraction mapping was proved. Finally, we demonstrated an application of main results in solving integral equation.

A semi-analytical exploration is carried out on the flow of blood-based Casson nanofluid through an elongating sheet having partial slip concerning the implications of thermophoresis and also Brownian motion. The controlling partial differential equations are shifted to ordinary differential equations with the utilization of the similarity transformation. The semi-analytical expressions for non-dimensional velocity, dimensionless concentration, and dimensionless temperature are attained by adopting the Modified Homotopy Analysis technique. Graphical diagrams represent the numerous physical aspects of the system and illustrate their impact. Furthermore, the dimensionless skin friction factor, dimensionless Sherwood number, and non-dimensional Nusselt number are interlined graphically and shown numerically in tables. The error percentage is computed to establish the efficiency and precision of the strategy presented.

This study examines a free convective flow of a trihybrid nanofluid across a truncated cone in a non-Darcy porous medium. Nanoparticles $Fe_3O_4$, $Zn$, and $Au$ are constituents of this trihybrid nanofluid model and these are present in a base fluid, blood. Solving the flow governing equations with related boundary conditions and analyzing fluid behavior and heat transfer include the application of the local linearization method (LLM) with local non-similarity approach. The effect of variation in nanoparticle volume fractions in trihybrid nanofluid on velocity and temperature profiles, also the generation of entropy and Nusselt number, are illustrated and thoroughly examined. The velocity profile increases with the increase in nanoparticle volume fractions for all the three types of nanoparticles. Additionally, the error analysis demonstrates the effectiveness of the previously described solution process. A potential comparison with the published results in the literature is also provided for certain parameter values. This method makes it easier to comprehend the linearized dynamics of a nonlinear system, which helps with prediction and optimization in a variety of domains, including engineering and medicine.

Detecting unhealthy conversations online presents significant challenges, especially regarding class imbalance and the nuanced features of social media language. Class imbalance can lead to biased models and poor performance, particularly in identifying minority class instances. Furthermore, existing methods often struggle to accurately detect unhealthy conversations due to the complexity of language nuances and the sheer volume of online discourse. To address these challenges, this paper presents the Stratified RoBERTa Enhanced Framework for detecting unhealthy conversations online. This framework employs stratified sampling during data pre-processing to ensure proper distribution and preservation of minority classes, effectively mitigating the negative impact of class imbalance. Additionally, we introduce a novel technique using the Hugging Face Auto Tokenizer to enhance tokenisation efficiency. The proposed approach utilizes a neural network architecture that integrates a pre-trained DistilRoBERTa-base model, followed by a hidden layer with ReLU activation. Fine-tuning with the Adam optimizer further enhances the model's adaptability to varying learning rates. Experimental results, illustrated through Receiver Operating Characteristic (ROC) graphs, demonstrate improved true positive rates and false positive rates, affirming the efficacy of the proposed framework in accurately detecting unhealthy conversations.

The internet is gradually becoming a riskier medium of communication for all types of highly sensitive information. Individuals, institutions, and organizations greater reliance on the internet to conduct business processes has created a fertile ground for intruders to launch various attacks on the system and network. The security of crucial secret information such as personal identifying information, credit card details, online transactions, and e-commerce is of the utmost importance, and it is dependent on cutting-edge cryptography. The article proposes an alternative cryptography algorithm based on mathematical objects known as elliptic curves. In this paper, firstly a new text encryption technique is described in which a private key can be created using the Haga's theorem. Secondly, a new image encryption technique that makes use of specific functions such as the Haga's function and the Montgomery Curve is introduced. The proposed algorithm's effectiveness is verified by experimental testing and the usage of modern security tools. The simulation results and comparison of the proposed approach to existing image encryption algorithms show that it provides a sufficient level of security.

The Magnetohydrodynamic Casson nanofluid with heat radiation is evaluated mathematically across an elongating or shrinking sheet within a porous substance exhibiting brownian and thermophoretic diffusion. By incorporating non-dimensional variables in the formulation, the controlling equations acquire dimensionless. The approximate analytical approach (ASM) and Modified Homotopy Analysis Methodology (MHAM) are utilized to attain the velocity, temperature and concentration explicitly in a dimensionless manner. We graphically portray the model's physical components to interpret their consequences. Furthermore, the semi-analytical expressions for the Sherwood number, Nusselt number, and dimensionless skin friction factor are assessed. Comparing the semi-analytical results with the numerical results, a great fit is confirmed.