This manuscript introduces and investigates a new class of operators, termedn-quasi-$[m,\mathcal{C}]$-symmetric operators, which generalize and extend the existing notions of $[m,\mathcal{C}]$-symmetric and n-quasi-$[m,\mathcal{C}]$-isometric operators. Specifically, given a conjugation $\mathcal{C}$ on a Hilbert space, an operator $\mathcal{Q}\in \mathcal{B}\left(\mathcal{K}\right)$ is said to be n-quasi-$[m,\mathcal{C}]$-symmetric if it satisfies the relation${\mathcal{Q}}^{\ast n}\left({\displaystyle \sum _{0\le j\le m}}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{m}{j}}\right)\mathcal{C}{\mathcal{Q}}^{m-j}\mathcal{C}{\mathcal{Q}}^j\right){\mathcal{Q}}^n=0$. Our study systematically explores the algebraic properties and structural characterization of n-quasi-$[m,\mathcal{C}]$-symmetric operators through matrix representations, providing a deeper understanding of their internal structure. Moreover, we establish sufficient conditions under which the powers and products of such operators inherit the n-quasi-$[m,\mathcal{C}]$-symmetric property. Additionally, we investigate the tensor products of n-quasi-$[m,\mathcal{C}]$-symmetric operators. Finally, we identify conditions that distinguish n-quasi-$[m,\mathcal{C}]$-symmetric operators from n-quasi-$[m-1;\mathcal{C}]$-symmetric operators.

This paper is devoted to the introduction and systematic study of $(P,Q)$-normal operators in the context of semi-Hilbertian spaces, where P and Q are non-constant complex polynomials in one variable. This class generalizes the well-known notion of polynomially normal operators and offers a natural setting to study their structural properties in spaces endowed with a semi-inner product induced by a positive operator. We establish fundamental properties of $(P,Q)$-normal operators, including conditions for commutativity with respect to the $\mathbf{A}$-adjoint and relations to other classes of $\mathbf{A}$-operators. Several examples are provided to illustrate the theory and demonstrate how $(P,Q)$-normality extends classical concepts in operator theory.

This study focuses on developing a mathematical model to analyse the effects of insecticide application on mosquito populations and the role of community awareness in controlling malaria transmission. The model incorporates factors such as insecticide resistance, mosquito life cycles, and human behaviour to optimize control strategies. The results demonstrate that a combination of targeted insecticide application and community education significantly decreases malaria transmission rates, providing a comprehensive approach to managing this public health challenge. The study revealed that mathematical modelling can effectively predict changes in insecticide susceptibility among mosquito populations. Optimal control strategies were identified, which included targeted insecticide application and community education programs. These strategies significantly reduced the transmission rates of malaria, highlighting the importance of integrating scientific models with public health initiatives.

This manuscript aims at the introduction of interpolative $\mathcal{FG}$-contractions and their application to derive fixed-point results. As an application, one of the obtained results is used to analyze the fractional-order Aizawa model. Moreover, we introduce a Hutchinson–Barnsley operator defined in terms of interpolative $\mathcal{FG}$-contractions and a related iterated function system to prove the existence of a unique fractal. The theoretical findings are supported with illustrative examples and graphical demonstrations. This manuscript also sheds light on a theoretical framework for analyzing material decomposition and recycling processes.

Let $ \mathcal{B} $ be an algebra over a commutative ring with identity $ S $, and let $ \sigma $: $ \mathcal{B}\to\mathcal{B} $ be an algebra homomorphism. In this paper, we study linear operators $ \Delta $: $ \mathcal{B}\to\mathcal{B} $ that are constrained by zero-product conditions involving the Jordan product $ u\circ v = uv + vu $. In particular, we consider mappings that satisfy $ uv = 0 \Rightarrow \Delta(u\circ v) = \Delta(u)\circ \sigma(v), \; \; \; uv = 0 \Rightarrow \Delta(u\circ v) = \sigma(u)\circ \Delta(v), $ and $ uv = 0 \Rightarrow \Delta(u\circ v) = \Delta(u)\circ\sigma(v) = \sigma(u)\circ\Delta(v). $ Assuming the endomorphism $ \sigma $ is bijective, we prove that the scenario essentially simplifies to the identity case $ \sigma = \mathrm{id}_{\mathcal{B}} $. Such a simplification enables a comprehensive the forms of these linear operators. As a result, we obtain precise expressions for these operators across diverse algebraic structures, including generalized matrix algebras, upper-triangular algebras, von Neumann algebras, standard operator algebras, and nest algebras. Moreover, this approach produces analogous results for Jordan $ \sigma $-centralizers, thus extending and integrating various prior findings in the field.

In light of recent advances, this paper introduces a novel class of multimappings within the framework of metric spaces that aims to deepen the understanding of the structure and interplay of symmetric commuting tuples of mappings. A central focus of our study is the analysis of the product of two symmetric commuting tuples of mappings (c.t.m): Specifically, we consider an $(m, q)$-symmetric c.t.m ${{\bf N}} = (\mathcal{N}_1, \dots, \mathcal{N}_p)$ and an $(n, q)$-symmetric c.t.m ${\bf W} = (\mathcal{W}_1, \dots, \mathcal{W}_r)$, and establish the precise conditions under which their product ${\bf N} \odot {\bf W}$ yields an $(m+n-1, q)$-symmetric c.t.m. In addition, we investigate the behavior of $(m, q)$-isometric commuting tuples under composition ${\bf N} \cdot {\bf W}$, deriving new results that generalize existing theorems in the literature, most notably extending of Theorem 2.14 in Bermúdez et al., J. Operat. Theor., 72(2) (2014), 313–329 to a broader setting. The analysis of mappings in metric spaces is inherently complex, as these spaces often lack algebraic structures such as vector addition or scalar multiplication, making the study of nonlinear and multimappings particularly challenging. Our findings contribute to this area by elucidating the subtle relationships among symmetry, isometry, and commutativity in mapping tuples, thereby enabling more structured approaches to higher-dimensional problems within this framework.